(original) (raw)

%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: paper.dvi %%Pages: 18 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o paper17.ps paper.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.06.19:1608 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N /vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N /rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N /@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{ /hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B /@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{ /urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known {userdict/md get type/dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{ itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack} if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{ noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{ Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale }if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState save N userdict maxlength dict begin/magscale true def normalscale currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts /psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR/showpage{}N/erasepage{}N/copypage{}N/p 3 def @MacSetUp}N/doclip{ psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N/endTexFig{end psf$SavedState restore}N/@beginspecial{SDict begin/SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count/ocount X/dcount countdictstack N}N/@setspecial{ CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR }{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if/showpage{}N/erasepage{}N/copypage{}N newpath}N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{end} repeat grestore SpecialSave restore end}N/@defspecial{SDict begin}N /@fedspecial{end}B/li{lineto}B/rl{rlineto}B/rc{rcurveto}B/np{/SaveX currentpoint/SaveY X N 1 setlinecap newpath}N/st{stroke SaveX SaveY moveto}N/fil{fill SaveX SaveY moveto}N/ellipse{/endangle X/startangle X /yrad X/xrad X/savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet TeXDict begin 40258431 52099146 1000 600 600 (paper.dvi) @start %DVIPSBitmapFont: Fa cmtt10 10 9 /Fa 9 118 df<3801FFF0000713FE001F6D7E15E048809038C01FF81407EC01FC381F80 000006C77EC8127EA3ECFFFE131F90B5FC1203120F48EB807E383FF800EA7FC090C7FC12 FE5AA47E007F14FEEB8003383FE01F6CB612FC6C15FE6C14BF0001EBFE1F3A003FF007FC 27247CA32C>97 DI100 DII<387FFFE0B57EA37EEA0003B3B3A5007FB61280B712C0A36C15 8022337BB22C>108 D111 D<131E133FA9007FB6FCB71280A36C1500D8003FC8FC B1ED03C0ED07E0A5EC800F011FEB1FC0ECE07F6DB51280160001035B6D13F89038003FE0 232E7EAD2C>116 D<3A7FF003FF80486C487FA3007F7F0001EB000FB3A3151FA2153F6D 137F3900FE03FF90B7FC6D15807F6D13CF902603FE07130029247FA32C>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb cmsy6 6 2 /Fb 2 87 df0 D<003E1407D87F80EB0F80486CEB1FC0EA1FE0 000715E06C7E0001140315016C6C130016C0A21378ED0180137C1503ED0700013C5B150E 5D5D15785D4A5A4A5A4A5A021FC7FC143E5C5CEB3FF05CEB7F8091C8FC137E137C137013 2023257DA127>86 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc cmti9 9 47 /Fc 47 125 df11 D39 D45 D48 D50 DI57 D<1370EA01FC1203A413F8EA00E01300B0121C127F5AA45A12 380E20779F18>I<161C163CA2167C16FCA21501821503A2ED077E150F150E151CA21538 A2157015F015E0EC01C0A2913803807F82EC0700A2140E141E141C5CA25CA25C49B6FCA2 5B913880003F49C7EA1F80A2130E131E131C133C13385B13F05B12011203D80FF0EC3FC0 D8FFFE903807FFFEA32F367BB539>65 D<0107B612C04915F017FC903A003F8000FE177F EF3F8092C7121FA24A15C0A2147EA214FE18804A143FA20101ED7F00177E4A5C16010103 EC03F04C5A4AEB1FC091B6C7FC495C9139F0007F804AEB0FC0707E010F6E7E834A1301A2 011F81A25CA2133F5F91C71203A2494A5AA2017E4A5A4C5A01FE4A5A4CC7FC49EB01FE00 01EC07FC007FB612F0B712C04BC8FC32337BB236>II<0107B612C04915F017FC903A003F8001FEEE007FEF1F8092C7EA0FC0EF 07E05CEF03F0147E170102FE15F8A25CA21301A25CA2130317035CA2130718F04A1407A2 130F18E04A140F18C0011F151F18805CEF3F00133F177E91C85AA2494A5A4C5A017E4A5A 4C5A01FE4A5A047EC7FC49495A0001EC0FF8007FB612E0B7C8FC15F835337BB23A>I<01 07B712F05B18E0903A003F80001F1707170392C7FC17015C18C0147EA214FEA24A130EA2 0101EC1E03041C13804A91C7FC163C13035E9138F001F891B5FC5B5EECE0011500130F5E 5C1707011F01015BEEC00E0280141E92C7121C133F173C91C812381778495DA2017E1401 4C5A01FE14074C5A49141F00014AB45A007FB7FCB8FC94C7FC34337CB234>I<92391FE0 01809238FFF8030207EBFE07913A1FF01F0F0091393F80079F9139FE0003DFD901F86DB4 FCD907F05C49481300495A4948147E49C8127C137E13FE485A48481578A2485AA2484815 70A2485A94C7FC123F5BA3127F90CBFCA400FE91383FFFFCA25F9238003F8094C7FCA200 7E5DA2167EA2007F15FE7E5E6C6C1301A26C6C495A6D13076C6CEB0F786C6C133E3A00FF 01FC3090387FFFF0011F01C0C8FCD903FEC9FC313775B43B>71 D<010FB51280A2160090 38003FC05DA292C7FCA25CA2147EA214FEA25CA21301A25CA21303A25CA21307A25CA213 0FA25CA2131FA25CA2133FA291C8FCA25BA2137EA213FEA25B1201B512F8A25C21337BB2 1E>73 D<91381FFFFE5C16FC9138003F80A31600A25D157EA315FE5DA314015DA314035D A314075DA3140F5DA3141F5DA3143F92C7FCA2121C007E5B00FE137EA214FE485BEAF801 00E05B495A387007E038780FC06C48C8FCEA1FFCEA07F0273579B228>I<0107B512C05B A29026003FC0C7FC5DA292C8FCA25CA2147EA214FEA25CA21301A25CA21303A25CA21307 A25CA2130FA25C17E0011F140117C05C1603013F1580160791C7FCEE0F005B5E017E143E A201FE5CED01FC4913030001EC1FF8007FB6FCB7FC5E2B337CB230>76 D<902607FFC0ED7FFC4917FF81D9003F4B1300611803023BED077CA2027BED0EFC610273 151C1838DAF1F01439F071F014E118E10101ED01C36102C1EC0383EF070301031607050E 5BEC80F8171C0107ED380F6102001470A249EDE01FDC01C090C7FC130EEE0380011E017C 5C933807003E011C140EA2013C4A137E187C01385C5E017816FC6F485B1370ED3FC001F0 EC80016000011500D807F81503277FFF803E90B512C0B5EB3C01151C46337BB245>I79 D<0107B612C04915F883903A003F8001FE EE003FEF1F8092C713C0170F5C18E0147EA214FEEF1FC05CA201011680173F4A1500177E 010315FE5F4AEB03F8EE07E00107EC3FC091B6C7FC16F802E0C9FC130FA25CA2131FA25C A2133FA291CAFCA25BA2137EA213FEA25B1201387FFFF0B5FCA233337CB234>II< 0107B512FE49ECFFC017F0903A003F8007F8EE01FCEE007E92C7127F835C1880147EA214 FEEF7F005CA2010115FE5F4A13015F01034A5AEE0FC04A495A04FEC7FC49B512F016C091 38E003E0ED01F8010F6D7E167C4A137EA2131FA25CA2013F14FEA291C7FCA24913015E13 7EEF01C001FE150318805B00011607277FFFF0001400B5ECFE0EEE7E1CC9EA1FF8EE07E0 32357BB238>I<913901FC018091380FFF03023F13C791387E07EF903A01F801FF004948 7E4A7F495A4948133E131F91C7FC5B013E143CA3137E1638A293C7FC137FA26D7E14E014 FE90381FFFC06D13F86D7F01017F6D6C7E020F7F1400153F6F7E150FA4120EA2001E5D12 1CA2151F003C92C7FCA2003E143E5D127E007F5C6D485A9038C007E039F3F80FC000F0B5 C8FC38E03FFC38C00FF029377AB42B>I<0003B812C05A1880903AF800FC003F260FC001 141F0180150F01005B001EEE07001403121C003C4A5BA200380107140E127800705CA202 0F141E00F0161CC74990C7FCA2141FA25DA2143FA292C9FCA25CA2147EA214FEA25CA213 01A25CA21303A25CA21307A25C497E001FB512F05AA2323374B237>I87 D97 D<137EEA0FFE121F5B1200A35BA21201A25BA21203A25BA21207A2EB C3E0EBCFF8380FDC3EEBF81F497E01E01380EA1FC0138015C013005AA2123EA2007E131F 1580127CA2143F00FC14005AA2147EA25CA2387801F85C495A6C485A495A6C48C7FCEA0F FCEA03F01A3578B323>I<14FCEB07FF90381F078090383E03C0EBFC013801F8033803F0 073807E00F13C0120F391F80070091C7FC48C8FCA35A127EA312FE5AA4007C14C0EC01E0 A2EC03C06CEB0F80EC1F006C137C380F81F03803FFC0C648C7FC1B2278A023>III<151FED7FC0EDF0E0020113F0EC03 E3A2EC07C316E0EDC1C091380FC0005DA4141F92C7FCA45C143E90381FFFFEA3D9007EC7 FC147CA414FC5CA513015CA413035CA413075CA3130FA25CA3131F91C8FCA35B133E1238 EA7E3CA2EAFE7812FC485AEA78E0EA3FC0000FC9FC244582B418>I<143FECFF80903803 E1E6903807C0FF90380F807FEB1F00133E017E133F49133EA24848137EA24848137CA215 FC12074913F8A21401A2D80FC013F0A21403120715E01407140F141F3903E03FC0000113 7FEBF0FF38007FCF90381F0F801300141FA21500A25C143E1238007E137E5C00FE5B4848 5A387803E0387C0F80D81FFFC7FCEA07F820317CA023>III<133FEA07FF5A13FEEA007EA3137CA213FCA213F8A21201A213F0A21203A2 13E0A21207A213C0A2120FA21380A2121FA21300A25AA2123EA2127EA2127C1318EAFC1C 133CEAF838A21378137012F013F0EAF8E01279EA3FC0EA0F00103579B314>108 D<2703C003F8137F3C0FF00FFE01FFC03C1E783C1F07C1E03C1C7CF00F8F01F03B3C3DE0 079E0026383FC001FC7FD97F805B007001005B5E137ED8F0FC90380FC00100E05FD860F8 148012000001021F130360491400A200034A13076049013E130FF081800007027EEC83C0 051F138049017C1403A2000F02FC1407053E130049495CEF1E0E001F01015D183C010049 EB0FF0000E6D48EB03E03A227AA03F>I<3903C007F0390FF01FFC391E787C1E391C7CF0 1F393C3DE00F26383FC01380EB7F8000781300EA707EA2D8F0FC131F00E01500EA60F812 0000015C153E5BA20003147E157C4913FCEDF8180007153C0201133801C013F0A2000F15 78EDE070018014F016E0001FECE1C015E390C7EAFF00000E143E26227AA02B>I<14FCEB 07FF90381F07C090383E03E09038FC01F0EA01F83903F000F8485A5B120F484813FCA248 C7FCA214014814F8127EA2140300FE14F05AA2EC07E0A2007CEB0FC01580141FEC3F006C 137E5C381F01F0380F83E03803FF80D800FCC7FC1E2278A027>I<011E137C90387F81FF 9039F3C387C09039E3EF03E03901E1FE01D9C1FC13F0EBC3F8000313F0018314F814E0EA 07871307000313C01200010F130316F01480A2011F130716E01400A249EB0FC0A2013EEB 1F80A2017EEB3F00017F133E5D5D9038FF81F09038FDC3E09038F8FF80027EC7FC000190 C8FCA25BA21203A25BA21207A25BB5FCA325307FA027>I<3903C00FC0390FF03FF0391E 78F078391C7DE03C393C3FC0FC00381380EB7F00007814F8D8707E13701500EAF0FC12E0 EA60F812001201A25BA21203A25BA21207A25BA2120FA25BA2121FA290C8FC120E1E227A A020>114 DI<1303EB0F80 A3131FA21400A25BA2133EA2137EA2137C387FFFF8A2B5FC3800F800A21201A25BA21203 A25BA21207A25BA2120FA25B1460001F13F014E01300130114C01303001E1380EB07005B EA0F1EEA07F8EA01E015307AAE19>II<01F01338D803FC13FCEA 0F1E120E121C123C0038147CEA783E0070143CA2137ED8F07C1338EA60FCC65A15780001 14705BA215F0000314E05BA2EC01C0A2EBC003158014071500EBE00EA26C6C5A3800F878 EB7FE0EB1F801E227AA023>I<011F137C90387FC1FF3A01E1E787803A03C0F703C09038 80FE0FEA07004813FC000E1580001E9038F80700001C91C7FC1301003C5B121812001303 5CA31307A25C1506010F130F150E14800038141ED87C1F131C00FC143C1538013F5B39F0 7FC0E03970F3C3C0393FE1FF80260F807EC7FC22227CA023>120 D<13F0D803FC1307D80F1E130F000E141F121C123C0038143FD8783E133E1270A2017E13 7ED8F07C137CEA60FCC65A15FC000114F85BA21401000314F013E0A2140315E0EA07C0A2 0003130715C0EBE00F141F0001133F9038F07F8038007FEFEB1F8FEB001F1500A25C003E 133E007E137E147C5C007C5BEA7001495A38380780D83C1FC7FCEA0FFCEA07F020317AA0 25>I124 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd cmsy7 7 4 /Fd 4 87 df0 D<1338A50060130C00F8133E00FC137E00FE13 FE383FBBF83807FFC000011300EA007C48B4FC000713C0383FBBF838FE38FE00FC137E00 F8133E0060130C00001300A517197B9A22>3 D<017F157F2601FFE0903803FFC0000701 F890380FF1F0260F83FC90381F0038261E00FF013C7F001890263F8078130C4890261FC0 E07F007090260FE1C07F0060EB07E3913803F780486DB4C7EA01806E5A157E157F81824B 7E0060DAF7E0EB0300913801E3F0DBC3F85B6C90260381FC13066C90260F00FE5B001C01 1E90387F803C6C017C90381FE0F82607C7F86DB45A2601FFE0010313C06C6CC86CC7FC39 1B7C9942>49 D<003F15E0D87FC0EB01F8486C14FCD81FF014FE000714036C6CEB01FF00 01EC007F6D141F0000150F6D1407017E1406A2133E160CA2013F141C1638A26D147016E0 A2ED01C0ED03801507ED0F00151E5D5D5D4A5AEC07C04A5A4AC7FC143E14FC5CEB3FE05C 5C91C8FC133C13381330282B7DA72A>86 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe cmex10 10 21 /Fe 21 113 df<1430147014E0EB01C01303EB0780EB0F00A2131E5BA25B13F85B12015B 1203A2485AA3485AA3121F90C7FCA25AA3123EA2127EA6127C12FCB3A2127C127EA6123E A2123FA37EA27F120FA36C7EA36C7EA212017F12007F13787FA27F7FA2EB0780EB03C013 01EB00E0147014301462738226>0 D<12C07E12707E123C7E7EA26C7E6C7EA26C7E7F12 007F1378137CA27FA37FA31480130FA214C0A31307A214E0A6130314F0B3A214E01307A6 14C0A2130FA31480A2131F1400A3133EA35BA2137813F85B12015B485AA2485A48C7FCA2 121E5A12385A5A5A14627C8226>I<1538EC01F8EC07E0EC1F80EC7E005CEB03F85C495A A2495AB3AB131F5CA249C7FC137E5BEA03F8EA07E0EA3F8000FCC8FCA2EA3F80EA07E0EA 03F8C67E137E7F6D7EA280130FB3AB6D7EA26D7E80EB00FC147EEC1F80EC07E0EC01F8EC 00381D62778230>8 D<12E012FCEA3F80EA07E0EA03F8C67E137E7F6D7EA280130FB3AB 6D7EA26D7E80EB00FC147EEC1F80EC07E0EC01F8A2EC07E0EC1F80EC7E005CEB03F85C49 5AA2495AB3AB131F5CA249C7FC137E5BEA03F8EA07E0EA3F8000FCC8FC12E01D62778230 >I<12F0B3B3B2043674811C>12 D<12F8B3B3B3B3B3B3B3B3B3B3ADB612C0A51AC66C82 30>22 DIII<161E167EED01FE1507ED0FF8ED3FE0ED7FC0EDFF80913801FE004A5A4A 5A5D140F4A5A5D143F5D147F92C7FCA25C5CB3B3B3A313015CA3495AA213075C495AA249 5A495A137F49C8FC485A485AEA07F0EA1FE0485AB4C9FC12FCA2B4FCEA3FC06C7EEA07F0 EA03FC6C7E6C7E6D7E133F6D7E6D7EA26D7E801303A26D7EA3801300B3B3B3A38080A281 143F81141F816E7E1407816E7E6E7E913800FF80ED7FC0ED3FE0ED0FF8ED07FE1501ED00 7E161E27C675823E>I40 D56 D58 D60 D62 D80 D<12F8B3B3B3B3B3B3B3B1B6FCA518946E822C>106 D<141FB3B3B3B3B3B3B3B1B6FCA518947F822C>I110 D<12F012FE6C7E13E0EA3F F0EA0FFCEA03FE6C7E6C6C7E6D7E6D7EA26D7E1307A2801303B3B3A76D7EA28013008080 816E7E6E7E6E7E6E7EEC01FC6EB4FCED3FC0150FA2153FEDFF00EC01FCEC07F84A5A4A5A 4A5A4A5A92C7FC5C5C13015CA2495AB3B3A713075CA2130F495AA2495A495A4848C8FC48 5AEA0FFCEA3FF0B45A138048C9FC12F02294768237>I<1B301B781BF8A2F201F0A2F203 E0A2F207C0A2F20F80A2F21F00A21A3EA262A262A24F5AA24F5AA24F5AA262190FA24FC7 FCA2193EA261A261A24E5AA24E5AA24E5AA24E5AA24EC8FCA2183EA260131001305E13F8 00014C5A1203D80FFC4B5A121DD838FE4B5A12F0D8407F4B5A12004DC9FC6D7E173E6D7E 5F6D7E5FA26D6C495AA26D6C495AA26D6C5C1607A26D6C495AA2027F49CAFCA291383F80 3EA25EEC1FC05EEC0FE0EDE1F0EC07F1EDF3E0A26EB45AA26E5BA26E90CBFCA25D157E15 7C15384D64788353>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmr5 5 4 /Ff 4 52 df<14E0B0B712C0A3C700E0C7FCB022237C9B2B>43 D<1360EA01E0120F12FF 12F11201B3A3387FFF80A2111C7B9B1C>49 DII E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmmi5 5 11 /Fg 11 117 df<146014E0130114C0A213031480130714005B130EA2131E131C133C1338 13781370A213F05B12015BA212035B120790C7FC5A120EA2121E121C123C123812781270 A212F05AA213297B9E1F>61 D<0003B612C0A239003E0007ED03801501A25BA2EC0181A2 4948C7FCA25C90B5FC485BEBF00EA2140648485AA291C8FCA2485AA4EAFFFEA2221C7C9B 24>70 D<91387F8040903907FFE0C090381FC07190393E001B8001F8130FEA01E0484813 074848140048C7FC5A123E15064891C7FCA35AA291381FFF805C913800F800A21278A26C 495A123E121E380F80073907F01E603901FFF82026003FC0C7FC221E7C9C2C>I78 D<3CFFF81FFF01FF8001F0495A3C1F0003F00078 00018015305F000F13075F020D495AA2021949C7FC913831F80701C0140602605B000701 E0131C02C01318D9C1805B13C302005B01C66D5A13E6D803ECEB7D8001F8137F93C8FC49 137EA249137C49137815386C481330311D7B9B35>87 D<9039FFF80FFEA290390FC003E0 D907E0130015066D6C5A5D6D6C5A15E0903800FDC0ECFF806EC7FC80A281147FECEFC0EB 01CF90380307E01306496C7E1318496C7E136048486C7E0007803A7FF007FFE012FF271C 7D9B2E>II<90 B512FEA23901F800FC9038E001F83903C003F090388007E09038000FC00006EB1F80EC3F 00147EC75A495A495A495A495A495A90383F0030137E491360485A484813E0484813C038 0FC001391F800380383F0007007E133FB61200A21F1C7B9B27>I<137013F8A213F013E0 1300A6EA0F80EA1FC0EA31E01261A2EAC3C01203EA0780A3EA0F001308EA1E18A2133013 70EA0FE0EA07800D1D7D9C16>105 D108 D<13C0EA01E0A3EA03C0A4EAFFFCA2 EA0780A2EA0F00A4121EA31304EA3C0CA213181370EA1FE0EA0F800E1A7D9917>116 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fh cmmi7 7 19 /Fh 19 122 df<16E015011503821507150F151FED1BF815331561EDC1FC14011581EC03 0002067F140E5C0218137F5C5C02E07F4A1480495A49C7FC49141F010E15C0130C49140F 4915E013701360491407484815F048C8FC481503000616F85A001FB7FC5A17FC5AB8FC2E 2A7CA937>1 D21 D<137001F81338157CA248485BA44848485AA44848485AA44848485AEDC180A3 001F90380F8300A2141F9038C03786393FE0E7CC9038FFC3FC393E7F00F090C9FC5AA45A A45A5A21267D9928>I<1403A21406A45CA45CA4903807FF80011F13E090387C30F0D801 F0133C3803C060D80780131ED80F00131F48140F003E13C0A25AA239F801801FA3151E90 3803003E153C157C1578D8780613F0EC01E0003CEB03C0001EEB0F80390F0C3E003807FF F8000113E0D8000CC7FC5BA45BA45BA220347CA728>30 D<013FB612F0A2903901FC0007 4A1301160001031560A25CA21307A25CED0180010F0103130093C7FC14C05D131F151EEC FFFEA290383F801E150C1400A249131C1518137E92C8FC13FEA25BA21201A25BA21203B5 12F0A22C287DA72A>70 D<4AB41308020FEBE01891397F80F038903A01F8001870D903E0 EB0CF0D90F80130749C71203013E15E05B491401485A484815C0485A120F5B001F168090 C8FC4892C7FCA2127EA4127C00FC91387FFFE0A2923800FE00127C5EA21501007E5D123E A27E6C6C495A6C6C13076C6C130FD801F8131CD800FEEBF06090393FFFC020D907FEC8FC 2D2A7DA834>I78 D87 D<903B3FFFE00FFFC0A2010190390001FC006D4814F017C0027F495A4CC7FC6E130E6F5A 021F5B6F5A5E91380FE1C0EDE380DA07F7C8FC15FE6E5A5D6E7EA2811403EC077F140E4A 7E02187FEC301F02607F14C049486C7EEB030001066D7E5B01386D7E5B01F06D7E485AD8 0FF0497ED8FFFC90381FFFE0A232287DA736>II<010FB612C05B9139E0003F800280EB7F00013EC712FE013C49 5A0138495A49495A4B5A0160495A01E0495A4949C7FC5D90C75A4A5A4A5A4A5A4A5A4A5A 4A5A4AC8FC14FE495A495A494813304948137049481360133F4A13E049C75A01FE130148 5A4848495A485A484813074848130F4848013FC7FC484848B4FCB7FC5D2A287CA72D>I< 130E131F5BA2133E131C90C7FCA7EA03E0487EEA0C78EA187C1230A212605B12C0A2EA01 F0A3485AA2485AA2EBC180EA0F81A2381F0300A213066C5A131CEA07F06C5A11287DA617 >105 D<133EEA07FEA2EA007CA213FCA25BA21201A25BA21203EC07809038E01FC0EC38 600007EB61E014C3EBC187EBC307D80FC613C09038CC038001B8C7FC13E0487E13FEEB3F 80EB0FC0486C7E1303003E1460A2127EECC0C0127CECC18012FC903801E30038F800FE00 70137C1B297CA723>107 D<137CEA0FFCA2EA00F8A21201A213F0A21203A213E0A21207 A213C0A2120FA21380A2121FA21300A25AA2123EA2127EA2EA7C18A3EAF830A21320EA78 6013C0EA3F80EA0F000E297EA715>I<3B07801FC007E03B0FE07FF01FF83B18F0E0F878 3C3B30F1807CE03E903AFB007D801ED860FEEB3F005B49133E00C14A133E5B1201A24848 495BA35F4848485A1830EE01F0A23C0F8003E003E060A218C0933801E180271F0007C013 E3933800FF00000E6D48137C341B7D993B>I<3907801FC0390FE07FF03918F0E0F83930 F1807CEBFB00D860FE133C5B5B00C1147C5B1201A248485BA34A5AEA07C01660EC03E0A2 3A0F8007C0C0A2EDC180913803C300D81F0013C7EC01FE000EEB00F8231B7D9929>I<90 38F007C03901FC1FF039031E78780006EBE03C90381FC01C000CEB801E14005B0018141F 133E1200137E153E137CA213FC157C5B1578000114F0A2EC01E0EC03C03903FC07809038 FE1F00EBE7FCEBE1F0D807E0C7FCA25BA2120FA25B121FEAFFF8A22025809922>112 D<131C133EA25BA45BA4485AB512E0A23801F000485AA4485AA4485AA448C7FC1460A214 C0123EEB0180EB0300EA1E06EA1F1CEA0FF8EA03E013267EA419>116 D121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fi cmmi6 6 20 /Fi 20 119 df11 DI21 D<13C03901E00180EC03C0A23903C00780A43907800F00A4380F001EA21508150C48 EB3C18A2147CECFC30393F839E609038FF0FE0393CFC078090C8FC5AA45AA45A1E217D95 25>I<90B612FCA2903807C000163C4948131CA2160CA249C7FCA21530A2013EEB6000A2 15E0140190387FFFC0A2EB7C03140101F85BA44848C8FCA4485AA31207B57EA226227CA1 27>70 D<91380FF0039138FFFE06903903F8070E90390FC0019E013FC712FE017C147C5B 4848143C485A48481438485A121F90C8FC481530123E007E1500A25AA4913803FFFCA291 380007C0A2ED0F80127CA27EED1F007E6C6C5BD807E0136F3903F803C6C6B51202D91FF8 C7FC28247CA22F>I78 D<3CFFFC01FFF007FF5C270FC0003FC712F849166018C07F00074AEB018003DF13031800 DA019F13061403031F5B0206141CEE80189026E00C0F5B0003131C02185C023014E05F02 60EB818014E002C00183C7FCD9E18013C601F11307D9F30013CCEA01F701F614D801FC14 F0A2495C5B5E495C15036C4891C8FC38237CA139>87 DII<013FB512F0A291388007E0D97C0013C00178EB0F800170EB1F0049133E495B5D484848 5A4A5A4A5AC7485A4AC7FC143E5C14FC495A495A495A90380FC006EB1F80EB3F00017E5B 5B4848131C4848131848481338485A48485B393F0001F0007E130FB65AA224227CA12A> I<1338137CA2137813701300A7EA0780EA1FC0EA38E01230EA60F0EAC1E0A3EA03C0A3EA 0780A2EA0F0013041306EA1E0CA21318121CEA1E70EA0FE0EA07800F237DA116>105 D<1418143C147CA214381400A7EB0780EB1FE01338EB60F013C0A2EA0180A2380001E0A4 EB03C0A4EB0780A4EB0F00A4131EA21238EA783CEAF8381378EA70F0EA7FC0001FC7FC16 2D81A119>I<13F8EA0FF0A21200A2485AA4485AA43807801E147FEB81C3EB8387380F06 0F495A1318EB700E4848C7FCA213FCEA1E7EEA3C0F80EB0781158039780F0300A21402EB 070600F0138CEB03F8386000F019247CA221>II<000F017E13FC3A1F81FF83FF3B31C383C707803A61EE03CC039026EC01F813C0D8C1 F813F013F001E013E00003903903C0078013C0A2EE0F003907800780A2EE1E041706270F 000F00130C163C1718A2001E011EEB1C70EE1FE0000C010CEB07802F177D9536>I<000F 13FC381FC3FF3931C707803861EC0301F813C0EAC1F0A213E03903C00780A3EC0F00EA07 80A2EC1E041506D80F00130C143C15181538001EEB1C70EC1FE0000CEB07801F177D9526 >I<3801E01F3903F07FC0390639C1E0390C3F80F0EB3E00001814F8013C137815F8C65A A49038F001F0A3EC03E0D801E013C0EBF00715809038F80F003803DC3CEBCFF8EBC7E001 C0C7FC485AA448C8FCA2EA7FF012FF1D20809520>112 D117 D<3807800E380FE01FEA 38F012300060130F12C01407EAC1E000011306EA03C0A33807800CA214081418A2143014 6014C0EA03C13801FF00EA007E18177D951F>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fj cmr6 6 11 /Fj 11 58 df<130C1338137013E0EA01C0EA038013005A120EA25AA25AA312781270A3 12F0AB1270A312781238A37EA27EA27E7E1380EA01C0EA00E013701338130C0E317AA418 >40 D<12C012707E7E7E7E7E1380EA01C0A2EA00E0A21370A313781338A3133CAB1338A3 13781370A313E0A2EA01C0A2EA038013005A120E5A5A5A12C00E317CA418>I<1438B2B7 12FEA3C70038C7FCB227277C9F2F>43 D<13FF000313C0380781E0380F00F0001E137848 133CA248131EA400F8131FAD0078131EA2007C133E003C133CA26C13786C13F0380781E0 3803FFC0C6130018227DA01E>48 D<13E01201120712FF12F91201B3A7487EB512C0A212 217AA01E>II<13FF000313C0380F03E0381C 00F014F8003E13FC147CA2001E13FC120CC712F8A2EB01F0EB03E0EB0FC03801FF00A238 0003E0EB00F01478147C143E143F1230127812FCA2143E48137E0060137C003813F8381E 03F0380FFFC00001130018227DA01E>I<14E01301A213031307A2130D131D1339133113 6113E113C1EA01811203EA07011206120C121C12181230127012E0B6FCA2380001E0A6EB 03F0EB3FFFA218227DA11E>I<00101330381E01F0381FFFE014C01480EBFE00EA1BF000 18C7FCA513FE381BFF80381F03C0381C01E0381800F014F8C71278A2147CA21230127812 F8A214784813F8006013F0387001E01238381E07803807FF00EA01F816227CA01E>II<13FE3803FFC0380781E0 380E0070481378003C133848133CA200F8131EA3141FA40078133FA26C137F121C380F01 DF3807FF9F3803FE1EC7FCA2143E143C001C1338003E13781470003C13E0381801C0381C 0780380FFE00EA03F818227DA01E>57 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fk cmsy10 10 23 /Fk 23 118 df<007FB81280B912C0A26C17803204799641>0 D<121C127FEAFF80A5EA 7F00121C0909799917>I3 D20 D<127012FCB4FCEA7FC0EA1FF0EA07FCEA01FF38007F C0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FCEC01FF9138007FC0ED1FF0ED07FCED 01FF9238007FC0EE1FF0EE07FCEE01FF9338007FC0171F177F933801FF80933807FC00EE 1FF0EE7FC04B48C7FCED07FCED1FF0ED7FC04A48C8FCEC07FCEC1FF0EC7FC04948C9FCEB 07FCEB1FF0EB7FC04848CAFCEA07FCEA1FF0EA7FC048CBFC12FC1270CCFCAD007FB81280 B912C0A26C1780324279B441>I<020FB6128091B712C01303010F1680D91FF8C9FCEB7F 8001FECAFCEA01F8485A485A485A5B48CBFCA2123EA25AA2127812F8A25AA87EA2127812 7CA27EA27EA26C7E7F6C7E6C7E6C7EEA00FEEB7F80EB1FF86DB71280010316C01300020F 1580323279AD41>26 D<181EA4181F84A285180785727EA2727E727E85197E85F11F80F1 0FC0F107F0007FBA12FCBCFCA26C19FCCCEA07F0F10FC0F11F80F13F00197E61614E5A4E 5AA24E5A61180F96C7FCA260181EA4482C7BAA53>33 D49 D<91381FFFFE91B6FC1303010F14FED91FF0C7FCEB7F8001FEC8FCEA 01F8485A485A485A5B48C9FCA2123EA25AA2127812F8A25AA2B712FE16FFA216FE00F0C9 FCA27EA21278127CA27EA27EA26C7E7F6C7E6C7E6C7EEA00FEEB7F80EB1FF06DB512FE01 0314FF1300021F13FE283279AD37>I69 D72 D76 DI<0203B5 12F8027FECFF8049B712F0010F8290273FC3F00313FED978039038003FFF2601E0070207 1380D803C06F13C0D807801500000F177FD81F00EE3FE0484A141F123E5A0078010F150F 12C0C7FC4B15C0A3021FED1F80A24B1500183EA2023F5D6092C85A4D5A4D5A4A4A5A027E 020EC7FC173C17F84AEB03E0EE3F80DB1FFEC8FC0101EB7FF89138F8FFC0DAF9FCC9FC02 F8CAFC495AA3495AA3495AA3495AA291CBFC5BA2137EA35B13F013C03B3D7FB83A>80 D83 D85 D<0060161800F0163CB3B26C167CA2007C16F8A26CED01F0003F15036C6CEC07E06C6CEC 0FC0D807F0EC3F80D803FE903801FF003A00FFC00FFC6DB55A011F14E0010391C7FC9038 007FF82E347CB137>91 DI102 D<12FCEAFFC0EA07F0EA01FCEA007E7F80131F80130FB3A7801307806D7E6D7EEB007EEC 1FF0EC07F8EC1FF0EC7E00495A495A495A5C130F5CB3A7131F5C133F91C7FC137E485AEA 07F0EAFFC000FCC8FC1D537ABD2A>I<126012F0B3B3B3B3A91260045377BD17>106 D<0060166000F016F0B3B3A9B8FCA36C16E02C327BB137>116 D<007FB712E0B812F0A3 00F0C9FCB3B3A9006016602C327BB137>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fl cmbx10 10 44 /Fl 44 123 df<913803FFC0027F13F00103B512FC010FEB00FED93FF8133FD97FE0EBFF 8049485A5A1480484A13C04A6C1380A36F1300167E93C7FCA592383FFFC0B8FCA4000390 C7FCB3ABB5D8FC3F13FFA4303A7EB935>12 D<912603FFC0EB7FF8027F9039F00FFFFE49 B5D8FC7F6D7E010F903B007FFFE01FC0D91FF8011F90380007E0D97FE0D97FFCEB1FF049 484948133F485C02805C484E7E02004A6D5AA281735A047F6E5A96C8FCA5953807FFF8BB FCA4000390C7397FE0001FB3ABB5D8FC1FB50087B512E0A44B3A7EB950>14 D45 DI<141E143E14FE1307133FB5FCA313CFEA000FB3B3A6007FB61280A421 3779B630>49 DIII<00 1C15C0D81F80130701F8137F90B61280A216005D5D15F05D15804AC7FC14F090C9FCA8EB 07FE90383FFFE090B512F89038FC07FC9038E003FFD98001138090C713C0120EC813E015 7F16F0A216F8A21206EA3F80EA7FE012FF7FA44914F0A26C4813FF90C713E0007C15C06C 5B6C491380D9C0071300390FF01FFE6CB512F8000114E06C6C1380D90FF8C7FC25387BB6 30>II<123C123EEA3FE090B71280A4 1700485D5E5E5EA25E007CC7EA0FC000784A5A4BC7FC00F8147E48147C15FC4A5A4A5AC7 485A5D140F4A5A143F92C8FC5C147E14FE1301A2495AA31307A2130F5CA2131FA5133FA9 6D5A6D5A6D5A293A7BB830>I65 D67 DI70 D73 D76 D78 D80 D83 D<003FB91280A4D9F800EBF003D87FC09238007FC049161F 007EC7150FA2007C1707A200781703A400F818E0481701A4C892C7FCB3AE010FB7FCA43B 387DB742>I97 D<13FFB5FCA412077EAF4AB47E020F13F0023F13FC9138FE 03FFDAF00013804AEB7FC00280EB3FE091C713F0EE1FF8A217FC160FA217FEAA17FCA3EE 1FF8A217F06E133F6EEB7FE06E14C0903AFDF001FF80903AF8FC07FE009039F03FFFF8D9 E00F13E0D9C00390C7FC2F3A7EB935>I<903801FFC0010F13FC017F13FFD9FF80138026 03FE0013C048485AEA0FF8121F13F0123F6E13804848EB7F00151C92C7FC12FFA9127FA2 7F123FED01E06C7E15036C6CEB07C06C6C14806C6C131FC69038C07E006DB45A010F13F0 0101138023257DA42A>II<903803FF80011F13F0017F13FC3901FF83FE3A03FE007F804848133F4848 14C0001FEC1FE05B003FEC0FF0A2485A16F8150712FFA290B6FCA301E0C8FCA4127FA36C 7E1678121F6C6C14F86D14F000071403D801FFEB0FE06C9038C07FC06DB51200010F13FC 010113E025257DA42C>II<161FD907FEEBFFC090387FFFE348B6EAEFE02607FE07138F260FF80113 1F48486C138F003F15CF4990387FC7C0EEC000007F81A6003F5DA26D13FF001F5D6C6C48 90C7FC3907FE07FE48B512F86D13E0261E07FEC8FC90CAFCA2123E123F7F6C7E90B512F8 EDFF8016E06C15F86C816C815A001F81393FC0000F48C8138048157F5A163FA36C157F6C 16006D5C6C6C495AD81FF0EB07FCD807FEEB3FF00001B612C06C6C91C7FC010713F02B37 7DA530>I<13FFB5FCA412077EAFED7FC0913803FFF8020F13FE91381F03FFDA3C011380 14784A7E4A14C05CA25CA291C7FCB3A3B5D8FC3F13FFA4303A7DB935>II<13FFB5FCA412077EAF92380FFFE0A4923803FC0016F0ED0FE0ED1F804BC7FC157E5D EC03F8EC07E04A5A141FEC7FE04A7E8181A2ECCFFEEC0FFF496C7F806E7F6E7F82157F6F 7E6F7E82150F82B5D8F83F13F8A42D3A7EB932>107 D<13FFB5FCA412077EB3B3ACB512 FCA4163A7DB91B>I<01FED97FE0EB0FFC00FF902601FFFC90383FFF80020701FF90B512 E0DA1F81903983F03FF0DA3C00903887801F000749DACF007F00034914DE6D48D97FFC6D 7E4A5CA24A5CA291C75BB3A3B5D8FC1FB50083B512F0A44C257DA451>I<01FEEB7FC000 FF903803FFF8020F13FE91381F03FFDA3C011380000713780003497E6D4814C05CA25CA2 91C7FCB3A3B5D8FC3F13FFA430257DA435>I<903801FFC0010F13F8017F13FFD9FF807F 3A03FE003FE048486D7E48486D7E48486D7EA2003F81491303007F81A300FF1680A9007F 1600A3003F5D6D1307001F5DA26C6C495A6C6C495A6C6C495A6C6C6CB45A6C6CB5C7FC01 1F13FC010113C029257DA430>I<9039FF01FF80B5000F13F0023F13FC9138FE07FFDAF0 0113800003496C13C00280EB7FE091C713F0EE3FF8A2EE1FFCA3EE0FFEAA17FC161FA217 F8163F17F06E137F6E14E06EEBFFC0DAF00313809139FC07FE0091383FFFF8020F13E002 0390C7FC91C9FCACB512FCA42F357EA435>I<9038FE03F000FFEB0FFEEC3FFF91387C7F 809138F8FFC000075B6C6C5A5CA29138807F80ED3F00150C92C7FC91C8FCB3A2B512FEA4 22257EA427>114 D<90383FF0383903FFFEF8000F13FF381FC00F383F0003007E130100 7C130012FC15787E7E6D130013FCEBFFE06C13FCECFF806C14C06C14F06C14F81203C614 FC131F9038007FFE140700F0130114007E157E7E157C6C14FC6C14F8EB80019038F007F0 90B512C000F8140038E01FF81F257DA426>I<130FA55BA45BA25B5BA25A1207001FEBFF E0B6FCA3000390C7FCB21578A815F86CEB80F014816CEBC3E090383FFFC06D1380903803 FE001D357EB425>I<01FFEC3FC0B5EB3FFFA4000714016C80B3A35DA25DA26C5C6E4813 E06CD9C03E13FF90387FFFFC011F13F00103138030257DA435>III121 D<003FB612C0A3D9F0031380EB800749481300003E5C003C495A 007C133F5D0078495A14FF5D495B5BC6485B92C7FC495A131F5C495A017FEB03C0EBFFF0 14E04813C05AEC80074813005A49EB0F80485A003F141F4848133F9038F001FFB7FCA322 257DA42A>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fm cmti10 10 49 /Fm 49 123 df12 D<150C151C153815F0EC01E0EC03C0EC0780EC0F00141E5C147C5C5C495A1303 495A5C130F49C7FCA2133EA25BA25BA2485AA212035B12075BA2120F5BA2121FA290C8FC A25AA2123EA2127EA2127CA412FC5AAD1278A57EA3121C121EA2120E7EA26C7E6C7EA212 001E5274BD22>40 D<140C140E80EC0380A2EC01C015E0A2140015F0A21578A4157C153C AB157CA715FCA215F8A21401A215F0A21403A215E0A21407A215C0140F1580A2141F1500 A2143EA25CA25CA2495AA2495A5C1307495A91C7FC5B133E133C5B5B485A12035B48C8FC 120E5A12785A12C01E527FBD22>I44 D<387FFFF8A2B5FCA214F0150579941E>I<120EEA3F80127F12FFA31300127E123C0909 778819>I<15181538157815F0140114031407EC0FE0141F147FEB03FF90383FEFC0148F EB1C1F13001580A2143FA21500A25CA2147EA214FEA25CA21301A25CA21303A25CA21307 A25CA2130FA25CA2131FA25CA2133FA291C7FC497EB61280A31D3877B72A>49 DII56 D<133C137E13FF5AA313FE13 FCEA00701300B2120EEA3F80127F12FFA31300127E123C102477A319>58 D65 D67 D<0107B8FCA3903A000FF000034BEB00 7F183E141F181E5DA2143FA25D181C147FA29238000380A24A130718004A91C7FC5E1301 5E4A133E167E49B512FEA25EECF8000107147C163C4A1338A2010F147818E04A13701701 011F16C016004A14031880013F150718004A5CA2017F151E173E91C8123C177C4915FC4C 5A4914070001ED7FF0B8FCA25F38397BB838>69 D<0107B712FEA3903A000FF000074B13 00187C021F153CA25DA2143FA25D1838147FA292C8FCEE03804A130718004A91C7FCA201 015CA24A131E163E010314FE91B5FC5EA2903807F800167C4A1378A2130FA24A1370A201 1F14F0A24A90C8FCA2133FA25CA2137FA291CAFCA25BA25B487EB6FCA337397BB836>I< 0103B5D8F80FB512E0A390260007F8C7381FE0004B5DA2020F153F615DA2021F157F96C7 FC5DA2023F5D605DA2027F14016092C7FCA24A1403605CA249B7FC60A202FCC712070103 150F605CA20107151F605CA2010F153F605CA2011F157F95C8FC5CA2013F5D5F5CA2017F 14015F91C7FC491403007FD9FE01B512F8B55BA243397CB83E>72 D<0103B512F8A390390007F8005DA2140FA25DA2141FA25DA2143FA25DA2147FA292C7FC A25CA25CA21301A25CA21303A25CA21307A25CA2130FA25CA2131FA25CA2133FA25CA213 7FA291C8FC497EB6FCA25C25397CB820>I<0107B512FCA25E9026000FF8C7FC5D5D141F A25DA2143FA25DA2147FA292C8FCA25CA25CA21301A25CA21303A25CA21307A25CA2130F 170C4A141CA2011F153C17384A1478A2013F157017F04A14E01601017F140317C091C712 07160F49EC1F80163F4914FF000102071300B8FCA25E2E397BB834>76 D<902607FFF8923807FFF0614F13E0D9000FEFF0004F5AA2021F167FF1EFC0141DDA1CFC EC01CF023C16DF9538039F800238ED071FA20278ED0E3F97C7FC0270151CA202F04B5AF0 707E14E0037E14E0010117FE4D485A02C0EC0380A20103ED0701610280140EA20107ED1C 0305385B14006F137049160705E05B010EEC01C0A2011E913803800F61011CEC0700A201 3C020E131F4C5C1338ED1FB80178163F04F091C8FC01705CA201F04A5B187E00015DD807 F816FEB500C09039007FFFFC151E150E4C397AB84A>I<0107B612F817FF1880903B000F F0003FE04BEB0FF0EF03F8141FEF01FC5DA2023F15FEA25DA2147FEF03FC92C7FCA24A15 F817074A15F0EF0FE01301EF1FC04AEC3F80EFFE0001034A5AEE0FF091B612C04CC7FCD9 07F8C9FCA25CA2130FA25CA2131FA25CA2133FA25CA2137FA291CAFCA25BA25B1201B512 FCA337397BB838>80 D<92383FC00E913901FFF01C020713FC91391FC07E3C91393F001F 7C027CEB0FF84A130749481303495A4948EB01F0A2495AA2011F15E091C7FCA34915C0A3 6E90C7FCA2806D7E14FCECFF806D13F015FE6D6D7E6D14E0010080023F7F14079138007F FC150F15031501A21500A2167C120EA3001E15FC5EA3003E4A5AA24B5AA2007F4A5A4B5A 6D49C7FC6D133ED8F9F013FC39F8FC03F839F07FFFE0D8E01F138026C003FCC8FC2F3D7A BA2F>83 D<0007B812E0A25AD9F800EB001F01C049EB07C0485AD900011403121E001C5C 003C17801403123800785C00701607140700F01700485CA2140FC792C7FC5DA2141FA25D A2143FA25DA2147FA292C9FCA25CA25CA21301A25CA21303A25CA21307A25CA2130FA25C EB3FF0007FB512F8B6FCA2333971B83B>I<003FB539800FFFFEA326007F80C7EA7F8091 C8EA3F00173E49153CA2491538A20001167817705BA2000316F05F5BA2000715015F5BA2 000F15035F5BA2001F150794C7FC5BA2003F5D160E5BA2007F151E161C90C8FCA2163C48 15385A16781670A216F04B5A5E1503007E4A5A4BC8FC150E6C143E6C6C5B15F0390FC003 E03907F01FC00001B5C9FC38007FFCEB1FE0373B70B83E>I91 D93 D<14F8EB07FE90381F871C90383E03FE137CEBF801120148486C5A485A120F EBC001001F5CA2EA3F801403007F5C1300A21407485C5AA2140F5D48ECC1C0A2141F1583 1680143F1587007C017F1300ECFF076C485B9038038F8E391F0F079E3907FE03FC3901F0 00F0222677A42A>97 D<133FEA1FFFA3C67E137EA313FE5BA312015BA312035BA31207EB E0F8EBE7FE9038EF0F80390FFC07C013F89038F003E013E0D81FC013F0A21380A2123F13 00A214075A127EA2140F12FE4814E0A2141F15C05AEC3F80A215005C147E5C387801F800 7C5B383C03E0383E07C0381E1F80D80FFEC7FCEA01F01C3B77B926>I<147F903803FFC0 90380FC1E090381F0070017E13784913383901F801F83803F003120713E0120FD81FC013 F091C7FC485AA2127F90C8FCA35A5AA45AA3153015381578007C14F0007EEB01E0003EEB 03C0EC0F806CEB3E00380F81F83803FFE0C690C7FC1D2677A426>II<147F903803 FFC090380FC1E090383F00F0017E13785B485A485A485A120F4913F8001F14F0383F8001 EC07E0EC1F80397F81FF00EBFFF891C7FC90C8FC5A5AA55AA21530007C14381578007E14 F0003EEB01E0EC03C06CEB0F806CEB3E00380781F83803FFE0C690C7FC1D2677A426>I< ED07C0ED1FF0ED3E38ED7C3CEDF8FC15F9140115F1020313F8EDF0F0160014075DA4140F 5DA4141F5D010FB512C05B16809039003F800092C7FCA45C147EA414FE5CA413015CA413 035CA413075CA4130F5CA3131F5CA391C8FC5B121CEA7E3EA2EAFE3C137C1378EAF8F012 78EA3FC0EA0F80264C82BA19>IIII107 DIII<147F903803FFC090380FC1F090381F00F8017E137C5B4848 137E4848133E0007143F5B120F485AA2485A157F127F90C7FCA215FF5A4814FEA2140115 FC5AEC03F8A2EC07F015E0140F007C14C0007EEB1F80003EEB3F00147E6C13F8380F83F0 3803FFC0C648C7FC202677A42A>I<9039078007C090391FE03FF090393CF0787C903938 F8E03E9038787FC00170497EECFF00D9F0FE148013E05CEA01E113C15CA2D80003143FA2 5CA20107147FA24A1400A2010F5C5E5C4B5A131F5EEC80035E013F495A6E485A5E6E48C7 FC017F133EEC70FC90387E3FF0EC0F8001FEC9FCA25BA21201A25BA21203A25B1207B512 C0A3293580A42A>I<3903C003F0390FF01FFC391E783C0F381C7C703A3C3EE03F803838 3FC0EB7F800078150000701300151CD8F07E90C7FCEAE0FE5BA2120012015BA312035BA3 12075BA3120F5BA3121F5BA3123F90C9FC120E212679A423>114 D<14FE903807FF8090380F83C090383E00E04913F00178137001F813F00001130313F0A2 15E00003EB01C06DC7FC7FEBFFC06C13F814FE6C7F6D13807F010F13C01300143F141F14 0F123E127E00FE1480A348EB1F0012E06C133E00705B6C5B381E03E06CB45AD801FEC7FC 1C267AA422>II<13F8D8 03FEEB01C0D8078FEB03E0390E0F8007121E121C0038140F131F007815C01270013F131F 00F0130000E015805BD8007E133FA201FE14005B5D120149137EA215FE120349EBFC0EA2 0201131E161C15F813E0163CD9F003133814070001ECF07091381EF8F03A00F83C78E090 393FF03FC090390FC00F00272679A42D>I<01F0130ED803FC133FD8071EEB7F80EA0E1F 121C123C0038143F49131F0070140FA25BD8F07E140000E08013FEC6485B150E12015B15 1E0003141C5BA2153C000714385B5DA35DA24A5A140300035C6D48C7FC0001130E3800F8 3CEB7FF8EB0FC0212679A426>I<01F01507D803FC903903801F80D8071E903907C03FC0 D80E1F130F121C123C0038021F131F49EC800F00701607A249133FD8F07E168000E0ED00 0313FEC64849130718000001147E5B03FE5B0003160E495BA2171E00070101141C01E05B 173C1738A217781770020314F05F0003010713016D486C485A000190391E7C07802800FC 3C3E0FC7FC90393FF81FFE90390FE003F0322679A437>I<903907E007C090391FF81FF8 9039787C383C9038F03E703A01E01EE0FE3803C01F018013C0D8070014FC481480000E15 70023F1300001E91C7FC121CA2C75AA2147EA214FEA25CA21301A24A1370A2010314F016 E0001C5B007E1401010714C000FEEC0380010F1307010EEB0F0039781CF81E9038387C3C 393FF03FF03907C00FC027267CA427>I<13F0D803FCEB01C0D8071EEB03E0D80E1F1307 121C123C0038140F4914C01270A249131FD8F07E148012E013FEC648133F160012015B5D 0003147E5BA215FE00075C5BA214015DA314035D14070003130FEBF01F3901F87FE03800 7FF7EB1FC7EB000F5DA2141F003F5C48133F92C7FC147E147C007E13FC387001F8EB03E0 6C485A383C1F80D80FFEC8FCEA03F0233679A428>I<903903C0038090380FF007D91FF8 1300496C5A017F130E9038FFFE1E9038F83FFC3901F007F849C65A495B1401C7485A4A5A 4AC7FC141E5C5C5C495A495A495A49C8FC131E5B49131C5B4848133C4848133849137800 0714F8390FF801F0391FFF07E0383E1FFFD83C0F5B00785CD8700790C7FC38F003FC38E0 00F021267BA422>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fn cmmi10 10 41 /Fn 41 122 df<170C171C173E177EA217FE4C7E5EA2EE067F040C7F161C1618EE303F04 607F16E016C0923801801FDB03007F5D15064B130F4B80153815305D4B6D7E14015D4AC7 FC4A14030206815C5C023814010230815C5C010115004A8149C9FC1306010E82010C1780 5B5B0170163F016017C05B485A0003171F90CA13E012065A001FB9FC19F05A5AA2BAFC3C 3C7CBB45>1 D<4BB4FC031F13F09238FE01FC913903F0007EDA07C0EB1F80DA1F80EB0F C0023EC7EA07E002FCEC03F0495A4948EC01F8495A4948EC00FC495A013F16FE49C9FC13 FE187F485A12035B12075B120F4916FF121F02C014C0EA3FC091B6FCA2D87F81EDC1FE17 81A34848ED83FC91C7EA0303A290C9EA07F8A218F0170F18E0171F18C0EF3F807EEF7F00 17FE5F6C4B5A6D1403001F4B5A4C5A6C6CEC1F806C6C4AC7FC167E6C6CEB01F86C6C495A D8007EEB0FC090263F807FC8FC903807FFF801001380383D7CBA3F>I13 D15 D<133F14C0EB07F06D7E801301A26D7EA3147FA36E7EA36E7EA36E7EA36E7EA36E7EA36E 7EA26E7EA214014A7E5C4A7E91381E3F80143C14784A6C7E1301EB03E049486C7EEB0F80 EB1F00496D7E137E5B48486D7E485A485A000F6E7E485A485A48C87E12FE167F48168000 70151F293B7CB930>21 DI<017E1438D83FFE147E16FEA2D801FC14FC1200000114 0116F85BED03F0120315074914E0150F000715C0ED1F805BED3F00000F147EA2495B4A5A 001F495A5D49485A4A5A003F49C7FC143EEB00F8495A48485AEB0F80D87E3EC8FC13F8EA FFE0138000F8C9FC27257CA429>I<1503A35DA21506A2150EA2150CA2151CA21518A215 38A21530A21570A2EC07FE91383FFFC0903901FCE3F0903907E0E0F890391F80C03ED93E 007FEB7C01D801F8EC0F80D803F0018013C0D807E014071403D80FC015E0D81F801300A2 48485AA2007E1306A2020E130F12FE48010C14C0A2021CEB1F80A20218EB3F00A2023813 7E007C5D1430007E4A5A003E90387003F06CEC07C09138600F80D80F80013FC7FC3903E0 E0FC3901F8E7F039007FFF80D90FFCC8FCEB01C0A25CA21303A291C9FCA25BA21306A213 0EA2130CA22B4B7CB931>30 D<121C127FEAFF80A5EA7F00121C0909798817>58 D<121C127FEAFF80A213C0A3127F121C1200A412011380A2120313005A1206120E5A5A5A 12600A19798817>II<150C151E153EA2153C157CA2157815F8A215F01401A215E01403A215C01407A2158014 0FA215005CA2141E143EA2143C147CA2147814F8A25C1301A25C1303A2495AA25C130FA2 91C7FC5BA2131E133EA2133C137CA2137813F8A25B1201A25B1203A25B1207A25B120FA2 90C8FC5AA2121E123EA2123C127CA2127812F8A25A12601F537BBD2A>I<126012FCB4FC EA7FC0EA1FF0EA07FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FC EC01FF9138007FC0ED1FF0ED07FCED01FF9238007FC0EE1FF0EE07FCEE01FF9338007F80 EF1FC0A2EF7F80933801FF00EE07FCEE1FF0EE7FC04B48C7FCED07FCED1FF0ED7FC04A48 C8FCEC07FCEC1FF0EC7FC04948C9FCEB07FCEB1FF0EB7FC04848CAFCEA07FCEA3FF0EA7F C048CBFC12FC1270323279AD41>I<1760177017F01601A21603A21607160FA24C7EA216 331673166316C3A2ED0183A2ED0303150683150C160115181530A21560A215C014011580 DA03007FA202061300140E140C5C021FB5FC5CA20260C7FC5C83495A8349C8FC1306A25B A25B13385B01F01680487E000716FFB56C013F13FF5EA2383C7DBB3E>65 D<0103B77E4916F018FC903B0007F80003FE4BEB00FFF07F80020FED3FC0181F4B15E0A2 141FA25DA2143F19C04B143F1980027F157F190092C812FE4D5A4A4A5AEF0FF04AEC1FC0 05FFC7FC49B612FC5F02FCC7B4FCEF3FC00103ED0FE0717E5C717E1307844A1401A2130F 17035CA2131F4D5A5C4D5A133F4D5A4A4A5A4D5A017F4BC7FC4C5A91C7EA07FC49EC3FF0 B812C094C8FC16F83B397DB83F>I<9339FF8001C0030F13E0037F9038F80380913A01FF 807E07913A07F8000F0FDA1FE0EB079FDA3F80903803BF0002FFC76CB4FCD901FC80495A 4948157E495A495A4948153E017F163C49C9FC5B1201484816385B1207485A1830121F49 93C7FCA2485AA3127F5BA312FF90CCFCA41703A25F1706A26C160E170C171C5F6C7E5F00 1F5E6D4A5A6C6C4A5A16076C6C020EC8FC6C6C143C6C6C5C6CB4495A90393FE00FC0010F B5C9FC010313FC9038007FC03A3D7CBA3B>I<0103B812F05BA290260007F8C7123F4B14 07F003E0020F150118005DA2141FA25D19C0143FA24B1330A2027F1470190092C7126017 E05C16014A495A160F49B6FCA25F9138FC000F01031407A24A6DC8FCA201075C18034A13 0660010F160693C7FC4A150E180C011F161C18184A1538A2013F5E18F04A4A5AA2017F15 074D5A91C8123F49913803FF80B9FCA295C7FC3C397DB83D>69 D<0103B812E05BA29026 0007F8C7123F4B140FF003C0140F18015DA2141FA25D1980143FA25D1760027F14E095C7 FC92C75AA24A1301A24A495A16070101141F91B6FC94C8FCA2903903FC001F824A130EA2 1307A24A130CA2010F141CA24A90C9FCA2131FA25CA2133FA25CA2137FA291CBFC497EB6 12C0A33B397DB835>II<902603FFF891381FFFF8 496D5CA2D90007030113006FEC007C02061678DA0EFF157081020C6D1460A2DA1C3F15E0 705CEC181F82023815016F6C5C1430150702706D1303030392C7FC02607FA2DAE0015C70 1306ECC0008201016E130EEF800C5C163F0103EDC01C041F131891C713E0160F49EDF038 18300106140717F8010E02031370EFFC60130CEE01FE011C16E004005B011815FF177F13 38600130153FA20170151F95C8FC01F081EA07FCB512E01706A245397DB843>78 D<4BB4FC031F13F09238FE01FC913903F0007EDA07C0EB1F80DA1F80EB0FC0023EC7EA07 E002FCEC03F0495A4948EC01F8495A4948EC00FC495A49C912FE49167E13FE49167F1201 485AA2485AA2120F5B001F17FFA2485AA34848ED01FEA400FFEE03FC90C9FCA2EF07F8A2 EF0FF0A218E0171F18C0EF3F806C167F180017FE4C5A6C6C5D1603001F4B5A6D4A5A000F ED1F806C6C4AC7FC6D147E0003EC01F8D801FC495AD8007EEB0FC090263F807FC8FC9038 07FFF801001380383D7CBA3F>I<0103B7FC4916E018F8903B0007F80007FC4BEB00FE18 7F020FED3F80F01FC05DA2021F16E0A25DA2143FF03FC05DA2027FED7F80A292C8130018 FE4A4A5A604AEC07F04D5A0101ED3FC04CB4C7FC91B612FC17E0D903FCCAFCA25CA21307 A25CA2130FA25CA2131FA25CA2133FA25CA2137FA291CBFC497EB6FCA33B397DB835>I< 0003B812FEA25A903AF8003FC00101C0913880007E4848163C90C7007F141C121E001C92 C7FCA2485CA200305C007017180060130112E0485CA21403C716005DA21407A25DA2140F A25DA2141FA25DA2143FA25DA2147FA292C9FCA25CA25CA21301A25CA21303A25CEB0FFC 003FB6FC5AA237397EB831>84 D<267FFFFC91383FFFC0B55DA2000390C83807FC006C48 ED03E06060000094C7FC5F17065FA25F6D5DA26D5D17E05F4C5AA24CC8FC6E1306A2013F 5C161C16185EA25E6E5BA2011F495A150393C9FC1506A25D6E5AA2010F5B157015605DA2 ECE18002E3CAFC14F3EB07F614FE5C5CA25C5CA26D5AA25C91CBFC3A3B7CB830>86 D<277FFFFC01B500F890B51280B5FC60000390C7D807FCC7380FF80001FC4BEC03E00001 6204035E98C7FC621A0604075DA2040F5DA2041B5D6216336D02735D1663000003C34A5A 83DB01834AC8FC04815CDB0301140603075D1506030C5DA203185D197003301560611560 6D01E04A5A15C090267F01804AC9FC17FEDA030014060400130E0206150C020E5D140C4A 5DA24A5D18E04A5D715A5C4A92CAFCA26DC85AA2013E157C1778133C1770133801301560 513B7CB84E>I<49B500F890387FFFF095B5FC1AE0D90003018090380FFC004BC713E002 01ED07804EC7FC6E6C140E606F5C705B606F6C485A4D5A031F91C8FCEEE0065F6F6C5A5F 03075B705A16F96FB45A94C9FC6F5AA36F7EA34B7FED037F9238063FC0150E4B6C7E1538 ED700F03E07F15C04A486C7EEC0300020613034A805C4A6D7E14704A1300494880495A49 C86C7E130E011E153F017E4B7ED803FF4B7E007F01E0011FEBFFC0B5FC6144397EB845> II<91B712FCA25B9239E00007F84AC7EA0FF0D9 03F8EC1FE04AEC3FC04AEC7F804A150049485C91C7485A4C5A010E4A5A4C5A010C4A5A01 1C4A5A01185D167F4CC7FC90C7485A4B5A4B5A4B5A5E151F4B5A4B5A4BC8FC4A5A4A5A4A 5A5D140F4A5A4A5A4A48130C4AC7FC495A4A141C01031518495A49481438494814304948 1470495A49C812F0495D000115014848140348484A5A4848140F4848141F4848EC7F8048 48EB07FF90B7FCB8FC94C7FC36397BB839>I<133FEA1FFFA3C67E137EA313FE5BA31201 5BA312035BA31207EBE0FCEBE3FF9038E707C0390FFE03E09038F801F001F013F8EBE000 485A15FC5BA2123F90C7FCA214015A127EA2140312FE4814F8A2140715F05AEC0FE0A215 C0EC1F80143F00781400007C137E5C383C01F86C485A380F07C06CB4C7FCEA01FC1E3B7C B924>98 DI<14E0EB03F8A21307A314F0EB01C090C7FCAB13F8EA03FEEA070F000E1380 121C121812381230EA701F1260133F00E0130012C05BEA007EA213FE5B1201A25B12035B A20007131813E01438000F133013C01470EB806014E014C01381EB838038078700EA03FE EA00F815397EB71D>105 D<150FED3F80A2157FA31600151C92C7FCABEC0F80EC3FE0EC F0F0903801C0F849487E14005B130E130C131CEB1801133801305BA2EB0003A25DA21407 A25DA2140FA25DA2141FA25DA2143FA292C7FCA25CA2147EA214FEA25CA21301001E5B12 3F387F83F0A238FF87E0495A00FE5BD87C1FC8FCEA707EEA3FF8EA0FC0214981B722>I< EB03F0EA01FFA3EA00075CA3130F5CA3131F5CA3133F91C8FCA35B017EEB07C0ED1FF0ED 783801FEEBE0F89039FC01C1FCEC0383EC07070001130ED9F81C13F891383803F0913870 01E0000349C7FCEBF1C0EBF38001F7C8FCEA07FEA2EBFFE0EBE7F8380FE0FEEBC07F6E7E 141F001F80D9800F1330A21670003F011F136001001380A216E04815C0007E1481020F13 80158300FE903807870048EB03FE0038EB00F8263B7CB92B>IIII<90390F8003F090391FE00FFC903939F03C1F903A70F8700F80903AE0FDE007C09038C0 FF80030013E00001491303018015F05CEA038113015CA2D800031407A25CA20107140FA2 4A14E0A2010F141F17C05CEE3F80131FEE7F004A137E16FE013F5C6E485A4B5A6E485A90 397F700F80DA383FC7FC90387E1FFCEC07E001FEC9FCA25BA21201A25BA21203A25B1207 B512C0A32C3583A42A>112 D116 D<13F8D803FE1438D8070F147C000E6D13FC121C1218003814011230D8701F5C 12601503EAE03F00C001005B5BD8007E1307A201FE5C5B150F1201495CA2151F120349EC 80C0A2153F1681EE0180A2ED7F0303FF130012014A5B3A00F8079F0E90397C0E0F1C9039 3FFC07F8903907F001F02A267EA430>I<01F8EB03C0D803FEEB07E0D8070F130F000E01 8013F0121C12180038140700301403D8701F130112601500D8E03F14E000C090C7FC5BEA 007E16C013FE5B1501000115805B150316001203495B1506150E150C151C151815385D00 015C6D485A6C6C485AD97E0FC7FCEB1FFEEB07F024267EA428>I<13F8D803FE1470D807 0F14F8000EEB8001121C121800381403003015F0EA701F1260013F130700E0010013E012 C05BD8007E130F16C013FE5B151F000115805BA2153F000315005BA25D157EA315FE5D14 01000113033800F80790387C1FF8EB3FF9EB0FE1EB00035DA2000E1307D83F805B007F49 5AA24A5A92C7FCEB003E007C5B00705B6C485A381E07C06CB4C8FCEA01FC25367EA429> 121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fo cmbxti10 10 43 /Fo 43 123 df<93381FFFC093B512FC030314FF923A0FF8007F80DB1FE0EB0FC04B48EB 3FE04B48137F15FF93C712FF5C19C04A5AF07F80F01E00020792C7FC5DA4140F5DA2013F B712FE5BA26D5E903A001FF00007A2170F60143F5D171F60A3027F143F4B5CA3177F02FF 5D5DA217FF6049EE83E092C7FC4C1387F007C0A25BF00F805CF01F00820107ED7F3E4AEC 3FFC715AEF07E094C8FC495AA2120F383F8FF0EA7FCFEAFFDF5CA2EB9FC0EBBF80013FCB FCEAFC7EEA7FFCEA1FF8EA07E03B4C81BA38>12 D<383FFFFE5AA3B5FCA214FC7E170878 9622>45 D<130FEB3FC0EBFFE0A25AA314C0A214806C1300137C90C7FCADEA07C0EA1FF0 123F487EA312FF5BA26C5A5B001FC7FC132576A41E>58 D<173E177FA25F5EA25EA25E5E 845EA25E5EA293B5FCA2ED01F7150304E77FED07C7ED0FC31683151F1603153E157E157C 03FC8015F8EC01F002037F15E0140715C0EC0F80141F15004A8192B6FC5C91B7FCA2D901 F0C7FCA2495A13075C494881A249C8127F5B133E48B4FC007F01F8017FB512F06E16F8B5 4816F07E3D3B7ABA48>65 D<0103B712F04916FEF0FF8019E0D9000790C713F0187FF03F F84A151F4B15FCA3141F5DA3023FED3FF85D19F0187F027FEDFFE05D4D13C04D138002FF 913807FE004BEB1FFCEFFFF092B612C0495E18F09239C0001FF8717E496F7E4B6D7E1980 A25B92C7FCA35B5C5FA2011F17004A5C60171F013F5E4A4A5A177F4D5A017F02035B4A01 0F1380B9C7FC17FC17F094C8FC3E3979B844>II<0103B712E04916FE72 7E19E0D90007D980037F9438007FF8F01FFC4A6F7E93C71207727EA24A6F13805D1AC0A2 023F815DA3027F5D5DA314FF5D60A24918805D60A24918005D60615B4B4A5AA26149163F 92C85B187F61494C5A4A4A5BA24D90C7FC013F4B5A4AEC1FFC4D5AEFFFE0017F02035B4A 011F90C8FCB812FC17F0178004F8C9FC423979B849>I<0103B812E04917F0A3D90007EB 8000F03FE0180F5C93C71207A35C4B15C0A3023F14F0EDFC01A21603027F168003F849C7 FC1607A202FF130F9238F03FC092B5FCA25B5FEDE07F163F49141F03C090C8FCA24C130F 495EED803E60043C133E4991C7FC92C8127E187C18FC495E4A1401170360013F15074A4A 5A171F177F017F4A485A4A130FB95AA395C7FC3C3979B83F>I<0103B812C04917E0A3D9 0007EB80019438007FC0181F5C93C7120FA35C4B1580A3143F4B13F017F81601027F1600 03F849C7FCA2160302FF13074B485A163F92B5FC5B5FA2EDE03F49141F03C05B160F161F 5B038090C8FCA349141E92CAFCA35B5CA3133F5CA3137F5CB612FE815DA23B397AB83C> I<0103B600C0B612F04903E115F805C115F0A2D900079026800001EBE000A2605C93C75C A2605C4B5EA260143F4B93C7FCA260147F4B5DA2183F14FF4B5D92B7FCA25B6103E0C712 7F18FF5B4B5DA25F5B4B5DA25F5B92C75CA25F5B4A5EA25F133F4A93C8FCA25F137F4A5D B6D8F83FB512FEA44D397AB84A>72 D<0103B612E017F0A217E0D90003EBC000A35C5EA3 5C93C7FCA35C5DA3143F5DA3147F5DA314FF5DA35B5DA35B5DA35B5DA35B92C8FCA35B5C A3133F5C007FB512FC81B65A7E2C397BB827>I<0103B612F04981A25FD900070180C7FC A35C93C8FCA35C5DA3143F5DA3147F5DA314FF5DA35B5DA35B5DA2EF078049150F5DA217 1F49160092C75A173E177E5B4A5CA21601013F4A5A4A1307160F4C5A017F147F9138F803 FFB85AA35F313979B83A>76 D<0103B500C0030FB51280496E4B14C0621D80D900074D90 C7FC97B5FC824A5E03BF923803EFFEA2F107DF141F033F92380F9FFCF11F1F6F6C153F02 3F163E023E047C5BA2F1F87F147E027CDB01F05BF003E06F6C15FF02FC4B5A02F84B485B A2F01F01130102F0033E5C6F6C137C61010316F802E0DA01F05CA24D485A0107ED07C002 C095C8FC923903FF0F8061010FED1F000280023E5CA24D131F011F5D02005F6F5B193F49 5D013E4B5CA24D137F017E92C7FCD801FE60B526FC00FE90B612F0A25E5E5A3979B859> I<0103B500C049B512F0496E4914F81BF082D9000792390007F80070EC03E019074A7F62 EDBFFE190F91381F9FFF031F5E6F7F191F4A6C7F023E94C7FC6F7F61DA7E017F027C163E 6F7F197E02FC6D7E4A167C707E19FC01016E7E4A5E701380188101036E13C14A5E7013E1 18E301076E13F34AEDFBE08218FF010F814A5E83A2011F8191C86C5BA2835B013E6F90C8 FCA283137ED801FE6F5AB512FC187EA2183C4D397AB84A>I<923803FF80033F13F84AB5 12FE0207903801FF80913A1FF8007FE0DA3FE06D7EDAFF806D7E4990C76C7E49481407D9 0FFC81494814034A81133F495A494816805A5C5A91C8FC5A5A5BA2121F495D123FA34848 4B1300A448484B5AA34D5AA26049157F6017FF605E604C5B6C7E4C90C7FC4C5A003F4B5A 6D4A5A001F4B5A6C6C4A5A6D4913802707FF800790C8FC6C9038E03FFCC690B512F0013F 14C0010F49C9FC010013E0393B72B947>I<0103B712E04916FC18FF19C0D90007D98003 13E0050013F04A157FF03FF893C7FCA24A16FCA25DA2143FF07FF85DA2027F16F018FF4B 15E0A202FF4A13C019804B4913004D5A49ED1FFCEFFFF892B612E018804903FCC7FC03E0 C9FC5DA25BA25DA25BA292CAFCA25BA25CA2133FA25CA2137FA25CB612F8A43E397AB841 >I<0103B7FC4916F018FC18FFD90007D9800F1380050113C07113E04AED7FF093C7FCA2 19F85C5DA3023FEDFFF05DA219E0027F5C4B15C04D1380190002FF4A5A4BEB1FFCEF3FF0 4CB45A4990B6C7FC17FC839238E00FFF4902037F4B6C7F848249825DA25E495E1500A25E 5B4A5DA3013F4A141C4AED803EA2197E017F6E147C4A16FCB6D8F801EBC0F870EBE3F094 387FFFE0051F13C0CA000313003F3A79B847>82 D<92393FF001C0913901FFFE030207EB FF07021F14CF913A3FE01FFF8091387F80079138FE000349487F49486D130013074A8013 0F177E495AA3013F157C80A26E91C7FC80ECFF8015F8EDFF8016F06D14FC82826D817F6D 817F6D7E020780EC007F1507150181167FA2D80F805DA2163F001F157F5FA294C7FC003F 5D5E6D13015E486C495A6D495A01FCEB1FE09039FF807FC000FE90B55AD8F83F49C8FCD8 F00F13F848C613C0323B78B936>I<48B9FC481880A21900489039807FF8039026FC00FF 130001F08248484A7F5B5CD81F80163E01005C5A5C123E007E5D127C4A157E00FC177C00 784B133CC793C7FC5CA293C9FCA25CA25DA2143FA25DA2147FA25DA214FFA25DA25BA25D A25BA25DA25BA25D007FB612F8B7FCA25E393871B742>I<007FB500E0B639C01FFFFE6F 6F5AB612E16C02E003C014FC00019026E0000301C0C71380F37E001B7C1BFC6E6F5C505A 6C6F1503634C1507634C4B5AA24C4BC7FC4C5D1A3E4C5D6E81047D5D017F02FC14014C5D 030115034C5D4B484A5A15074C4A5A030F151F4C92C8FC6E4848EBF83EA2033E017F5B01 3F017E15FC037C5D4B14F961DAFDF0ECFBE002FF15FF4B5DA24B5D4B92C9FCA26D90C76C 5AA24A5D4A5DA24A5DA24A5DA24A5D4A5DA26D486ECAFC573A6EB860>87 D97 DIIII<167E923803FFC04B13E092381FE3F092 383FC7F816CFED7F9FA215FF17F05CEE1FE0EE078093C7FCA25C5DA414075D011FB61280 17C01780A29026000FF8C7FCA5141F5DA4143F5DA4147FA25DA414FF5DA45B92C8FCA449 5AA45C1307120F383F87F8EA7FC7EAFFCF5CA2EB8FE0EB9FC0EB1F80D8FC3FC9FCEA7FFE EA1FF8EA07E02D4C81BA21>III<143C147F495A15805B1500A25C6D5AEB00 7091C7FCAB133FEBFFC000037F3807C7F0380F87F8EA1F07A2EA3E0FA2127C131F5C12FC EAF83F00005B137F5CA213FF5CA25A91C7FC5A5BEC0F801207EBFC1F1500120F495A143E 5C13F000075BEBF1F06CB45AC65B013EC7FC193C79BA1E>I107 DIIII<90390FC003FC903A3FF00FFF8090267FFC3F13E0903A7DFEFE0FF0903AF8FF F807F800019138F003FC4913C001F115FED803E113801500A2EA07E313C35CEA00030107 1407A25CA2010F140F17FC5CA2011F141F17F85CEE3FF0133F17E04AEB7FC0A2017FECFF 806E4813004B5A6E485A9039FFFC0FF091B512C0029F90C7FCEC87F8480180C8FCA291C9 FCA25AA25BA21207A25B387FFFE0B57EA25C2F367EA531>I<91383F803C903901FFE1FC 010713FB90381FE0FF90383FC07F9039FF803FF848EB001F485A5B000715F0485AA2001F 143F4914E0123FA2007F147F4914C0A300FF14FF491480A35C01801400A35C5DA2007F13 07140F003F495AEBC03F381FE0FF6CB5FC0003EBEFF838007F0F1300141F5DA3143F5DA3 147F5D90381FFFFE497FA26D5B263677A52C>I<01FCEB7F803A03FF01FFF04801C713FC 3A0F9FEFC0FE3A1F0FFF003F4A13FF003E495A131F4A5A007C13F0A216FED8FC3FEB01FC 00F89038E000F000001500A2137F5CA313FF5CA35A91C8FCA35A5BA312075BA35B5BEA01 E0282779A52A>II<14F0EB 03F8130780495AA3131FA25CA2133FA25CA2137FA2B6128015C0A21580C6EB8000A25AA2 91C7FCA25AA25BA21207A25BA2120FA25BA2121FA29038F007C0A2003F130F1580EBE01F 1500143E147EEBC07C6C6C5A380FE3F0EBFFE000031380C648C7FC1A3778B520>I<133F D9FFC0130F000301F0EB1FC02607E7F8133FD80F83147F1387381F07FC003E15FFA2D87E 0F1580007C5B011F5B00FC13F000F81600EA003F4A5AA2017F5C14C0150713FF02805BA2 150F5A0200EBF83EA2031F137E177C16F0A2033F13F8A26C91387FF1F0EC80FF903A7F83 F7F3E0903A3FFFC3FFC0010F01811380903A01FC007E002F2779A534>I<017EEB01E03A 01FF8003F8489038E007FC3907C7F00FD80F87EB1FFEEA1F0F14F8123E150F397C1FF007 1503013F130100FC9038E000FC12F8EA007F5C16F813FF5C15014815F01400A2ED03E05A 5BED07C0A2ED0F80A2ED1F00151E0001143E6D5B6C5C90387F81F06DB45A010F1380D901 FEC7FC272779A52C>I<133FD9FFC0130F000301F0EB1FC02607E7F8133FD80F83147F13 87381F07FC003E15FF1780EA7E0F007C5B011F5B00FC01F0140012F8EA003F4A5A5E137F 14C0150701FF5C1480A2150F485D1400A2151F5EA3153F5E6C147FEC80FFEB7F836DB55A 130F903801FC7F90C712FF5EEA03C0260FF00190C7FC121F003F495AA24A5A4A5A01E05B 49485A9038007F80261FC0FFC8FC380FFFFC000313F0C613802A3779A52F>121 D<02FC131FEB03FF010FEB803F49EBC07E49EBE07C49EBF8FC90B612F816F04815E09039 F8000FC049EB1F80C8EA3F00157E5D4A5A4A5A4A5AEC1FC04AC7FC147E5C495A495AEB0F E0EB1F8049C7127C137E4914FC4848EB01F84848130748B612F05A16E0D81F8314C0D83F 011480D87E001400007CEB3FFE00FCEB1FF848EB07E028277BA529>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fp cmbx12 12 35 /Fp 35 119 df45 D49 DII<163FA25E5E5D5DA25D5D5D5DA25D92B5FCEC01F7EC03E7140715C7EC0F 87EC1F07143E147E147C14F8EB01F0EB03E0130714C0EB0F80EB1F00133E5BA25B485A48 5A485A120F5B48C7FC123E5A12FCB91280A5C8000F90C7FCAC027FB61280A531417DC038 >I<0007150301E0143F01FFEB07FF91B6FC5E5E5E5E5E16804BC7FC5D15E092C8FC01C0 C9FCAAEC3FF001C1B5FC01C714C001DF14F09039FFE03FFC9138000FFE01FC6D7E01F06D 13804915C0497F6C4815E0C8FC6F13F0A317F8A4EA0F80EA3FE0487E12FF7FA317F05B5D 6C4815E05B007EC74813C0123E003F4A1380D81FC0491300D80FF0495AD807FEEBFFFC6C B612F0C65D013F1480010F01FCC7FC010113C02D427BC038>I58 D65 D67 DIII73 D77 D80 D82 DI86 D<903801FFE0011F13FE017F6D7E48B612E03A03FE007FF84848 EB1FFC6D6D7E486C6D7EA26F7FA36F7F6C5A6C5AEA00F090C7FCA40203B5FC91B6FC1307 013F13F19038FFFC01000313E0000F1380381FFE00485A5B127F5B12FF5BA35DA26D5B6C 6C5B4B13F0D83FFE013EEBFFC03A1FFF80FC7F0007EBFFF86CECE01FC66CEB8007D90FFC C9FC322F7DAD36>97 DIIIII<137C48B4FC4813804813C0A24813E0A56C13C0A26C13806C1300EA007C90C7FC AAEB7FC0EA7FFFA512037EB3AFB6FCA518467CC520>105 D108 D<90277F8007FEEC0FFCB590263FFFC090387FFF8092 B5D8F001B512E002816E4880913D87F01FFC0FE03FF8913D8FC00FFE1F801FFC0003D99F 009026FF3E007F6C019E6D013C130F02BC5D02F86D496D7EA24A5D4A5DA34A5DB3A7B600 81B60003B512FEA5572D7CAC5E>I<90397F8007FEB590383FFF8092B512E0028114F891 3987F03FFC91388F801F000390399F000FFE6C139E14BC02F86D7E5CA25CA35CB3A7B600 83B512FEA5372D7CAC3E>II<90397FC00FF8B590B57E02C314E002CF14F89139DFC03FFC9139FF001FFE000301FCEB 07FF6C496D13804A15C04A6D13E05C7013F0A2EF7FF8A4EF3FFCACEF7FF8A318F017FFA2 4C13E06E15C06E5B6E4913806E4913006E495A9139DFC07FFC02CFB512F002C314C002C0 91C7FCED1FF092C9FCADB67EA536407DAC3E>I<90387F807FB53881FFE0028313F0028F 13F8ED8FFC91389F1FFE000313BE6C13BC14F8A214F0ED0FFC9138E007F8ED01E092C7FC A35CB3A5B612E0A5272D7DAC2E>114 D<90391FFC038090B51287000314FF120F381FF0 03383FC00049133F48C7121F127E00FE140FA215077EA27F01E090C7FC13FE387FFFF014 FF6C14C015F06C14FC6C800003806C15806C7E010F14C0EB003F020313E0140000F0143F A26C141F150FA27EA26C15C06C141FA26DEB3F8001E0EB7F009038F803FE90B55A00FC5C D8F03F13E026E007FEC7FC232F7CAD2C>IIII E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fq cmmi9 9 30 /Fq 30 119 df<147F903803FFE090380FC0F890383F007C017C017E1360497F484815E0 484890381F80C0120748481481EEC1804848130F003F15C390C7140016C74815C6007E15 CE16DC16D816F8485D5E5E127CA3151F6C143F037713C06C903801E7E03A0F800783E13B 07C07E03E3803B01FFF801FF003A007F80007C2B227EA031>11 DI<90 3801FF80130F013F130001FFC7FCEA01F8485A485A485A485A48C8FCA2127E387FFFF880 B5FC00FCC8FCA9127CA27EA26C1307380F800E3803E07C3801FFF038003F8019217D9F1F >15 D<137CEB7F80EB1FE0130F6D7EA26D7EA36D7EA36D7EA28080A26E7EA36E7EA28114 0FA26E7EA381140F141FEC3DFC1479ECF8FEEB01F0EB03E0903807C07FEB0F80EB1F0001 3EEB3F80137E4914C04848131F485A4848EB0FE0EA1FC0123F4848EB07F048C7FC4815F8 48140348EC01FC48140026357CB32D>21 D<1307D90FC01338011F147C16FC5CA2013F13 01A202005BA2491303A2017E5CA201FE1307A2495CA20001140FA2495C17800003021F13 C016C149EC8180A20007EC3F836D017F130016034B5A3A0FFC03CF869039FE070F8E9039 DFFE07FC9039C3F801F0D81FC0C9FCA25BA2123FA290CAFCA25AA2127EA212FEA25A1238 2A327FA02E>I<123C127E12FFA4127E123C08087A8715>58 D<123C127EB4FCA21380A2 127F123D1201A412031300A25A1206120E120C121C5A5A126009177A8715>I<171C177E EE01FEEE07FCEE1FF0EE7FC0923801FF00ED07FCED1FF0ED7FC04A48C7FCEC07FCEC1FF0 EC7FC04948C8FCEB07FCEB1FF0EB7FC04848C9FCEA07FCEA1FF0EA7FC048CAFCA2EA7FC0 EA1FF0EA07FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FCEC01FF 9138007FC0ED1FF0ED07FCED01FF9238007FC0EE1FF0EE07FCEE01FEEE007E171C2F2E7A A93C>I<1530157815F8A215F01401A215E01403A215C01407A21580140FA215005CA214 3EA2143C147CA2147814F8A25C1301A25C1303A25C1307A2495AA291C7FC5BA2131E133E A2133C137CA2137813F8A25B1201A25B1203A2485AA25B120FA290C8FC5AA2121E123EA2 123C127CA2127812F8A25A12601D4B7CB726>I<127012FCB4FCEA7FC0EA1FF0EA07FCEA 01FF38007FC0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FCEC01FF9138007FC0ED1F F0ED07FCED01FF9238007FC0EE1FF0EE07FCEE01FEA2EE07FCEE1FF0EE7FC0923801FF00 ED07FCED1FF0ED7FC04A48C7FCEC07FCEC1FF0EC7FC04948C8FCEB07FCEB1FF0EB7FC048 48C9FCEA07FCEA1FF0EA7FC048CAFC12FC12702F2E7AA93C>I<16035E5EA24C7EA2163F 167FA216FFA2ED01BFED033F831506161F150C1518A215301570156015C083EC01800203 130F15001406A25C141C14184A80A2027FB5FC91B6FCA2903901800007A249C7FC130683 5B16035B5B1370136013E01201D807F04A7EB549B512F0A25B34367DB53A>65 D<010FB612F017FEEFFF80903B003FC0003FE0EF0FF017074B14F81703027F15FCA292C7 FCA25C18F84A140718F00101150F18E04AEC1FC0EF3F800103ED7F00EE01FE4AEB07F891 B612E04915809139F8001FF04AEB03FCEE00FE010F157FA24AEC3F80A2011F16C0A25CA2 133F18804A147FA2017FEDFF005F91C712014C5A494A5A4C5A49EC3FE00001913801FF80 B748C7FC16F816C036337DB23A>I<010FB712FCA218F8903A003FC00007170018785D18 38147F183092C8FCA25CA25C16060101020E1370040C13604A1500A20103141C5E5C16F8 49B5FCA25EECF001010F130016605CA2011F14E05E5CA2013F91C8FCA25CA2137FA291CA FCA25BA25B487EB6FCA336337DB231>70 DI<90260FFFE049B5FCA281D9001F9138000FE04A6CEC 07801900DA33FC1406A2DA71FE140E180C146081DAE07F141C701318ECC03F8201011638 6F6C133014806F7E010316706F6C136014001503496E13E003015C0106801500010EECFF 0160010CEC7F81A2011CEC3FC395C7FC0118EC1FE3A20138EC0FF717F60130140717FE01 7014035F01601401A213E0705A1201D807F01578B57E1730A240337DB23D>78 D86 DI<0103B539C007FF FC5BA29026000FFCC713804BECFC00020715F0606E6C495A4D5A02014AC7FC6F130E5F6E 6C5B5F92387F80605F92383F818004C3C8FC16C6ED1FEC16F86F5AA2150782A282150FED 1DFE153915704B7E4A5A4A486C7E150002066D7E5C4A131F4A805C4A6D7E495A49C76C7E 1306010E1403013C81137CD803FE4A7EB500C090387FFFFCA2603E337EB23F>I<49B712 C05BA292C7EA7F80D907F8ECFF0002E0495A4A130349485C91C7485A010E4A5A011E4A5A 011C4A5A01184A5A01384AC7FC4B5A0130495A150701705C90C7485A4B5A4B5A4B5A4BC8 FC4A5A4A5A4A5A4A5AA24A5A4A5A4A5A4AC71260494814E049485C495A49481301495A5F 49481303495A49C748C7FC485A48485C4848141E4848143E4848147E4848495A150F48B6 FCB7FC5E32337CB234>90 D<133FEA1FFFA25B1200A35BA21201A25BA21203A25BA21207 A2EBE0F8EBE3FF390FEF07809038FC03C001F813E0EBF001D81FE013F013C0138015F812 3FA21300A248130315F0127EA2140700FE14E05AA2EC0FC0A2EC1F80007C14005C147E00 3C137C003E5B381E01F0380F07C06CB4C7FCEA00FC1D357EB321>98 D<147F903803FFC090380FC0F090383F0038137C4913F83801F0013803E0031207EA0FC0 90388001F0001F90C7FC123F90C8FCA25A127EA45AA3127C150C151C15386C147015E06C EB03C0390F800F003807C07E3801FFF038007F801E227EA021>I105 D<151C157E15FEA315FC15781500AA143FECFFC0903801C3E0EB038390380701F0130EEB 0C03131C1338133014071370012013E01300140FA215C0A2141FA21580A2143FA21500A2 5CA2147EA214FEA25CA21301A25CA21303001C5B127F495AA238FE0FC0495AD8783FC7FC EA707CEA3FF0EA0FC01F4281B11F>IIIII<011F131F90397FC07FE09039E3E1E0F09039C3E380783A01 C1F7007CD981FE133CD983FC133E00035BEB03F0163FEA0707120600025B1200010F147F 167E5CA2011F14FE16FC5CA2013FEB01F8A291380003F016E0491307ED0FC002801380ED 1F009038FFC03E9038FEE0F89038FC7FE0EC1F80000190C8FCA25BA21203A25BA21207A2 5BB57EA3283083A027>112 D<13F8D803FEEB01C0D8070FEB03E0000EEB8007121C0018 13C00038140FEA301F0070018013C01260013F131F00E0130000401580C65A017E133F13 FE491400A25D120149137E1602EDFE0716064913FCA2160E0201130C9039F803F81C1618 000090380F7C38D97C1C137090393FF81FE0903907E0078028227EA02C>117 D<01F0130ED803FC131FD8071EEB3F80EA0E1F121C0038EB801F0030140F013F13070070 1300006014035BD8E07E14001240EA00FE495B000114065BA2150E0003140C5B151C1518 1538491330157015606D13E04A5A0001495A6D48C7FC3800FC1EEB3FF8EB07E021227EA0 25>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fr cmbx9 9 19 /Fr 19 117 df<120FEA3FC0EA7FE0EAFFF0A6EA7FE0EA3FC0EA0F000C0C7A8B19>46 D<147814F81303131FEA03FFB5FCA3EAFC1F1200B3B2007FB512FEA41F317AB02C>49 DII<151F5D5DA25D5C5C5C5CA25C143D14 7D14F9EB01F114E1EB03C1EB0781130FEB1F01133E133C137813F01201EA03E0EA07C013 80EA0F00121E123E5A5AB712FEA4C700031300A80103B512FEA427317EB02C>I<000C14 0ED80FE013FE90B5FC5D5D5D5D5D92C7FC14FC14F091C8FC1380A6EB87FE9038BFFFC090 B512F09038FC0FF89038E003FE01C07F497E01001480000E6D13C0C8FCA216E0A3121FEA 7F807F487EA316C05B5CD87F801480D87C0014006C5B393F8007FE391FE01FFC0007B512 F06C14C0C691C7FCEB1FF823327CB02C>II<123C123F90B612F8A44815F016E016C0168016005D 007CC7127E00785C4A5A00F8495A48495A4A5A4A5AC7FC4AC7FC147E14FE5C13015C1303 A2495AA2130FA2131FA25C133FA4137FA96D5AA2010FC8FC25337BB12C>II65 D70 D97 DI<903807FF80013F13F090B512FC3903FE 01FE4848487EEA0FF8EA1FF0EA3FE0A2007F6D5A496C5A153000FF91C7FCA9127F7FA200 3FEC07807F6C6C130F000FEC1F00D807FE133E3903FF80FCC6EBFFF8013F13E0010790C7 FC21217DA027>I<16F890390FFC07FE90387FFF9F48B6127F3907FC0FFC380FF003001F 14FED9E001133E003FECFF1C1600A6001F5CEBF003000F5C3907FC0FF890B512E0486C13 80D90FFCC7FC48C9FCA37F7F90B512F015FE6CECFF8016E06C15F06C15F84815FC121F39 3F80001F48C7EA03FE481401481400A46C14016C6CEB03FC6C6CEB07F86C6CEB0FF0D80F FCEB7FE00003B61280C6ECFE00010F13E028327EA12C>103 D105 D<3901F81F8000FFEB7FF0ECFFF89038F9E3FC9038FBC7FE380FFF876C1307A213FEEC03 FCEC01F8EC0060491300B1B512F0A41F217EA024>114 D<9038FFE1C0000713FF5A383F 803F387E000F14075A14037EA26C6CC7FC13FCEBFFE06C13FC806CEBFF80000F14C06C14 E0C6FC010F13F0EB007F140F00F0130714037EA26C14E06C13076CEB0FC09038C01F8090 B5120000F913FC38E03FE01C217DA023>I<133CA5137CA313FCA21201A212031207001F B51280B6FCA3D807FCC7FCB0EC03C0A79038FE078012033901FF0F006C13FEEB3FFCEB0F F01A2F7EAE22>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fs cmtt9 9 22 /Fs 22 119 df<120FEA3FC013E0EA7FF0A213F8A2123FA2120F120113F01203EA07E012 1FEA7FC0EAFF8013005A12700D14738927>44 D<121EEA7F80A2EAFFC0A4EA7F80A2EA1E 000A0A728927>46 D64 D<007FB512E0B612F0A36C14E039001F8000B3B2007FB512E0B612F0A36C14E01C2E7BAD 27>73 D<90387FC0E03901FFF1F0000713FF5A5AEA3FE0EB801F387F000F007E130712FE 5A1403A3EC01E06C90C7FC127E127FEA3FC013F86CB47E6C13F86C13FE6CEBFF80C614C0 010F13E0010013F0140FEC07F81403140115FC1400127812FCA46CEB01F8A26C13039038 8007F09038F01FE090B5FC15C0150000F85B38701FF81E307CAE27>83 D<3803FFC0000F13F04813FC4813FF811380EC1FC0381F000F000480C71207A2EB0FFF13 7F0003B5FC120F5A383FFC07EA7FC0130012FE5AA46C130F007F131FEBC0FF6CB612806C 15C07E000313F1C69038807F8022207C9F27>97 DIIII<153F90391FC0FF80D97FF313C048B612E05A4814EF390FF07F873A1FC01FC3C0 EDC000EB800F48486C7EA66C6C485AEBC01FA2390FF07F8090B5C7FC5C485BEB7FF0EB1F C090C9FCA27F6CB5FC15E015F84814FE4880EB8001007EC7EA3F80007C140F00FC15C048 1407A46C140F007C1580007F143F6C6CEB7F009038F807FF6CB55A000714F86C5CC614C0 D90FFCC7FC23337EA027>103 DI<130F497E497EA46D5A6DC7FC90C8FCA7383FFF80487FA37EEA000FB3A4007F B512F0B6FC15F815F07E1D2F7BAE27>I<387FFF80B57EA37EEA000FB3B2007FB512F8B6 12FCA36C14F81E2E7CAD27>108 D<397F07C01F3AFF9FF07FC09039FFF9FFE091B57E7E 3A0FFC7FF1F89038F03FC001E0138001C01300A3EB803EB03A7FF0FFC3FF486C01E31380 01F913E701F813E36C4801C313002920819F27>I111 D<387FE0FFD8FFF313C090B512F0816C800003EB81FE49C67E 49EB3F8049131F16C049130FA216E01507A6150F16C07F151F6DEB3F80157F6DEBFF0090 38FF83FEECFFFC5D5D01F313C0D9F0FEC7FC91C8FCAC387FFF80B57EA36C5B23317F9F27 >I<397FFC03FC39FFFE0FFF023F13804A13C0007F90B5FC39007FFE1F14F89138F00F80 9138E002004AC7FC5CA291C8FCA2137EAD007FB57EB67EA36C5C22207E9F27>114 D<9038FFF3800007EBFFC0121F5A5AEB803F38FC000F5AA2EC07806C90C7FCEA7F8013FC 383FFFF06C13FC000713FF00011480D8000F13C09038003FE014070078EB03F000FC1301 A27E14036CEB07E0EBE01F90B512C01580150000FB13FC38707FF01C207B9F27>I<133C 137EA8007FB512F0B612F8A36C14F0D8007EC7FCAE1518157EA415FE6D13FC1483ECFFF8 6D13F06D13E0010313C0010013001F297EA827>I<397FE01FF8486C487EA3007F131F00 031300B21401A21403EBFC0F6CB612E016F07EEB3FFE90390FF87FE024207F9F27>I<3A 7FFC0FFF80486C4813C0A36C486C13803A07C000F800EBE00100035CA2EBF00300015CA2 EBF80700005CA390387C0F80A36D48C7FCA3EB3F3FEB1F3EA214FE6D5AA36D5AA26D5A22 207E9F27>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ft cmsy9 9 13 /Ft 13 113 df<007FB712FCB812FEA26C16FC2F047A943C>0 D<171C177EEE01FEEE07 FCEE1FF0EE7FC0923801FF00ED07FCED1FF0ED7FC04A48C7FCEC07FCEC1FF0EC7FC04948 C8FCEB07FCEB1FF0EB7FC04848C9FCEA07FCEA1FF0EA7FC048CAFCA2EA7FC0EA1FF0EA07 FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FCEC01FF9138007FC0 ED1FF0ED07FCED01FF9238007FC0EE1FF0EE07FCEE01FEEE007E171C1700AC007FB712FC B812FEA26C16FC2F3E7AB03C>20 D<127012FCB4FCEA7FC0EA1FF0EA07FCEA01FF38007F C0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FCEC01FF9138007FC0ED1FF0ED07FCED 01FF9238007FC0EE1FF0EE07FCEE01FEA2EE07FCEE1FF0EE7FC0923801FF00ED07FCED1F F0ED7FC04A48C7FCEC07FCEC1FF0EC7FC04948C8FCEB07FCEB1FF0EB7FC04848C9FCEA07 FCEA1FF0EA7FC048CAFC12FC1270CBFCAC007FB712FCB812FEA26C16FC2F3E7AB03C>I< 91383FFFF849B512FC1307011F14F8D93FE0C7FC01FFC8FCEA01FCEA03F0485A485A5B48 C9FC5A123E5AA21278A212F8A25AB712F816FCA216F800F0C9FC7EA21278A2127CA27E12 3F7E6C7E7F6C7E6C7EEA01FC6CB4FCEB3FE06DB512F8010714FC1301D9003F13F8262E7A A933>50 D69 D76 D<03071880031F17014B 17035D1B071B0F70EE1F006363A203FF5F704B5A1A0303DF16071A0F03CF161F02016D15 3E158F505A1AF891260387F0EC01F00307ED03E1F107C1F10F8191260603F8EC1F01073F 5B197E020E167C91260C01FCECF803F001F0021CED03E00218ED07C06F6CEB0F804AED1F 00183E70017E5C4A6D5B4D5A02E0DA83F013074A90383F87E0EFCFC04948ECDF8070B4C7 FC01035D91C76C5A495D01066E5AD8300E5DD87C1C6E5AD87FFC5D484891C914E0F3FBC0 49F1FF8049F1FE0063D83FC0F003F0D80F8095C8FC53397EB45C>I83 D85 D<0060ED018000F0ED03C0B3AF 6C1507A2007CED0F80A26CED1F00003F5D6C6C147ED80FE0495AD807F8EB07F83A01FF80 7FE06C90B55A013F91C7FC010F13FC010013C02A307CAD33>91 D102 D<12FCEAFFC0EA07F0EA01FC6C7E137F7F80131FB3A580130F6D7E6D7EEB01FC9038007F C0EC1FE0EC7FC0903801FC00EB03F0495A495A131F5CB3A5133F91C7FC5B13FE485AEA07 F0EAFFC000FCC8FC1B4B7BB726>I<1930197819F8A2F001F0A2F003E0A2F007C0A2F00F 80A2F01F00A2183EA260A260A24D5AA24D5AA24D5AA24D5AA24DC7FCA2173EA25FA25FA2 4C5A13C000014B5AEA07E0000F4B5AEA3FF000734B5AEAE3F800C14BC8FCEA01FC000015 3E7F017E5C137F6D5CA26E485A131F6E485A130F6E485A13076E485A13036E48C9FC1301 153E14FC01005B14FEEC7EF8147F6E5AA26E5AA26E5AA26E5A92CAFC3D4C7B8340>112 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fu cmr9 9 75 /Fu 75 124 df<91393FE00FE0903A01FFF83FF8903A07E01EF83C903A1F800FF07E903A 3F001FE0FE017E133F4914C0485A1738484890381F8000ACB812C0A33B03F0001F8000B3 A7486C497EB50083B5FCA32F357FB42D>11 DI14 D<137813FCA212011203EA07F813E0EA0FC0EA1F801300 123C5A5A12400E0E71B326>19 D<14C01301EB0380EB0F00130E5B133C5B5BA2485A485A A212075B120F90C7FC5AA2121E123EA3123C127CA55AB0127CA5123C123EA3121E121FA2 7E7F12077F1203A26C7E6C7EA213787F131C7F130FEB0380EB01C01300124A79B71E>40 D<12C07E1270123C121C7E120F6C7E6C7EA26C7E6C7EA27F1378137C133C133EA2131E13 1FA37F1480A5EB07C0B0EB0F80A514005BA3131E133EA2133C137C137813F85BA2485A48 5AA2485A48C7FC120E5A123C12705A5A124A7CB71E>I<156015F0B3A4007FB812C0B912 E0A26C17C0C800F0C8FCB3A4156033327CAB3C>43 D<123C127EB4FCA21380A2127F123D 1201A412031300A25A1206120E120C121C5A5A126009177A8715>II<123C127E12FFA4127E123C08087A8715>I48 D<13075B5B137FEA07FFB5FC13BFEAF83F1200B3B3A2497E007F B51280A319327AB126>IIII<000C14C0380FC00F90B5128015005C5C14F014 C0D80C18C7FC90C8FCA9EB0FC0EB7FF8EBF07C380FC03F9038001F80EC0FC0120E000CEB 07E0A2C713F01403A215F8A41218127E12FEA315F0140712F8006014E01270EC0FC06C13 1F003C14806CEB7F00380F80FE3807FFF8000113E038003F801D347CB126>I<14FE9038 07FF80011F13E090383F00F0017C13703901F801F8EBF003EA03E01207EA0FC0EC01F048 48C7FCA248C8FCA35A127EEB07F0EB1FFC38FE381F9038700F809038E007C039FFC003E0 018013F0EC01F8130015FC1400A24814FEA5127EA4127F6C14FCA26C1301018013F8000F 14F0EBC0030007EB07E03903E00FC03901F81F806CB51200EB3FFCEB0FE01F347DB126> I<1230123C003FB6FCA34814FEA215FC0070C7123800601430157015E04814C01401EC03 80C7EA07001406140E5C141814385CA25CA2495A1303A3495AA2130FA3131F91C7FCA25B A55BA9131C20347CB126>III<123C127E12FFA4127E123C1200B0123C127E12FFA4127E12 3C08207A9F15>I<123C127E12FFA4127E123C1200B0123C127E12FE12FFA3127F123F12 03A412071206A3120E120C121C1238123012701260082F7A9F15>I<007FB812C0B912E0 A26C17C0CCFCAC007FB812C0B912E0A26C17C033147C9C3C>61 D<15E0A34A7EA24A7EA3 4A7EA3EC0DFE140CA2EC187FA34A6C7EA202707FEC601FA202E07FECC00FA2D901807F15 07A249486C7EA301066D7EA2010E80010FB5FCA249800118C77EA24981163FA2496E7EA3 496E7EA20001821607487ED81FF04A7ED8FFFE49B512E0A333367DB53A>65 DIIIII72 DI<017FB5FCA39038003FE0EC1FC0B3B1127EB4FCA4EC3F805A006014 0000705B6C13FE6C485A380F03F03803FFC0C690C7FC20357DB227>IIII< D8FFFE91381FFFF87F80C6030013006E143CD9DFE01418EBCFF0A2EBC7F8EBC3FCA2EBC1 FEEBC0FF6E7EA26E7E6E7EA26E7E6E7E6E7EA26E7E6E7EA2ED7F80ED3FC0ED1FE0A2ED0F F0ED07F8A2ED03FCED01FEED00FFA2EE7F98EE3FD8A2EE1FF8160F1607A216031601A248 6C1400D807F81578B500C01438A2171835337EB23A>III82 D<90381FE00390387FFC 0748B5FC3907F01FCF390F8003FF48C7FC003E80814880A200788000F880A46C80A27E92 C7FC127F13C0EA3FF013FF6C13F06C13FF6C14C06C14F0C680013F7F01037F9038003FFF 140302001380157F153FED1FC0150F12C0A21507A37EA26CEC0F80A26C15006C5C6C143E 6C147E01C05B39F1FC03F800E0B512E0011F138026C003FEC7FC22377CB42B>I<007FB7 12FEA390398007F001D87C00EC003E0078161E0070160EA20060160600E01607A3481603 A6C71500B3AB4A7E011FB512FCA330337DB237>IIII89 D<003FB612FCA39039F80007F813C090C7EA0FF0003EEC1FE0123C 0038EC3FC00078EC7F801270EDFF004A5AA20060495AA24A5A4A5AC7FC4A5A4A5AA24A5A 4AC7FCA2495A495AA2495A495AA24948130C495AA2495A49C7FCA24848141CA2485A485A 1638485A4848147816F84848130148481307153FB7FCA326337CB22F>II93 D97 DII<153FEC0FFFA3EC007F81AEEB07F0EB3FFCEBFC0F3901F003BF39 07E001FF48487E48487F8148C7FCA25A127E12FEAA127E127FA27E6C6C5BA26C6C5B6C6C 4813803A03F007BFFC3900F81E3FEB3FFCD90FE0130026357DB32B>III<151F90391FC07F809039FFF8E3C03901F07FC73907E03F033A0FC01F8380 9039800F8000001F80EB00074880A66C5CEB800F000F5CEBC01F6C6C48C7FCEBF07C380E FFF8380C1FC0001CC9FCA3121EA2121F380FFFFEECFFC06C14F06C14FC4880381F000100 3EEB007F4880ED1F8048140FA56C141F007C15006C143E6C5C390FC001F83903F007E0C6 B51280D91FFCC7FC22337EA126>IIIIII<2703F01FE013FF00FF90267FF80313C0903BF1E07C0F03E0 903BF3803E1C01F02807F7003F387FD803FE1470496D486C7EA2495CA2495CB3486C496C 487EB53BC7FFFE3FFFF0A33C217EA041>I<3903F01FC000FFEB7FF09038F1E0FC9038F3 807C3907F7007EEA03FE497FA25BA25BB3486CEB7F80B538C7FFFCA326217EA02B>II<3903F03F8000FFEBFFE09038 F3C0F89038F7007ED807FE7F6C48EB1F804914C049130F16E0ED07F0A3ED03F8A9150716 F0A216E0150F16C06D131F6DEB3F80160001FF13FC9038F381F89038F1FFE0D9F07FC7FC 91C8FCAA487EB512C0A325307EA02B>I<903807F00390383FFC07EBFC0F3901F8038F38 07E001000F14DF48486CB4FC497F123F90C77E5AA25A5AA9127FA36C6C5B121F6D5B000F 5B3907E003BF3903F0073F3800F81EEB3FF8EB0FE090C7FCAAED7F8091380FFFFCA32630 7DA029>I<3803E07C38FFE1FF9038E38F809038E71FC0EA07EEEA03ECA29038FC0F8049 C7FCA35BB2487EB512E0A31A217FA01E>II<1330A51370A313F0A21201A212031207381FFFFEB5FCA23803F000AF1403 A814073801F806A23800FC0EEB7E1CEB1FF8EB07E0182F7FAD1E>IIIII<3A7FFF807FF8A33A07F8001FC00003EC0F800001EC070015066C6C5BA26D131C017E 1318A26D5BA2EC8070011F1360ECC0E0010F5BA2903807E180A214F3010390C7FC14FBEB 01FEA26D5AA31478A21430A25CA214E05CA2495A1278D8FC03C8FCA21306130EEA701CEA 7838EA1FF0EA0FC025307F9F29>I<003FB512F0A2EB000F003C14E00038EB1FC00030EB 3F800070137F1500006013FE495A13035CC6485A495AA2495A495A49C7FC153013FE485A 12035B48481370485A001F14604913E0485A387F000348130F90B5FCA21C207E9F22>I< B712F8A22502809426>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fv cmr7 7 13 /Fv 13 62 df22 D<1306130C13181330136013E0EA01C0EA03 80A2EA07005A120E121EA2121C123CA35AA512F85AAB7E1278A57EA3121C121EA2120E12 0F7EEA0380A2EA01C0EA00E0136013301318130C13060F3B7AAB1A>40 D<12C012607E7E7E120E7EEA0380A2EA01C013E0120013F0A213701378A3133CA5133E13 1EAB133E133CA51378A3137013F0A213E0120113C0EA0380A2EA0700120E120C5A5A5A5A 0F3B7DAB1A>I<140EB3A2B812E0A3C7000EC8FCB3A22B2B7DA333>43 D48 D<13381378EA01F8121F12FE12E01200B3AB487EB512F8A215267BA521 >I<13FF000313E0380E03F0381800F848137C48137E00787F12FC6CEB1F80A4127CC7FC 15005C143E147E147C5C495A495A5C495A010EC7FC5B5B903870018013E0EA0180390300 030012065A001FB5FC5A485BB5FCA219267DA521>I<13FF000313E0380F01F8381C007C 0030137E003C133E007E133FA4123CC7123E147E147C5C495AEB07E03801FF8091C7FC38 0001E06D7E147C80143F801580A21238127C12FEA21500485B0078133E00705B6C5B381F 01F03807FFC0C690C7FC19277DA521>I<1438A2147814F81301A2130313071306130C13 1C131813301370136013C012011380EA03005A120E120C121C5A12305A12E0B612E0A2C7 EAF800A7497E90383FFFE0A21B277EA621>I<0018130C001F137CEBFFF85C5C1480D819 FCC7FC0018C8FCA7137F3819FFE0381F81F0381E0078001C7F0018133EC7FC80A21580A2 1230127C12FCA3150012F00060133E127000305B001C5B380F03E03803FFC0C648C7FC19 277DA521>II<137F3801FFC03807C1E0380F0070001E1378003E7F003C133E007C131EA200FC131FA4 1580A4007C133FA2123C003E137F121E380F01DF3807FF9F3801FE1FD8001013001300A2 143E123C007E133CA25C5C007C5B383003C0381C0780D80FFFC7FCEA03F819277DA521> 57 D<007FB712C0B812E0A2CBFCABB812E0A26C16C02B117D9633>61 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fw cmr10 10 73 /Fw 73 124 df11 DI14 D22 D<001C131C007F137F39FF80FF80A26D13C0A3007F137F001C131C00001300A400011301 01801380A20003130301001300485B00061306000E130E485B485B485B006013601A197D B92A>34 D<121C127FEAFF80A213C0A3127F121C1200A412011380A2120313005A120612 0E5A5A5A12600A1979B917>39 D<146014E0EB01C0EB0380EB0700130E131E5B5BA25B48 5AA2485AA212075B120F90C7FCA25A121EA2123EA35AA65AB2127CA67EA3121EA2121F7E A27F12077F1203A26C7EA26C7E1378A27F7F130E7FEB0380EB01C0EB00E01460135278BD 20>I<12C07E12707E7E7E120F6C7E6C7EA26C7E6C7EA21378A2137C133C133E131EA213 1F7FA21480A3EB07C0A6EB03E0B2EB07C0A6EB0F80A31400A25B131EA2133E133C137C13 78A25BA2485A485AA2485A48C7FC120E5A5A5A5A5A13527CBD20>I<15301578B3A6007F B812F8B912FCA26C17F8C80078C8FCB3A6153036367BAF41>43 D<121C127FEAFF80A213 C0A3127F121C1200A412011380A2120313005A1206120E5A5A5A12600A19798817>II<121C127FEAFF80A5EA7F00121C0909798817>I48 DIII<1538A2157815F8 A2140114031407A2140F141F141B14331473146314C313011483EB030313071306130C13 1C131813301370136013C01201EA038013005A120E120C5A123812305A12E0B712F8A3C7 3803F800AB4A7E0103B512F8A325397EB82A>I<0006140CD80780133C9038F003F890B5 FC5D5D158092C7FC14FC38067FE090C9FCABEB07F8EB3FFE9038780F803907E007E09038 8003F0496C7E12066E7EC87EA28181A21680A4123E127F487EA490C71300485C12E00060 5C12700030495A00385C6C1303001E495A6C6C485A3907E03F800001B5C7FC38007FFCEB 1FE0213A7CB72A>II<12301238123E003FB612E0A3 16C05A168016000070C712060060140E5D151800E01438485C5D5DC712014A5A92C7FC5C 140E140C141C5CA25CA214F0495AA21303A25C1307A2130FA3495AA3133FA5137FA96DC8 FC131E233B7BB82A>III<121C12 7FEAFF80A5EA7F00121CC7FCB2121C127FEAFF80A5EA7F00121C092479A317>I<121C12 7FEAFF80A5EA7F00121CC7FCB2121C127F5A1380A4127F121D1201A412031300A25A1206 A2120E5A121812385A1260093479A317>I<007FB812F8B912FCA3CCFCAEB912FCA36C17 F836167B9F41>61 D<1538A3157CA315FEA34A7EA34A6C7EA202077FEC063FA2020E7FEC 0C1FA2021C7FEC180FA202387FEC3007A202707FEC6003A202C07F1501A2D901807F81A2 49C77F167FA20106810107B6FCA24981010CC7121FA2496E7EA3496E7EA3496E7EA213E0 707E1201486C81D80FFC02071380B56C90B512FEA3373C7DBB3E>65 DI<913A01FF800180020FEBE003027F13F8903A01FF807E07903A03 FC000F0FD90FF0EB039F4948EB01DFD93F80EB00FF49C8127F01FE153F12014848151F48 48150FA248481507A2485A1703123F5B007F1601A35B00FF93C7FCAD127F6DED0180A312 3F7F001F160318006C7E5F6C7E17066C6C150E6C6C5D00001618017F15386D6C5CD91FE0 5C6D6CEB03C0D903FCEB0F80902701FF803FC7FC9039007FFFFC020F13F002011380313D 7BBA3C>I69 DI72 DI<013FB512E0A39039001FFC00EC07F8B3B3A3123F EA7F80EAFFC0A44A5A1380D87F005B0070131F6C5C6C495A6C49C7FC380781FC3801FFF0 38007F80233B7DB82B>I76 DIIII82 DI<003FB812E0A3D9C003EB001F273E0001FE130348EE01F00078160000701770A3006017 30A400E01738481718A4C71600B3B0913807FF80011FB612E0A335397DB83C>II87 D91 D<3901800180000313033907000700000E130E485B001813180038133800301330007013 7000601360A200E013E0485BA400CE13CE39FF80FF806D13C0A3007F137FA2393F803F80 390E000E001A1974B92A>II97 DIIII<147E903803FF8090380FC1E0EB1F8790383F0F F0137EA213FCA23901F803C091C7FCADB512FCA3D801F8C7FCB3AB487E387FFFF8A31C3B 7FBA19>IIIIIII<2703F00FF0EB1FE000FFD93FFCEB7FF8913AF03F01E07E903B F1C01F83803F3D0FF3800FC7001F802603F70013CE01FE14DC49D907F8EB0FC0A2495CA3 495CB3A3486C496CEB1FE0B500C1B50083B5FCA340257EA445>I<3903F00FF000FFEB3F FCECF03F9039F1C01F803A0FF3800FC03803F70013FE496D7EA25BA35BB3A3486C497EB5 00C1B51280A329257EA42E>II<3903F01FE000FFEB7FF89038F1E07E 9039F3801F803A07F7000FC0D803FEEB07E049EB03F04914F849130116FC150016FEA316 7FAA16FEA3ED01FCA26DEB03F816F06D13076DEB0FE001F614C09039F7803F009038F1E0 7E9038F0FFF8EC1FC091C8FCAB487EB512C0A328357EA42E>II<3807E01F00FFEB7F C09038E1E3E09038E387F0380FE707EA03E613EE9038EC03E09038FC0080491300A45BB3 A2487EB512F0A31C257EA421>II<1318A51338A31378A313F8120112031207001FB5FCB6FCA2D801 F8C7FCB215C0A93800FC011580EB7C03017E13006D5AEB0FFEEB01F81A347FB220>IIIIII<003FB512FCA2EB8003D83E0013F8003CEB07F00038EB0FE012300070EB1FC0EC3F80 0060137F150014FE495AA2C6485A495AA2495A495A495AA290387F000613FEA2485A485A 0007140E5B4848130C4848131CA24848133C48C7127C48EB03FC90B5FCA21F247EA325> II E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fx cmbx12 14.4 26 /Fx 26 118 df45 D<171F4D7E4D7EA24D7EA34C7FA24C7FA34C 7FA34C7FA24C7FA34C8083047F80167E8304FE804C7E03018116F8830303814C7E030781 16E083030F814C7E031F81168083033F8293C77E4B82157E8403FE824B800201835D8402 03834B800207835D844AB87EA24A83A3DA3F80C88092C97E4A84A2027E8202FE844A8201 0185A24A820103854A82010785A24A82010F855C011F717FEBFFFCB600F8020FB712E0A5 5B547BD366>65 D<932601FFFCEC01C0047FD9FFC013030307B600F81307033F03FE131F 92B8EA803F0203DAE003EBC07F020F01FCC7383FF0FF023F01E0EC0FF94A01800203B5FC 494848C9FC4901F8824949824949824949824949824990CA7E494883A2484983485B1B7F 485B481A3FA24849181FA3485B1B0FA25AA298C7FC5CA2B5FCAE7EA280A2F307C07EA36C 7FA21B0F6C6D1980A26C1A1F6C7F1C006C6D606C6D187EA26D6C606D6D4C5A6D6D16036D 6D4C5A6D6D4C5A6D01FC4C5A6D6DEE7F806D6C6C6C4BC7FC6E01E0EC07FE020F01FEEC1F F80203903AFFE001FFF0020091B612C0033F93C8FC030715FCDB007F14E0040101FCC9FC 525479D261>67 DI70 D<932601FFFCEC01C0047FD9FFC0 13030307B600F81307033F03FE131F92B8EA803F0203DAE003EBC07F020F01FCC7383FF0 FF023F01E0EC0FF94A01800203B5FC494848C9FC4901F882494982494982494982494982 4990CA7E494883A2484983485B1B7F485B481A3FA24849181FA3485B1B0FA25AA298C8FC 5CA2B5FCAE6C057FB712E0A280A36C94C7003FEBC000A36C7FA36C7FA27E6C7FA26C7F6C 7FA26D7E6D7F6D7F6D6D5E6D7F6D01FC93B5FC6D13FF6D6C6D5C6E01F0EC07FB020F01FE EC1FF10203903AFFF001FFE0020091B6EAC07F033FEE001F030703FC1307DB007F02E013 01040149CAFC5B5479D26A>II77 D80 D<91260FFF80130791B500F85B010702FF5B011FEDC03F49EDF0 7F9026FFFC006D5A4801E0EB0FFD4801800101B5FC4848C87E48488149150F001F824981 123F4981007F82A28412FF84A27FA26D82A27F7F6D93C7FC14C06C13F014FF15F86CECFF 8016FC6CEDFFC017F06C16FC6C16FF6C17C06C836C836D826D82010F821303010082021F 16801400030F15C0ED007F040714E01600173F050F13F08383A200788200F882A3187FA2 7EA219E07EA26CEFFFC0A27F6D4B13806D17006D5D01FC4B5A01FF4B5A02C04A5A02F8EC 7FF0903B1FFFC003FFE0486C90B65AD8FC0393C7FC48C66C14FC48010F14F048D9007F90 C8FC3C5479D24B>83 D97 DI<4DB47E0407B5FCA5EE001F1707B3A4913801FFE0021F13FC91B6FC 010315C7010F9038E03FE74990380007F7D97FFC0101B5FC49487F4849143F484980485B 83485B5A91C8FC5AA3485AA412FFAC127FA36C7EA37EA26C7F5F6C6D5C7E6C6D5C6C6D49 B5FC6D6C4914E0D93FFED90FEFEBFF80903A0FFFC07FCF6D90B5128F0101ECFE0FD9003F 13F8020301C049C7FC41547CD24B>100 D<913803FFC0023F13FC49B6FC010715C04901 817F903A3FFC007FF849486D7E49486D7E4849130F48496D7E48178048497F18C0488191 C7FC4817E0A248815B18F0A212FFA490B8FCA318E049CAFCA6127FA27F7EA218E06CEE01 F06E14037E6C6DEC07E0A26C6DEC0FC06C6D141F6C6DEC3F806D6CECFF00D91FFEEB03FE 903A0FFFC03FF8010390B55A010015C0021F49C7FC020113F034387CB63D>III<137F497E000313E0487FA2487FA7 6C5BA26C5BC613806DC7FC90C8FCADEB3FF0B5FCA512017EB3B3A6B612E0A51B547BD325 >105 D108 DII<913801FFE0021F13FE91B6 12C0010315F0010F9038807FFC903A1FFC000FFED97FF86D6C7E49486D7F48496D7F4849 6D7F4A147F48834890C86C7EA24883A248486F7EA3007F1880A400FF18C0AC007F1880A3 003F18006D5DA26C5FA26C5F6E147F6C5F6C6D4A5A6C6D495B6C6D495B6D6C495BD93FFE 011F90C7FC903A0FFF807FFC6D90B55A010015C0023F91C8FC020113E03A387CB643>I< 903A3FF001FFE0B5010F13FE033FEBFFC092B612F002F301017F913AF7F8007FFE0003D9 FFE0EB1FFFC602806D7F92C76C7F4A824A6E7F4A6E7FA2717FA285187F85A4721380AC1A 0060A36118FFA2615F616E4A5BA26E4A5B6E4A5B6F495B6F4990C7FC03F0EBFFFC9126FB FE075B02F8B612E06F1480031F01FCC8FC030313C092CBFCB1B612F8A5414D7BB54B>I< 90397FE003FEB590380FFF80033F13E04B13F09238FE1FF89139E1F83FFC0003D9E3E013 FEC6ECC07FECE78014EF150014EE02FEEB3FFC5CEE1FF8EE0FF04A90C7FCA55CB3AAB612 FCA52F367CB537>114 D<903903FFF00F013FEBFE1F90B7FC120348EB003FD80FF81307 D81FE0130148487F4980127F90C87EA24881A27FA27F01F091C7FC13FCEBFFC06C13FF15 F86C14FF16C06C15F06C816C816C81C681013F1580010F15C01300020714E0EC003F0307 13F015010078EC007F00F8153F161F7E160FA27E17E07E6D141F17C07F6DEC3F8001F8EC 7F0001FEEB01FE9039FFC00FFC6DB55AD8FC1F14E0D8F807148048C601F8C7FC2C387CB6 35>I<143EA6147EA414FEA21301A313031307A2130F131F133F13FF5A000F90B6FCB8FC A426003FFEC8FCB3A9EE07C0AB011FEC0F8080A26DEC1F0015806DEBC03E6DEBF0FC6DEB FFF86D6C5B021F5B020313802A4D7ECB34>II E %EndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: Letter %%EndSetup %%Page: 1 1 1 0 bop 614 448 a Fx(A)44 b(Closed-F)-11 b(orm)45 b(Solution)h(for)f (Mapping)f(General)698 598 y(Distributions)i(to)f(Minimal)h(PH)f (Distributions)1140 891 y Fw(T)-7 b(ak)i(a)n(yuki)27 b(Osogami)1814 861 y Fv(1)1877 891 y Fw(and)g(Mor)g(Harc)n(hol-Balter) 2753 861 y Fv(1)921 1065 y Fu(Departmen)n(t)d(of)i(Computer)f(Science,) h(Carnegie)h(Mellon)g(Univ)n(ersit)n(y)1137 1156 y(5000)g(F)-6 b(orb)r(es)26 b(Av)n(en)n(ue,)f(Pittsburgh,)h(P)-6 b(A)25 b(15213,)j(USA)1396 1248 y Ft(f)p Fs(osogami,)41 b(harchol)p Ft(g)p Fs(@cs.cmu.edu)759 1626 y Fr(Abstract.)i Fu(Appro)n(ximating)16 b(general)j(distributions)f(b)n(y)e(phase-t)n(yp)r(e)h(\(PH\))g(dis-) 759 1717 y(tributions)26 b(is)g(a)g(p)r(opular)g(tec)n(hnique)f(in)h (queueing)f(analysis,)i(since)f(the)f(Mark)n(o-)759 1809 y(vian)37 b(prop)r(ert)n(y)f(of)i(PH)e(distributions)i(often)f(allo)n (ws)i(analytical)f(tractabilit)n(y)-6 b(.)759 1900 y(This)27 b(pap)r(er)g(prop)r(oses)g(an)f(algorithm)h(for)g(mapping)e(a)i (general)g(distribution)g Fq(G)759 1991 y Fu(to)j(a)f(PH)g (distribution)g(where)g(the)g(goal)h(is)g(to)f(\014nd)f(a)i(PH)e (distribution)i(whic)n(h)759 2083 y(matc)n(hes)e(the)g(\014rst)g(three) g(momen)n(ts)e(of)j Fq(G)p Fu(.)g(Since)f(e\016ciency)g(of)h(the)f (algorithm)759 2174 y(is)j(of)f(primary)f(imp)r(ortance,)h(w)n(e)g (\014rst)g(de\014ne)f(a)h(particular)h(subset)f(of)g(the)g(PH)759 2265 y(distributions,)23 b(whic)n(h)f(w)n(e)h(refer)g(to)f(as)h(EC)g (distributions.)g(The)f(class)i(of)f(EC)g(dis-)759 2357 y(tributions)i(has)h(v)n(ery)e(few)i(free)f(parameters,)h(whic)n(h)f (narro)n(ws)h(do)n(wn)f(the)f(searc)n(h)759 2448 y(space,)j(making)e (the)g(algorithm)i(e\016cien)n(t)f({)g(In)f(fact)h(w)n(e)h(pro)n(vide)e (a)h(closed-form)759 2539 y(solution)i(for)f(the)g(parameters)f(of)i (the)e(EC)i(distribution.)f(Our)f(solution)i(is)f(gen-)759 2631 y(eral)39 b(in)e(that)g(it)h(applies)g(to)g(an)n(y)f(distribution) g(whose)i(\014rst)e(three)g(momen)n(ts)759 2722 y(can)30 b(b)r(e)f(matc)n(hed)f(b)n(y)h(a)g(PH)h(distribution.)f(Also,)i(our)e (resulting)h(EC)g(distribu-)759 2813 y(tion)g(requires)h(a)f(nearly)g (minimal)f(n)n(um)n(b)r(er)f(of)j(phases,)f(alw)n(a)n(ys)h(within)g (one)f(of)759 2905 y(the)d(minimal)f(n)n(um)n(b)r(er)f(of)j(phases)g (required)f(b)n(y)f(an)n(y)g(acyclic)i(PH)f(distribution.)759 2996 y(Lastly)-6 b(,)26 b(w)n(e)g(discuss)h(n)n(umerical)e(stabilit)n (y)h(of)g(our)g(solution.)523 3297 y Fp(1)112 b(In)m(tro)s(duction)523 3520 y Fo(Motivation)48 b Fw(There)33 b(is)g(a)g(v)n(ery)f(large)g(b)r (o)r(dy)h(of)h(literature)e(on)h(the)h(topic)f(of)g(appro)n(xi-)523 3620 y(mating)21 b(general)e(distributions)i(b)n(y)f(phase-t)n(yp)r(e)g (\(PH\))i(distributions,)e(whose)h(Mark)n(o)n(vian)523 3719 y(prop)r(erties)k(mak)n(e)f(them)i(far)f(more)g(analytically)g (tractable.)f(Muc)n(h)i(of)f(this)h(researc)n(h)e(has)523 3819 y(fo)r(cused)i(on)g(the)g(sp)r(eci\014c)g(problem)g(of)g (\014nding)g(an)g(algorithm)f(whic)n(h)h(maps)f(an)n(y)g(general)523 3919 y(distribution,)g Fn(G)p Fw(,)h(to)f(a)g(PH)g(distribution,)g Fn(P)12 b Fw(,)25 b(where)g Fn(P)37 b Fw(and)25 b Fn(G)g Fw(agree)f(on)h(the)g(\014rst)g(three)523 4018 y(momen)n(ts.)33 b(Throughout)g(this)h(pap)r(er)f(w)n(e)g(sa)n(y)f(that)i Fn(G)g Fw(is)g Fm(wel)t(l-r)l(epr)l(esente)l(d)g Fw(b)n(y)g Fn(P)45 b Fw(if)34 b Fn(P)523 4118 y Fw(and)29 b Fn(G)h Fw(agree)e(on)h(their)g(\014rst)g(three)g(momen)n(ts.)g(W)-7 b(e)30 b(c)n(ho)r(ose)e(to)i(limit)g(our)e(discussion)h(in)523 4217 y(this)22 b(pap)r(er)g(to)g(three-momen)n(t)g(matc)n(hing,)f(b)r (ecause)h(matc)n(hing)g(the)g(\014rst)g(three)g(momen)n(ts)523 4317 y(of)28 b(an)h(input)g(distribution)f(has)g(b)r(een)h(sho)n(wn)f (to)g(b)r(e)h(e\013ectiv)n(e)f(in)h(predicting)f(mean)g(p)r(er-)523 4417 y(formance)h(for)h(v)-5 b(ariet)n(y)30 b(of)g(man)n(y)g(computer)f (system)i(mo)r(dels)f([4,)13 b(5,)h(19,)f(23,)f(27].)30 b(Clearly)-7 b(,)523 4516 y(ho)n(w)n(ev)n(er,)19 b(three)i(momen)n(ts)f (migh)n(t)h(not)g(alw)n(a)n(ys)e(su\016ce)i(for)g(ev)n(ery)e(problem,)i (and)g(w)n(e)f(lea)n(v)n(e)523 4616 y(the)28 b(problem)f(of)h(matc)n (hing)f(more)g(momen)n(ts)g(to)g(future)h(w)n(ork.)648 4720 y(Momen)n(t-matc)n(hing)e(algorithms)g(are)h(ev)-5 b(aluated)27 b(along)g(four)g(di\013eren)n(t)g(measures:)533 4825 y Fl(The)k(n)m(um)m(b)s(er)f(of)h(momen)m(ts)d(matc)m(hed)e Fw({)h(In)g(general)e(matc)n(hing)i(more)f(momen)n(ts)g(is)618 4924 y(more)g(desirable.)p eop %%Page: 2 2 2 1 bop 537 448 a Fl(The)36 b(computational)e(e\016ciency)i(of)g(the)f (algorithm)29 b Fw({)i(It)g(is)g(desirable)f(that)i(the)618 548 y(algorithm)d(ha)n(v)n(e)h(short)g(running)h(time.)g(Ideally)-7 b(,)31 b(one)f(w)n(ould)h(lik)n(e)f(a)h(closed-form)e(so-)618 648 y(lution)e(for)g(the)h(parameters)e(of)i(the)g(matc)n(hing)f(PH)g (distribution.)539 759 y Fl(The)38 b(generalit)m(y)f(of)h(the)g (solution)31 b Fw({)h(Ideally)h(the)g(algorithm)f(should)g(w)n(ork)f (for)i(as)618 858 y(broad)26 b(a)h(class)g(of)g(distributions)h(as)f(p) r(ossible.)528 969 y Fl(The)e(minimalit)m(y)d(of)j(the)g(n)m(um)m(b)s (er)g(of)g(phases)d Fw({)f(It)h(is)g(desirable)f(that)h(the)h(matc)n (hing)618 1069 y(PH)32 b(distribution,)h Fn(P)12 b Fw(,)33 b(ha)n(v)n(e)f(v)n(ery)g(few)h(phases.)f(Recall)h(that)g(the)g(goal)f (is)h(to)g(\014nd)g Fn(P)618 1169 y Fw(whic)n(h)26 b(can)g(replace)f (the)i(input)g(distribution)f Fn(G)h Fw(in)f(some)g(queueing)g(mo)r (del,)g(allo)n(wing)618 1268 y(a)k(Mark)n(o)n(v)g(c)n(hain)g(represen)n (tation)g(of)h(the)h(problem.)f(Since)h(it)f(is)h(desirable)e(that)i (the)618 1368 y(state)25 b(space)g(of)h(this)g(resulting)f(Mark)n(o)n (v)e(c)n(hain)i(b)r(e)h(k)n(ept)g(small,)f(w)n(e)h(w)n(an)n(t)f(to)g(k) n(eep)h(the)618 1468 y(n)n(um)n(b)r(er)h(of)g(phases)g(in)h Fn(P)40 b Fw(lo)n(w.)648 1579 y(This)22 b(pap)r(er)h(prop)r(oses)e(a)h (momen)n(t-matc)n(hing)g(algorithm)g(whic)n(h)h(p)r(erforms)f(v)n(ery)f (w)n(ell)523 1678 y(along)26 b(all)g(four)g(of)h(these)g(measures.)e (Our)h(solution)g(matc)n(hes)g(three)h(momen)n(ts,)f(pro)n(vides)523 1778 y(a)e(closed)g(form)g(represen)n(tation)f(of)i(the)g(parameters)e (of)h(the)h(matc)n(hing)f(PH)h(distribution,)523 1878 y(applies)j(to)h(all)f(distributions)g(whic)n(h)h(can)f(b)r(e)h(w)n (ell-represen)n(ted)e(b)n(y)h(a)g(PH)h(distribution,)523 1977 y(and)e(is)h(nearly)f(minimal)g(in)h(the)g(n)n(um)n(b)r(er)g(of)f (phases)g(required.)648 2088 y(The)d(general)f(approac)n(h)g(in)i (designing)e(momen)n(t-matc)n(hing)h(algorithms)f(in)i(the)f(liter-)523 2188 y(ature)i(is)g(to)f(start)h(b)n(y)g(de\014ning)g(a)g(subset)g Fk(S)32 b Fw(of)26 b(the)h(PH)f(distributions,)g(and)g(then)g(matc)n(h) 523 2287 y(eac)n(h)36 b(input)i(distribution)f Fn(G)g Fw(to)f(a)h(distribution)g(in)g Fk(S)6 b Fw(.)37 b(The)g(reason)e(for)h (limiting)i(the)523 2387 y(solution)c(to)g(a)g(distribution)g(in)g Fk(S)41 b Fw(is)34 b(that)h(this)f(narro)n(ws)e(the)j(searc)n(h)d (space)i(and)g(th)n(us)523 2487 y(impro)n(v)n(es)f(the)i(computational) f(e\016ciency)h(of)g(the)g(algorithm.)f(Observ)n(e)f(that)i Fn(n)p Fw(-phase)523 2586 y(PH)g(distributions)g(ha)n(v)n(e)f Fn(\002)r Fw(\()p Fn(n)1516 2556 y Fv(2)1554 2586 y Fw(\))h(free)g (parameters)e([16])h(\(see)h(Figure)g(1\),)g(while)g Fk(S)41 b Fw(can)523 2686 y(b)r(e)35 b(de\014ned)g(to)f(ha)n(v)n(e)f (far)h(few)n(er)g(free)g(parameters.)f(F)-7 b(or)33 b(all)h (computationally)g(e\016cien)n(t)523 2786 y(algorithms)28 b(in)i(the)h(literature,)e Fk(S)36 b Fw(w)n(as)29 b(c)n(hosen)g(to)g(b) r(e)h(some)g(subset)f(of)h(the)g(acyclic)f(PH)523 2885 y(distributions,)20 b(where)g(an)f(acyclic)h(PH)f(distribution)i(is)e (a)h(PH)g(distribution)g(whose)f(under-)523 2985 y(lying)26 b(con)n(tin)n(uous)f(time)i(Mark)n(o)n(v)d(c)n(hain)i(has)f(no)h (transition)g(from)g(state)g Fn(i)g Fw(to)g(state)g Fn(j)32 b Fw(for)523 3085 y(all)25 b Fn(i)e(>)g(j)5 b Fw(.)25 b(One)h(has)f(to)g(b)r(e)h(careful)f(in)h(ho)n(w)f(one)g(de\014nes)h (the)g(subset)f Fk(S)6 b Fw(,)26 b(ho)n(w)n(ev)n(er.)e(If)i Fk(S)32 b Fw(is)523 3184 y(to)r(o)25 b(small)f(it)i(ma)n(y)e(limit)i (the)f(space)f(of)h(distributions)g(whic)n(h)g(can)f(b)r(e)i(w)n (ell-represen)n(ted.)3370 3154 y Fv(1)523 3284 y Fw(Also,)h(if)g Fk(S)33 b Fw(is)27 b(to)r(o)g(small)f(it)i(ma)n(y)e(exclude)h (solutions)f(with)h(minimal)g(n)n(um)n(b)r(er)g(of)g(phases.)648 3395 y(In)k(this)h(pap)r(er)e(w)n(e)h(de\014ne)h(a)f(subset)g(of)g(the) h(PH)f(distributions,)g(whic)n(h)h(w)n(e)f(call)f(EC)523 3494 y(distributions.)d(EC)f(distributions)g(ha)n(v)n(e)f(only)i(six)f (free)g(parameters)f(whic)n(h)h(allo)n(ws)g(us)g(to)523 3594 y(deriv)n(e)k(a)g(closed-form)g(solution)g(for)g(these)h (parameters)e(in)i(terms)g(of)g(the)g(input)g(distri-)523 3694 y(bution)i Fn(G)p Fw(.)g(The)f(set)h(of)f(EC)g(distributions)h(is) f(general)f(enough,)h(ho)n(w)n(ev)n(er,)f(that)h(for)g(all)523 3793 y(distributions)e Fn(G)h Fw(that)f(can)g(b)r(e)g(w)n(ell-represen) n(ted)f(b)n(y)g(a)h(PH)g(distribution,)g(there)g(exists)523 3893 y(an)f(EC)h(distribution,)f Fn(E)5 b Fw(,)30 b(suc)n(h)g(that)f Fn(G)h Fw(is)g(w)n(ell-represen)n(ted)e(b)n(y)h Fn(E)5 b Fw(.)30 b(F)-7 b(urthermore,)29 b(the)523 3993 y(class)d(of)g(EC)g (distributions)h(is)f(broad)f(enough)h(suc)n(h)h(that)f(for)g(an)n(y)g (distribution)h Fn(G)p Fw(,)g(that)523 4092 y(is)35 b(w)n(ell-represen) n(ted)f(b)n(y)g(an)h Fn(n)p Fw(-phase)g(acyclic)f(PH)h(distribution,)h (there)f(exists)g(an)f(EC)523 4192 y(distribution)24 b Fn(E)29 b Fw(with)c(at)f(most)g Fn(n)11 b Fw(+)g(1)23 b(phases,)g(suc)n(h)h(that)g Fn(G)h Fw(is)f(w)n(ell-represen)n(ted)e(b) n(y)i Fn(E)5 b Fw(.)3368 4162 y Fv(2)p 523 4291 473 4 v 546 4345 a Fj(1)606 4376 y Fu(F)-6 b(or)33 b(example,)e(let)i Fq(G)f Fu(b)r(e)h(a)f(distribution)h(whose)g(\014rst)f(three)g(momen)n (ts)f(are)i(1,)g(2,)g(and)f(12.)606 4468 y(The)k(system)f(of)i (equations)f(for)h(matc)n(hing)e Fq(G)h Fu(to)g(a)g(2-phase)g(Co)n (xian)2760 4436 y Fj(+)2847 4468 y Fu(distribution)g(\(see)606 4559 y(Figure)23 b(2\))g(with)g(three)g(parameters)g(\()p Fq(\025)1772 4567 y Fj(1)1806 4559 y Fu(,)g Fq(\025)1895 4567 y Fj(2)1929 4559 y Fu(,)g Fq(p)p Fu(\))f(results)i(in)e(either)h Fq(\025)2652 4567 y Fj(1)2709 4559 y Fu(or)g Fq(\025)2845 4567 y Fj(2)2902 4559 y Fu(b)r(eing)g(negativ)n(e.)606 4650 y(As)h(another)f(example,)h(it)g(can)g(b)r(e)f(sho)n(wn)h(that)g (the)f(generalized)i(Erlang)g(distribution)f(is)g(not)606 4742 y(general)j(enough)e(to)g(w)n(ell-represen)n(t)h(all)h(the)e (distributions)h(with)f(lo)n(w)i(v)l(ariabilit)n(y)e(\(see)h([17]\).) 546 4801 y Fj(2)606 4833 y Fu(Ideally)-6 b(,)22 b(one)h(w)n(ould)f(lik) n(e)g(to)h(ev)l(aluate)f(the)g(n)n(um)n(b)r(er)e(of)j(phases)g(with)f (resp)r(ect)h(to)f(the)g(minimal)606 4924 y(\(p)r(ossibly-cyclic\))35 b(PH)f(distribution,)h(i.e.,)h(the)e(PH)g(distribution)h(is)g(not)f (restricted)h(to)g(b)r(e)p eop %%Page: 3 3 3 2 bop 854 365 a 17524246 8051672 0 0 47889121 29207101 startTexFig 854 365 a %%BeginDocument: ph2.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: ph2.eps %%Creator: fig2dev Version 3.2 Patchlevel 3d %%CreationDate: Thu Jun 19 09🔞18 2003 %%For: osogami@gs57.sp.cs.cmu.edu (Takayuki Osogami) %%BoundingBox: 0 0 728 444 %%Magnification: 1.0000 %%EndComments /$F2psDict 200 dict def F2psDictbeginF2psDict begin F2psDictbeginF2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save newpath 0 444 moveto 0 0 lineto 728 0 lineto 728 444 lineto closepath clip newpath -39.2 572.8 translate 1 -1 scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def F2psBegin10setmiterlimit0.060000.06000scF2psBegin 10 setmiterlimit 0.06000 0.06000 sc % % Fig objects follow % /Times-Roman ff 270.00 scf sf 3450 6375 m gs 1 -1 sc (1) col0 sh gr /Symbol ff 360.00 scf sf 3225 6300 m gs 1 -1 sc (l) col0 sh gr /Times-Roman ff 270.00 scf sf 5700 6375 m gs 1 -1 sc (2) col0 sh gr /Symbol ff 360.00 scf sf 5475 6300 m gs 1 -1 sc (l) col0 sh gr /Times-Roman ff 270.00 scf sf 7875 6300 m gs 1 -1 sc (3) col0 sh gr /Symbol ff 360.00 scf sf 7650 6225 m gs 1 -1 sc (l) col0 sh gr /Times-Roman ff 270.00 scf sf 10125 6300 m gs 1 -1 sc (4) col0 sh gr /Symbol ff 360.00 scf sf 9900 6225 m gs 1 -1 sc (l) col0 sh gr /Times-Roman ff 360.00 scf sf 10500 6375 m gs 1 -1 sc (p) col0 sh gr /Times-Roman ff 270.00 scf sf 10650 6525 m gs 1 -1 sc (45) col0 sh gr /Times-Roman ff 360.00 scf sf 5325 5025 m gs 1 -1 sc (p) col0 sh gr /Times-Roman ff 270.00 scf sf 5475 5175 m gs 1 -1 sc (13) col0 sh gr /Times-Roman ff 360.00 scf sf 4425 5775 m gs 1 -1 sc (p) col0 sh gr /Times-Roman ff 270.00 scf sf 4575 5925 m gs 1 -1 sc (12) col0 sh gr /Times-Roman ff 360.00 scf sf 6525 5775 m gs 1 -1 sc (p) col0 sh gr /Times-Roman ff 270.00 scf sf 6675 5925 m gs 1 -1 sc (23) col0 sh gr /Times-Roman ff 360.00 scf sf 8700 5775 m gs 1 -1 sc (p) col0 sh gr /Times-Roman ff 270.00 scf sf 8850 5925 m gs 1 -1 sc (34) col0 sh gr /Times-Roman ff 360.00 scf sf 8775 6600 m gs 1 -1 sc (p) col0 sh gr /Times-Roman ff 270.00 scf sf 8925 6750 m gs 1 -1 sc (43) col0 sh gr /Times-Roman ff 360.00 scf sf 6450 6600 m gs 1 -1 sc (p) col0 sh gr /Times-Roman ff 270.00 scf sf 6600 6750 m gs 1 -1 sc (32) col0 sh gr /Times-Roman ff 360.00 scf sf 7725 7350 m gs 1 -1 sc (p) col0 sh gr /Times-Roman ff 270.00 scf sf 7875 7500 m gs 1 -1 sc (42) col0 sh gr /Times-Roman ff 360.00 scf sf 5475 7200 m gs 1 -1 sc (p) col0 sh gr /Times-Roman ff 270.00 scf sf 5625 7350 m gs 1 -1 sc (31) col0 sh gr /Times-Roman ff 360.00 scf sf 4500 6600 m gs 1 -1 sc (p) col0 sh gr /Times-Roman ff 270.00 scf sf 4650 6750 m gs 1 -1 sc (21) col0 sh gr /Times-Roman ff 360.00 scf sf 9150 9225 m gs 1 -1 sc (p) col0 sh gr /Times-Roman ff 270.00 scf sf 9300 9375 m gs 1 -1 sc (15) col0 sh gr /Times-Roman ff 360.00 scf sf 9600 8250 m gs 1 -1 sc (p) col0 sh gr /Times-Roman ff 270.00 scf sf 9750 8400 m gs 1 -1 sc (25) col0 sh gr /Times-Roman ff 360.00 scf sf 10200 7125 m gs 1 -1 sc (p) col0 sh gr /Times-Roman ff 270.00 scf sf 10350 7275 m gs 1 -1 sc (35) col0 sh gr /Times-Roman ff 360.00 scf sf 2250 6375 m gs 1 -1 sc (p) col0 sh gr /Times-Roman ff 270.00 scf sf 2400 6525 m gs 1 -1 sc (01) col0 sh gr /Times-Roman ff 360.00 scf sf 2400 5475 m gs 1 -1 sc (p) col0 sh gr /Times-Roman ff 270.00 scf sf 2550 5625 m gs 1 -1 sc (02) col0 sh gr /Times-Roman ff 360.00 scf sf 2550 4575 m gs 1 -1 sc (p) col0 sh gr /Times-Roman ff 270.00 scf sf 2700 4725 m gs 1 -1 sc (03) col0 sh gr /Times-Roman ff 360.00 scf sf 2625 3750 m gs 1 -1 sc (p) col0 sh gr /Times-Roman ff 270.00 scf sf 2775 3900 m gs 1 -1 sc (04) col0 sh gr /Times-Roman ff 360.00 scf sf 2700 2925 m gs 1 -1 sc (p) col0 sh gr /Times-Roman ff 270.00 scf sf 2850 3075 m gs 1 -1 sc (05) col0 sh gr /Times-Roman ff 360.00 scf sf 6600 8175 m gs 1 -1 sc (p) col0 sh gr /Times-Roman ff 270.00 scf sf 6750 8325 m gs 1 -1 sc (41) col0 sh gr /Times-Roman ff 360.00 scf sf 6375 4350 m gs 1 -1 sc (p) col0 sh gr /Times-Roman ff 270.00 scf sf 6525 4500 m gs 1 -1 sc (14) col0 sh gr /Times-Roman ff 360.00 scf sf 7500 4950 m gs 1 -1 sc (p) col0 sh gr /Times-Roman ff 270.00 scf sf 7650 5100 m gs 1 -1 sc (24) col0 sh gr 15.000 slw % Ellipse n 5625 6075 375 375 0 360 DrawEllipse gs col0 s gr % Ellipse n 7800 6000 375 375 0 360 DrawEllipse gs col0 s gr % Ellipse n 10050 6000 375 375 0 360 DrawEllipse gs col0 s gr % Ellipse n 3375 6075 375 375 0 360 DrawEllipse gs col0 s gr % Polyline 2 slj gs clippath 9746 6136 m 9782 6022 l 9507 5936 l 9719 6065 l 9472 6050 l cp eoclip n 8175 6075 m 8176 6075 l 8179 6073 l 8184 6071 l 8191 6068 l 8202 6064 l 8216 6059 l 8232 6052 l 8252 6045 l 8275 6037 l 8299 6028 l 8327 6019 l 8355 6009 l 8386 6000 l 8418 5991 l 8452 5981 l 8488 5973 l 8525 5964 l 8565 5956 l 8607 5949 l 8652 5943 l 8700 5937 l 8752 5932 l 8807 5928 l 8865 5926 l 8925 5925 l 8981 5926 l 9036 5928 l 9090 5931 l 9140 5935 l 9189 5940 l 9235 5946 l 9278 5953 l 9320 5960 l 9359 5967 l 9398 5975 l 9434 5983 l 9470 5992 l 9504 6001 l 9537 6009 l 9568 6018 l 9598 6027 l 9625 6035 l 9651 6043 l 9673 6050 l 9693 6056 l 9710 6062 l 9724 6066 l 9734 6070 l 9750 6075 l gs col0 s gr gr % arrowhead 0 slj n 9472 6050 m 9719 6065 l 9507 5936 l 9472 6050 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj gs clippath 5247 6136 m 5281 6021 l 5005 5940 l 5219 6066 l 4971 6055 l cp eoclip n 3750 6075 m 3751 6075 l 3753 6073 l 3757 6071 l 3764 6068 l 3773 6064 l 3784 6059 l 3799 6052 l 3816 6045 l 3835 6037 l 3856 6028 l 3880 6019 l 3905 6009 l 3931 6000 l 3960 5991 l 3990 5981 l 4021 5973 l 4055 5964 l 4090 5956 l 4129 5949 l 4170 5943 l 4214 5937 l 4262 5932 l 4313 5928 l 4368 5926 l 4425 5925 l 4479 5926 l 4532 5928 l 4583 5931 l 4633 5935 l 4680 5940 l 4725 5946 l 4769 5953 l 4810 5960 l 4850 5967 l 4888 5975 l 4925 5983 l 4961 5992 l 4996 6001 l 5030 6009 l 5062 6018 l 5092 6027 l 5121 6035 l 5147 6043 l 5170 6050 l 5191 6056 l 5208 6062 l 5223 6066 l 5233 6070 l 5250 6075 l gs col0 s gr gr % arrowhead 0 slj n 4971 6055 m 5219 6066 l 5005 5940 l 4971 6055 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj gs clippath 7417 6136 m 7460 6024 l 7193 5920 l 7395 6063 l 7149 6031 l cp eoclip n 6000 6150 m 6001 6149 l 6003 6148 l 6008 6145 l 6015 6141 l 6024 6135 l 6036 6128 l 6051 6119 l 6069 6108 l 6089 6097 l 6112 6085 l 6136 6072 l 6162 6059 l 6189 6045 l 6218 6032 l 6249 6019 l 6281 6006 l 6314 5994 l 6350 5982 l 6388 5971 l 6429 5960 l 6472 5951 l 6519 5942 l 6568 5935 l 6621 5929 l 6675 5925 l 6730 5923 l 6783 5923 l 6835 5925 l 6884 5929 l 6930 5934 l 6974 5940 l 7016 5947 l 7055 5955 l 7093 5963 l 7130 5972 l 7165 5982 l 7198 5992 l 7230 6002 l 7261 6013 l 7290 6023 l 7317 6032 l 7341 6042 l 7362 6050 l 7381 6057 l 7396 6063 l 7407 6068 l 7425 6075 l gs col0 s gr gr % arrowhead 0 slj n 7149 6031 m 7395 6063 l 7193 5920 l 7149 6031 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj gs clippath 3758 6163 m 3713 6274 l 3979 6383 l 3780 6237 l 4024 6272 l cp eoclip n 5250 6225 m 5249 6225 l 5246 6227 l 5241 6229 l 5233 6232 l 5222 6236 l 5208 6241 l 5190 6248 l 5169 6255 l 5146 6263 l 5120 6272 l 5092 6281 l 5062 6291 l 5031 6300 l 4998 6309 l 4964 6319 l 4928 6327 l 4890 6336 l 4850 6344 l 4808 6351 l 4764 6357 l 4717 6363 l 4666 6368 l 4613 6372 l 4557 6374 l 4500 6375 l 4443 6374 l 4387 6372 l 4334 6368 l 4283 6363 l 4236 6357 l 4192 6351 l 4150 6344 l 4110 6336 l 4072 6327 l 4036 6319 l 4002 6309 l 3969 6300 l 3937 6291 l 3908 6281 l 3880 6272 l 3854 6263 l 3831 6255 l 3810 6248 l 3792 6241 l 3778 6236 l 3767 6232 l 3750 6225 l gs col0 s gr gr % arrowhead 0 slj n 4024 6272 m 3780 6237 l 3979 6383 l 4024 6272 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj gs clippath 6005 6163 m 5965 6276 l 6237 6372 l 6031 6236 l 6277 6259 l cp eoclip n 7500 6150 m 7499 6151 l 7497 6152 l 7493 6155 l 7486 6159 l 7477 6165 l 7466 6172 l 7451 6181 l 7434 6192 l 7415 6203 l 7394 6215 l 7370 6228 l 7345 6241 l 7319 6255 l 7290 6268 l 7260 6281 l 7229 6294 l 7195 6306 l 7160 6318 l 7121 6329 l 7080 6340 l 7036 6349 l 6988 6358 l 6937 6365 l 6882 6371 l 6825 6375 l 6771 6377 l 6718 6377 l 6667 6375 l 6617 6372 l 6570 6368 l 6525 6363 l 6481 6357 l 6440 6350 l 6400 6342 l 6362 6334 l 6325 6326 l 6289 6317 l 6254 6307 l 6220 6298 l 6188 6288 l 6158 6279 l 6129 6270 l 6103 6261 l 6080 6253 l 6059 6246 l 6042 6240 l 6027 6235 l 6017 6231 l 6000 6225 l gs col0 s gr gr % arrowhead 0 slj n 6277 6259 m 6031 6236 l 6237 6372 l 6277 6259 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj gs clippath 3784 6248 m 3695 6328 l 3887 6542 l 3772 6324 l 3977 6462 l cp eoclip n 7500 6225 m 7499 6226 l 7498 6227 l 7496 6231 l 7492 6236 l 7487 6243 l 7480 6252 l 7470 6265 l 7459 6280 l 7445 6298 l 7428 6319 l 7410 6343 l 7389 6369 l 7365 6399 l 7339 6430 l 7311 6464 l 7281 6500 l 7249 6538 l 7215 6577 l 7180 6618 l 7143 6659 l 7104 6701 l 7064 6743 l 7023 6785 l 6980 6828 l 6937 6870 l 6892 6911 l 6845 6952 l 6798 6992 l 6749 7031 l 6699 7069 l 6647 7107 l 6594 7142 l 6539 7177 l 6482 7210 l 6422 7241 l 6361 7271 l 6297 7299 l 6231 7324 l 6162 7348 l 6090 7369 l 6016 7387 l 5940 7402 l 5861 7413 l 5781 7421 l 5700 7425 l 5618 7424 l 5538 7420 l 5458 7411 l 5380 7400 l 5303 7384 l 5229 7366 l 5158 7346 l 5088 7323 l 5021 7298 l 4956 7271 l 4893 7242 l 4831 7211 l 4772 7179 l 4714 7145 l 4658 7110 l 4603 7074 l 4549 7037 l 4497 6999 l 4445 6960 l 4395 6920 l 4346 6880 l 4298 6840 l 4252 6799 l 4206 6759 l 4162 6718 l 4120 6679 l 4079 6640 l 4040 6602 l 4004 6566 l 3969 6531 l 3937 6498 l 3907 6468 l 3880 6439 l 3855 6414 l 3833 6391 l 3815 6370 l 3798 6353 l 3785 6338 l 3774 6326 l 3765 6317 l 3759 6310 l 3750 6300 l gs col0 s gr gr % arrowhead 0 slj n 3977 6462 m 3772 6324 l 3887 6542 l 3977 6462 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj gs clippath 8104 6163 m 8066 6277 l 8339 6367 l 8131 6235 l 8377 6253 l cp eoclip n 9675 6150 m 9674 6151 l 9672 6152 l 9668 6155 l 9662 6159 l 9654 6165 l 9643 6172 l 9629 6181 l 9613 6192 l 9595 6203 l 9574 6215 l 9552 6228 l 9527 6241 l 9501 6255 l 9474 6268 l 9445 6281 l 9413 6294 l 9380 6306 l 9344 6318 l 9306 6329 l 9264 6340 l 9219 6349 l 9170 6358 l 9116 6365 l 9060 6371 l 9000 6375 l 8947 6377 l 8895 6377 l 8843 6376 l 8793 6373 l 8745 6370 l 8699 6365 l 8655 6360 l 8612 6354 l 8570 6347 l 8530 6340 l 8491 6332 l 8453 6324 l 8417 6315 l 8381 6307 l 8346 6298 l 8313 6289 l 8281 6280 l 8251 6271 l 8223 6263 l 8198 6256 l 8175 6249 l 8155 6243 l 8139 6237 l 8126 6233 l 8115 6230 l 8100 6225 l gs col0 s gr gr % arrowhead 0 slj n 8377 6253 m 8131 6235 l 8339 6367 l 8377 6253 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj gs clippath 1815 6135 m 1815 6015 l 1528 6015 l 1768 6075 l 1528 6135 l cp eoclip n 675 6075 m 1800 6075 l gs col0 s gr gr % arrowhead 0 slj n 1528 6135 m 1768 6075 l 1528 6015 l 1528 6135 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj [68] 0 sd gs clippath 3015 6135 m 3015 6015 l 2728 6015 l 2968 6075 l 2728 6135 l cp eoclip n 1800 6075 m 3000 6075 l gs col0 s gr gr [] 0 sd % arrowhead 0 slj n 2728 6135 m 2968 6075 l 2728 6015 l 2728 6135 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj [90] 0 sd gs clippath 11565 6135 m 11565 6015 l 11278 6015 l 11518 6075 l 11278 6135 l cp eoclip n 10425 6075 m 11550 6075 l gs col0 s gr gr [] 0 sd % arrowhead 0 slj n 11278 6135 m 11518 6075 l 11278 6015 l 11278 6135 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj gs clippath 12765 6135 m 12765 6015 l 12478 6015 l 12718 6075 l 12478 6135 l cp eoclip n 11625 6075 m 12750 6075 l gs col0 s gr gr % arrowhead 0 slj n 12478 6135 m 12718 6075 l 12478 6015 l 12478 6135 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj [90] 0 sd gs clippath 11686 6151 m 11571 6119 l 11494 6396 l 11616 6181 l 11610 6428 l cp eoclip n 5850 6450 m 5851 6451 l 5852 6452 l 5854 6454 l 5858 6457 l 5863 6463 l 5870 6470 l 5880 6479 l 5892 6490 l 5907 6504 l 5924 6521 l 5944 6540 l 5967 6561 l 5993 6586 l 6022 6613 l 6055 6643 l 6090 6676 l 6128 6712 l 6170 6750 l 6214 6790 l 6260 6832 l 6310 6877 l 6362 6924 l 6416 6972 l 6473 7022 l 6531 7073 l 6592 7125 l 6654 7178 l 6717 7232 l 6782 7287 l 6849 7342 l 6916 7397 l 6985 7452 l 7055 7507 l 7125 7562 l 7197 7616 l 7269 7670 l 7341 7724 l 7415 7776 l 7489 7828 l 7564 7879 l 7640 7928 l 7716 7977 l 7793 8024 l 7871 8070 l 7950 8115 l 8029 8158 l 8110 8200 l 8191 8240 l 8274 8278 l 8358 8315 l 8442 8349 l 8528 8381 l 8615 8412 l 8703 8439 l 8792 8464 l 8882 8487 l 8973 8506 l 9065 8523 l 9157 8536 l 9250 8545 l 9342 8551 l 9434 8553 l 9525 8550 l 9624 8542 l 9720 8529 l 9813 8512 l 9903 8490 l 9990 8464 l 10073 8435 l 10153 8401 l 10230 8365 l 10303 8326 l 10373 8284 l 10439 8240 l 10503 8194 l 10563 8145 l 10621 8094 l 10676 8041 l 10729 7987 l 10779 7931 l 10828 7874 l 10874 7815 l 10918 7754 l 10961 7693 l 11002 7630 l 11041 7566 l 11079 7502 l 11115 7437 l 11150 7371 l 11184 7305 l 11216 7238 l 11248 7172 l 11278 7106 l 11306 7040 l 11334 6975 l 11360 6910 l 11386 6847 l 11410 6786 l 11432 6726 l 11454 6668 l 11474 6612 l 11493 6559 l 11510 6509 l 11526 6461 l 11541 6417 l 11554 6376 l 11567 6339 l 11577 6306 l 11587 6275 l 11595 6249 l 11602 6226 l 11608 6207 l 11613 6191 l 11617 6178 l 11620 6168 l 11622 6161 l 11625 6150 l gs col0 s gr gr [] 0 sd % arrowhead 0 slj n 11610 6428 m 11616 6181 l 11494 6396 l 11610 6428 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj gs clippath 9715 5976 m 9804 5896 l 9612 5682 l 9728 5901 l 9522 5762 l cp eoclip n 6000 6075 m 6000 6074 l 6002 6072 l 6004 6068 l 6007 6063 l 6012 6054 l 6019 6043 l 6027 6028 l 6038 6011 l 6051 5989 l 6066 5965 l 6083 5937 l 6102 5906 l 6124 5871 l 6148 5834 l 6174 5794 l 6201 5751 l 6231 5707 l 6262 5661 l 6295 5613 l 6330 5565 l 6366 5515 l 6403 5466 l 6442 5416 l 6482 5366 l 6523 5316 l 6565 5267 l 6609 5219 l 6653 5172 l 6700 5125 l 6748 5080 l 6797 5036 l 6848 4993 l 6901 4952 l 6956 4913 l 7014 4876 l 7073 4840 l 7135 4807 l 7200 4776 l 7268 4747 l 7338 4722 l 7411 4700 l 7486 4681 l 7564 4666 l 7644 4656 l 7725 4650 l 7803 4649 l 7881 4653 l 7958 4660 l 8034 4672 l 8109 4687 l 8181 4705 l 8252 4726 l 8321 4750 l 8388 4776 l 8452 4804 l 8516 4834 l 8577 4867 l 8637 4901 l 8695 4936 l 8752 4973 l 8808 5012 l 8863 5052 l 8916 5092 l 8969 5134 l 9020 5177 l 9071 5220 l 9120 5264 l 9168 5309 l 9216 5353 l 9262 5398 l 9306 5442 l 9350 5485 l 9391 5528 l 9431 5570 l 9469 5610 l 9505 5649 l 9539 5685 l 9570 5720 l 9599 5752 l 9626 5782 l 9649 5808 l 9670 5832 l 9688 5853 l 9704 5871 l 9717 5886 l 9727 5898 l 9735 5908 l 9741 5915 l 9750 5925 l gs col0 s gr gr % arrowhead 0 slj n 9522 5762 m 9728 5901 l 9612 5682 l 9522 5762 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj gs clippath 7387 5974 m 7481 5899 l 7301 5675 l 7405 5900 l 7208 5750 l cp eoclip n 3750 6000 m 3751 5999 l 3752 5997 l 3754 5994 l 3757 5988 l 3762 5980 l 3769 5969 l 3778 5955 l 3789 5938 l 3802 5918 l 3817 5894 l 3835 5868 l 3854 5837 l 3876 5804 l 3901 5769 l 3927 5730 l 3955 5690 l 3985 5647 l 4017 5603 l 4051 5558 l 4086 5511 l 4122 5464 l 4160 5416 l 4199 5369 l 4239 5321 l 4281 5274 l 4324 5227 l 4368 5181 l 4413 5136 l 4459 5092 l 4507 5049 l 4557 5007 l 4608 4967 l 4661 4928 l 4716 4891 l 4773 4856 l 4832 4822 l 4894 4791 l 4958 4762 l 5025 4736 l 5094 4713 l 5166 4692 l 5241 4676 l 5317 4663 l 5395 4654 l 5475 4650 l 5552 4651 l 5628 4656 l 5703 4665 l 5777 4677 l 5850 4693 l 5920 4712 l 5989 4734 l 6056 4758 l 6120 4784 l 6183 4813 l 6244 4844 l 6304 4876 l 6361 4910 l 6418 4946 l 6473 4983 l 6526 5021 l 6579 5061 l 6630 5102 l 6680 5143 l 6730 5186 l 6778 5229 l 6825 5272 l 6871 5316 l 6916 5360 l 6960 5404 l 7003 5448 l 7044 5491 l 7084 5533 l 7122 5574 l 7158 5614 l 7192 5652 l 7225 5689 l 7254 5723 l 7282 5754 l 7307 5783 l 7329 5810 l 7349 5833 l 7367 5854 l 7381 5872 l 7394 5886 l 7403 5898 l 7411 5908 l 7417 5915 l 7425 5925 l gs col0 s gr gr % arrowhead 0 slj n 7208 5750 m 7405 5900 l 7301 5675 l 7208 5750 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj gs clippath 5887 6325 m 5793 6400 l 5973 6624 l 5870 6400 l 6066 6549 l cp eoclip n 9750 6150 m 9750 6151 l 9748 6152 l 9746 6156 l 9743 6161 l 9738 6168 l 9731 6178 l 9722 6191 l 9712 6207 l 9699 6225 l 9683 6247 l 9666 6272 l 9646 6300 l 9624 6331 l 9599 6365 l 9573 6401 l 9544 6440 l 9514 6480 l 9481 6523 l 9447 6567 l 9411 6612 l 9374 6658 l 9335 6705 l 9295 6752 l 9254 6800 l 9212 6847 l 9168 6894 l 9123 6941 l 9078 6987 l 9030 7032 l 8982 7077 l 8932 7121 l 8881 7163 l 8829 7204 l 8774 7244 l 8718 7283 l 8660 7320 l 8600 7355 l 8538 7389 l 8474 7421 l 8407 7450 l 8337 7477 l 8265 7502 l 8191 7524 l 8114 7542 l 8036 7557 l 7956 7568 l 7875 7575 l 7794 7577 l 7713 7576 l 7633 7570 l 7555 7560 l 7479 7547 l 7405 7531 l 7333 7512 l 7263 7490 l 7195 7466 l 7129 7440 l 7066 7412 l 7004 7382 l 6944 7350 l 6885 7316 l 6828 7281 l 6773 7245 l 6718 7208 l 6665 7169 l 6613 7129 l 6563 7089 l 6513 7047 l 6464 7006 l 6417 6963 l 6370 6921 l 6325 6879 l 6281 6837 l 6239 6795 l 6199 6754 l 6160 6715 l 6122 6676 l 6087 6639 l 6055 6604 l 6024 6571 l 5996 6540 l 5970 6512 l 5947 6487 l 5927 6464 l 5910 6444 l 5895 6427 l 5882 6412 l 5872 6401 l 5864 6392 l 5858 6385 l 5850 6375 l gs col0 s gr gr % arrowhead 0 slj n 6066 6549 m 5870 6400 l 5973 6624 l 6066 6549 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj [90] 0 sd gs clippath 11686 6070 m 11568 6049 l 11517 6332 l 11619 6107 l 11635 6353 l cp eoclip n 3675 6225 m 3676 6226 l 3678 6228 l 3682 6231 l 3686 6236 l 3693 6242 l 3701 6250 l 3712 6260 l 3724 6273 l 3740 6287 l 3758 6305 l 3778 6325 l 3802 6347 l 3829 6373 l 3858 6401 l 3891 6432 l 3927 6466 l 3966 6503 l 4008 6542 l 4054 6585 l 4102 6630 l 4154 6678 l 4208 6728 l 4265 6780 l 4325 6835 l 4387 6892 l 4452 6951 l 4519 7012 l 4589 7074 l 4660 7138 l 4734 7203 l 4809 7270 l 4886 7337 l 4965 7405 l 5045 7474 l 5126 7543 l 5209 7613 l 5292 7683 l 5377 7753 l 5462 7823 l 5549 7893 l 5636 7962 l 5724 8031 l 5812 8100 l 5901 8168 l 5991 8235 l 6081 8301 l 6172 8367 l 6263 8432 l 6355 8495 l 6448 8558 l 6541 8620 l 6635 8680 l 6729 8739 l 6824 8796 l 6919 8852 l 7016 8907 l 7113 8960 l 7210 9012 l 7309 9061 l 7408 9109 l 7508 9155 l 7609 9199 l 7711 9241 l 7814 9281 l 7917 9318 l 8022 9353 l 8127 9385 l 8232 9414 l 8338 9440 l 8444 9463 l 8551 9483 l 8657 9499 l 8763 9512 l 8868 9520 l 8972 9525 l 9075 9525 l 9194 9520 l 9310 9508 l 9423 9492 l 9531 9470 l 9636 9443 l 9737 9411 l 9833 9375 l 9926 9335 l 10014 9292 l 10099 9245 l 10179 9195 l 10256 9141 l 10329 9085 l 10399 9026 l 10465 8965 l 10529 8901 l 10589 8835 l 10646 8767 l 10701 8696 l 10753 8624 l 10803 8550 l 10850 8475 l 10896 8398 l 10939 8319 l 10981 8240 l 11021 8159 l 11059 8077 l 11095 7994 l 11130 7910 l 11163 7825 l 11195 7741 l 11225 7655 l 11255 7570 l 11283 7485 l 11309 7401 l 11335 7317 l 11359 7234 l 11382 7152 l 11404 7071 l 11425 6993 l 11445 6916 l 11463 6841 l 11480 6769 l 11497 6700 l 11512 6634 l 11526 6571 l 11539 6511 l 11551 6456 l 11562 6404 l 11572 6356 l 11581 6312 l 11588 6272 l 11595 6236 l 11601 6204 l 11607 6177 l 11611 6153 l 11615 6133 l 11618 6117 l 11620 6103 l 11622 6093 l 11623 6086 l 11625 6075 l gs col0 s gr gr [] 0 sd % arrowhead 0 slj n 11635 6353 m 11619 6107 l 11517 6332 l 11635 6353 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj gs clippath 3639 6327 m 3543 6399 l 3715 6629 l 3619 6401 l 3811 6557 l cp eoclip n 9750 6225 m 9750 6226 l 9749 6227 l 9747 6230 l 9744 6234 l 9740 6241 l 9735 6249 l 9727 6260 l 9718 6274 l 9707 6291 l 9694 6311 l 9679 6334 l 9662 6360 l 9642 6390 l 9620 6422 l 9596 6458 l 9569 6497 l 9541 6539 l 9510 6583 l 9476 6630 l 9441 6680 l 9404 6732 l 9365 6785 l 9324 6841 l 9282 6897 l 9238 6955 l 9192 7014 l 9145 7074 l 9097 7135 l 9047 7195 l 8996 7256 l 8944 7317 l 8890 7378 l 8836 7438 l 8780 7497 l 8723 7556 l 8665 7615 l 8606 7672 l 8545 7728 l 8483 7784 l 8420 7838 l 8355 7890 l 8288 7942 l 8219 7992 l 8149 8040 l 8077 8086 l 8003 8131 l 7926 8174 l 7847 8215 l 7766 8254 l 7682 8290 l 7596 8324 l 7507 8355 l 7416 8383 l 7322 8408 l 7226 8429 l 7128 8447 l 7029 8461 l 6927 8470 l 6825 8475 l 6726 8475 l 6627 8471 l 6529 8463 l 6431 8451 l 6336 8435 l 6242 8416 l 6150 8394 l 6059 8369 l 5971 8341 l 5885 8311 l 5801 8278 l 5719 8243 l 5638 8206 l 5560 8168 l 5483 8127 l 5408 8084 l 5335 8040 l 5263 7995 l 5192 7948 l 5123 7900 l 5055 7850 l 4988 7799 l 4922 7748 l 4858 7695 l 4794 7641 l 4731 7587 l 4670 7532 l 4609 7476 l 4550 7420 l 4492 7364 l 4434 7308 l 4378 7252 l 4324 7196 l 4270 7140 l 4218 7085 l 4167 7031 l 4119 6978 l 4071 6926 l 4026 6876 l 3983 6828 l 3942 6781 l 3903 6736 l 3866 6694 l 3832 6654 l 3800 6616 l 3770 6582 l 3743 6550 l 3719 6521 l 3697 6494 l 3678 6471 l 3661 6451 l 3647 6433 l 3635 6418 l 3625 6406 l 3617 6396 l 3611 6389 l 3606 6383 l 3600 6375 l gs col0 s gr gr % arrowhead 0 slj n 3811 6557 m 3619 6401 l 3715 6629 l 3811 6557 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj [90] 0 sd gs clippath 7476 5982 m 7548 5886 l 7318 5714 l 7474 5906 l 7246 5810 l cp eoclip n 1650 6075 m 1650 6074 l 1651 6072 l 1651 6069 l 1652 6063 l 1654 6055 l 1657 6045 l 1660 6031 l 1663 6013 l 1668 5993 l 1674 5968 l 1681 5940 l 1688 5908 l 1697 5873 l 1707 5834 l 1718 5791 l 1731 5746 l 1744 5697 l 1759 5646 l 1775 5592 l 1792 5536 l 1810 5478 l 1830 5418 l 1851 5357 l 1873 5296 l 1897 5233 l 1922 5170 l 1948 5107 l 1976 5044 l 2005 4982 l 2036 4919 l 2069 4858 l 2104 4797 l 2140 4738 l 2178 4679 l 2219 4622 l 2262 4566 l 2307 4511 l 2356 4458 l 2407 4407 l 2461 4358 l 2518 4311 l 2579 4266 l 2644 4223 l 2713 4183 l 2785 4145 l 2862 4110 l 2944 4079 l 3030 4051 l 3121 4027 l 3216 4007 l 3315 3991 l 3418 3980 l 3525 3975 l 3615 3975 l 3705 3978 l 3797 3984 l 3890 3994 l 3982 4007 l 4074 4022 l 4167 4040 l 4258 4061 l 4349 4084 l 4439 4109 l 4529 4136 l 4617 4165 l 4705 4196 l 4792 4228 l 4878 4263 l 4963 4298 l 5047 4335 l 5131 4373 l 5214 4413 l 5296 4454 l 5378 4496 l 5459 4539 l 5539 4582 l 5619 4627 l 5698 4673 l 5777 4719 l 5855 4766 l 5933 4814 l 6009 4862 l 6086 4911 l 6161 4960 l 6236 5009 l 6309 5058 l 6382 5107 l 6454 5156 l 6524 5204 l 6593 5252 l 6660 5300 l 6725 5346 l 6789 5392 l 6851 5437 l 6910 5480 l 6967 5522 l 7022 5562 l 7073 5600 l 7122 5637 l 7169 5671 l 7212 5704 l 7252 5734 l 7288 5762 l 7322 5787 l 7352 5810 l 7379 5831 l 7403 5850 l 7424 5866 l 7442 5880 l 7457 5891 l 7469 5901 l 7479 5908 l 7486 5914 l 7492 5919 l 7500 5925 l gs col0 s gr gr [] 0 sd % arrowhead 0 slj n 7246 5810 m 7474 5906 l 7318 5714 l 7246 5810 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj [90] 0 sd gs clippath 5299 5906 m 5374 5812 l 5150 5633 l 5300 5830 l 5075 5726 l cp eoclip n 1725 6075 m 1725 6074 l 1726 6072 l 1728 6069 l 1731 6063 l 1736 6056 l 1741 6045 l 1749 6032 l 1758 6015 l 1769 5996 l 1782 5973 l 1797 5947 l 1814 5919 l 1833 5887 l 1853 5853 l 1876 5817 l 1900 5779 l 1926 5740 l 1953 5699 l 1982 5657 l 2012 5614 l 2044 5571 l 2077 5528 l 2111 5485 l 2147 5441 l 2184 5399 l 2222 5357 l 2262 5316 l 2304 5275 l 2347 5236 l 2392 5198 l 2440 5161 l 2489 5126 l 2541 5091 l 2596 5059 l 2653 5028 l 2714 5000 l 2778 4973 l 2845 4949 l 2915 4927 l 2988 4909 l 3065 4894 l 3144 4882 l 3225 4875 l 3300 4872 l 3376 4873 l 3451 4877 l 3526 4884 l 3599 4894 l 3671 4907 l 3742 4922 l 3811 4939 l 3879 4958 l 3944 4979 l 4009 5002 l 4072 5027 l 4133 5052 l 4194 5080 l 4253 5108 l 4311 5138 l 4369 5168 l 4425 5200 l 4480 5232 l 4535 5265 l 4589 5299 l 4641 5333 l 4693 5368 l 4744 5402 l 4793 5437 l 4841 5471 l 4888 5505 l 4933 5539 l 4977 5571 l 5018 5603 l 5057 5633 l 5094 5662 l 5128 5689 l 5160 5714 l 5189 5737 l 5214 5758 l 5237 5777 l 5257 5793 l 5274 5808 l 5288 5819 l 5300 5829 l 5309 5836 l 5315 5842 l 5325 5850 l gs col0 s gr gr [] 0 sd % arrowhead 0 slj n 5075 5726 m 5300 5830 l 5150 5633 l 5075 5726 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj [90] 0 sd gs clippath 11526 6057 m 11597 5960 l 11365 5791 l 11524 5981 l 11294 5888 l cp eoclip n 1575 6000 m 1575 5999 l 1575 5998 l 1575 5995 l 1575 5991 l 1576 5984 l 1576 5975 l 1577 5964 l 1577 5949 l 1578 5931 l 1579 5910 l 1581 5886 l 1583 5858 l 1585 5826 l 1587 5790 l 1590 5750 l 1593 5707 l 1597 5659 l 1601 5608 l 1606 5553 l 1611 5495 l 1617 5433 l 1624 5368 l 1631 5300 l 1639 5229 l 1648 5156 l 1657 5080 l 1668 5001 l 1679 4921 l 1691 4840 l 1705 4756 l 1719 4672 l 1735 4587 l 1751 4500 l 1769 4414 l 1788 4327 l 1809 4239 l 1831 4152 l 1854 4065 l 1879 3979 l 1905 3893 l 1933 3808 l 1963 3723 l 1995 3640 l 2029 3557 l 2065 3476 l 2103 3396 l 2144 3318 l 2187 3241 l 2233 3166 l 2281 3092 l 2332 3020 l 2387 2950 l 2444 2882 l 2505 2816 l 2570 2753 l 2638 2692 l 2710 2633 l 2786 2577 l 2867 2523 l 2952 2473 l 3041 2425 l 3135 2381 l 3234 2341 l 3338 2304 l 3447 2272 l 3560 2243 l 3679 2219 l 3803 2200 l 3931 2186 l 4063 2178 l 4200 2175 l 4312 2177 l 4426 2182 l 4541 2191 l 4658 2204 l 4776 2219 l 4894 2238 l 5013 2260 l 5132 2284 l 5252 2311 l 5371 2341 l 5490 2373 l 5609 2407 l 5727 2444 l 5846 2483 l 5963 2523 l 6081 2566 l 6197 2610 l 6313 2656 l 6429 2704 l 6544 2753 l 6659 2804 l 6773 2856 l 6886 2909 l 6999 2964 l 7112 3020 l 7224 3077 l 7335 3136 l 7447 3196 l 7558 3256 l 7668 3318 l 7778 3381 l 7888 3444 l 7997 3509 l 8106 3574 l 8214 3640 l 8322 3706 l 8430 3774 l 8537 3842 l 8643 3910 l 8750 3979 l 8855 4048 l 8960 4117 l 9064 4187 l 9167 4257 l 9270 4327 l 9371 4396 l 9471 4466 l 9571 4535 l 9669 4604 l 9765 4672 l 9860 4740 l 9954 4807 l 10046 4873 l 10136 4938 l 10224 5001 l 10310 5064 l 10393 5125 l 10475 5185 l 10553 5244 l 10630 5300 l 10703 5355 l 10774 5408 l 10841 5458 l 10906 5507 l 10968 5553 l 11026 5597 l 11081 5639 l 11133 5679 l 11181 5716 l 11227 5750 l 11268 5782 l 11307 5812 l 11342 5839 l 11374 5864 l 11403 5886 l 11429 5906 l 11452 5923 l 11472 5939 l 11489 5952 l 11503 5964 l 11516 5973 l 11526 5981 l 11533 5987 l 11539 5992 l 11550 6000 l gs col0 s gr gr [] 0 sd % arrowhead 0 slj n 11294 5888 m 11524 5981 l 11365 5791 l 11294 5888 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj [90] 0 sd gs clippath 9648 6055 m 9724 5963 l 9504 5779 l 9650 5979 l 9427 5871 l cp eoclip n 1575 6150 m 1575 6149 l 1575 6148 l 1576 6145 l 1577 6140 l 1578 6133 l 1580 6123 l 1582 6111 l 1585 6096 l 1588 6077 l 1592 6055 l 1597 6029 l 1603 6000 l 1610 5966 l 1617 5929 l 1625 5888 l 1635 5843 l 1645 5795 l 1657 5742 l 1669 5687 l 1682 5628 l 1697 5566 l 1713 5502 l 1729 5434 l 1747 5365 l 1766 5293 l 1786 5219 l 1807 5144 l 1830 5068 l 1853 4990 l 1878 4912 l 1904 4833 l 1931 4753 l 1960 4674 l 1990 4594 l 2021 4515 l 2054 4436 l 2088 4358 l 2124 4280 l 2162 4203 l 2201 4128 l 2242 4053 l 2285 3980 l 2330 3908 l 2378 3837 l 2427 3768 l 2480 3701 l 2534 3635 l 2592 3572 l 2652 3510 l 2716 3450 l 2783 3393 l 2853 3338 l 2927 3285 l 3004 3235 l 3086 3188 l 3171 3144 l 3261 3103 l 3354 3066 l 3452 3032 l 3554 3003 l 3661 2977 l 3771 2957 l 3886 2941 l 4004 2930 l 4125 2925 l 4230 2925 l 4336 2929 l 4443 2937 l 4552 2949 l 4660 2964 l 4769 2982 l 4877 3003 l 4986 3028 l 5094 3055 l 5201 3085 l 5308 3117 l 5414 3152 l 5519 3188 l 5623 3227 l 5727 3268 l 5830 3311 l 5931 3356 l 6033 3402 l 6133 3450 l 6233 3499 l 6331 3550 l 6430 3603 l 6527 3656 l 6624 3711 l 6720 3767 l 6816 3825 l 6911 3883 l 7006 3942 l 7100 4003 l 7193 4064 l 7286 4126 l 7379 4189 l 7470 4252 l 7562 4316 l 7652 4380 l 7742 4445 l 7831 4510 l 7919 4575 l 8006 4641 l 8092 4706 l 8178 4771 l 8261 4835 l 8344 4900 l 8425 4963 l 8504 5026 l 8582 5088 l 8658 5149 l 8732 5208 l 8803 5266 l 8873 5323 l 8940 5378 l 9004 5431 l 9066 5482 l 9125 5531 l 9181 5578 l 9234 5622 l 9284 5665 l 9331 5704 l 9374 5741 l 9415 5775 l 9452 5807 l 9486 5836 l 9516 5862 l 9544 5886 l 9568 5907 l 9590 5926 l 9608 5942 l 9624 5956 l 9637 5967 l 9648 5976 l 9657 5984 l 9663 5990 l 9675 6000 l gs col0 s gr gr [] 0 sd % arrowhead 0 slj n 9427 5871 m 9650 5979 l 9504 5779 l 9427 5871 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj gs clippath 9714 5825 m 9805 5746 l 9616 5530 l 9729 5751 l 9525 5609 l cp eoclip n 3750 6000 m 3750 5999 l 3751 5998 l 3753 5995 l 3756 5991 l 3761 5985 l 3767 5976 l 3774 5965 l 3784 5952 l 3796 5935 l 3809 5916 l 3826 5894 l 3844 5868 l 3865 5840 l 3888 5808 l 3914 5774 l 3942 5736 l 3972 5696 l 4005 5654 l 4040 5609 l 4076 5562 l 4115 5513 l 4156 5462 l 4198 5410 l 4243 5356 l 4288 5302 l 4336 5247 l 4384 5191 l 4434 5135 l 4486 5079 l 4538 5023 l 4592 4967 l 4647 4911 l 4704 4856 l 4761 4802 l 4820 4748 l 4880 4695 l 4942 4643 l 5005 4592 l 5069 4542 l 5135 4494 l 5203 4446 l 5273 4400 l 5345 4356 l 5419 4313 l 5495 4272 l 5574 4233 l 5655 4196 l 5738 4161 l 5825 4128 l 5913 4098 l 6005 4070 l 6099 4045 l 6195 4024 l 6294 4006 l 6395 3991 l 6497 3981 l 6600 3975 l 6700 3973 l 6800 3976 l 6899 3982 l 6997 3992 l 7093 4005 l 7187 4021 l 7280 4041 l 7370 4063 l 7459 4087 l 7545 4114 l 7629 4143 l 7711 4175 l 7792 4208 l 7870 4243 l 7947 4279 l 8022 4317 l 8095 4357 l 8167 4398 l 8237 4441 l 8306 4484 l 8374 4529 l 8441 4575 l 8506 4621 l 8571 4669 l 8634 4717 l 8696 4766 l 8757 4816 l 8818 4866 l 8877 4916 l 8934 4966 l 8991 5017 l 9046 5067 l 9100 5116 l 9153 5165 l 9203 5214 l 9252 5261 l 9300 5307 l 9345 5352 l 9388 5395 l 9429 5436 l 9468 5475 l 9504 5513 l 9537 5547 l 9569 5580 l 9597 5610 l 9623 5637 l 9646 5662 l 9666 5684 l 9684 5703 l 9700 5720 l 9713 5734 l 9723 5745 l 9732 5755 l 9738 5762 l 9743 5767 l 9750 5775 l gs col0 s gr gr % arrowhead 0 slj n 9525 5609 m 9729 5751 l 9616 5530 l 9525 5609 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj [90] 0 sd gs clippath 11685 6088 m 11578 6034 l 11450 6291 l 11611 6104 l 11557 6345 l cp eoclip n 8100 6225 m 8101 6226 l 8102 6227 l 8105 6230 l 8109 6235 l 8115 6241 l 8124 6250 l 8135 6262 l 8148 6276 l 8165 6293 l 8184 6313 l 8205 6335 l 8230 6360 l 8258 6388 l 8288 6418 l 8321 6451 l 8356 6486 l 8393 6522 l 8433 6560 l 8474 6599 l 8517 6640 l 8562 6681 l 8608 6722 l 8655 6764 l 8704 6806 l 8754 6848 l 8804 6889 l 8856 6930 l 8908 6970 l 8961 7010 l 9015 7048 l 9071 7086 l 9127 7122 l 9184 7157 l 9242 7190 l 9302 7222 l 9363 7253 l 9425 7281 l 9489 7308 l 9554 7332 l 9621 7354 l 9689 7374 l 9760 7391 l 9831 7405 l 9904 7416 l 9977 7423 l 10051 7426 l 10125 7425 l 10201 7419 l 10275 7409 l 10347 7395 l 10417 7377 l 10484 7356 l 10548 7332 l 10609 7305 l 10667 7276 l 10723 7244 l 10776 7210 l 10827 7175 l 10875 7137 l 10922 7099 l 10966 7058 l 11009 7017 l 11050 6974 l 11089 6930 l 11127 6885 l 11164 6839 l 11200 6793 l 11234 6746 l 11267 6699 l 11299 6652 l 11330 6604 l 11359 6558 l 11388 6512 l 11415 6467 l 11440 6423 l 11464 6381 l 11486 6341 l 11507 6303 l 11526 6268 l 11544 6235 l 11559 6206 l 11573 6179 l 11585 6156 l 11595 6136 l 11603 6119 l 11610 6105 l 11615 6095 l 11619 6087 l 11625 6075 l gs col0 s gr gr [] 0 sd % arrowhead 0 slj n 11557 6345 m 11611 6104 l 11450 6291 l 11557 6345 l cp gs 0.00 setgray ef gr col0 s /Times-Roman ff 330.00 scf sf 3150 6000 m gs 1 -1 sc (Exp) col0 sh gr /Times-Roman ff 330.00 scf sf 5400 6000 m gs 1 -1 sc (Exp) col0 sh gr /Times-Roman ff 330.00 scf sf 7575 5925 m gs 1 -1 sc (Exp) col0 sh gr /Times-Roman ff 330.00 scf sf 9825 5925 m gs 1 -1 sc (Exp) col0 sh gr F2psBegin10setmiterlimit0.060000.06000scF2psEnd rs %%EndDocument endTexFig 523 1560 a Fr(Fig.)15 b(1.)35 b Fu(A)h(PH)g(distribution)g(is)g(the)g (distribution)g(of)h(the)f(absorption)g(time)g(in)g(\014nite)f(state) 523 1651 y(con)n(tin)n(uous)24 b(time)g(Mark)n(o)n(v)g(c)n(hain.)h(The) g(\014gure)f(sho)n(ws)h(a)g(4-phase)g(PH)f(distribution.)g(There)h(are) 523 1742 y Fq(n)d Fu(=)f(4)26 b(states,)h(where)f(the)f Fq(i)p Fu(th)h(state)g(has)g(exp)r(onen)n(tially-distributed)f(so)t (journ)i(time)e(with)g(rate)523 1834 y Fq(\025)568 1842 y Fi(i)594 1834 y Fu(.)31 b(With)f(probabilit)n(y)g Fq(p)1285 1842 y Fj(0)p Fi(i)1372 1834 y Fu(w)n(e)h(start)g(in)f(the)g Fq(i)p Fu(th)g(state,)h(and)f(the)g(next)g(state)h(is)g(state)g Fq(j)k Fu(with)523 1925 y(probabilit)n(y)19 b Fq(p)948 1933 y Fi(ij)1002 1925 y Fu(.)h(Eac)n(h)f(state)h Fq(i)f Fu(has)g(probabilit)n(y)g Fq(p)2009 1933 y Fi(i)p Fj(5)2085 1925 y Fu(that)f(the)h(next)f(state)i(will)g(b)r(e)f(the)g(absorbing) 523 2016 y(state.)27 b(The)e(absorption)i(time)e(is)h(the)f(sum)g(of)h (the)f(times)h(sp)r(en)n(t)f(in)h(eac)n(h)f(of)i(the)e(states.)523 2319 y Fo(Pr)-5 b(eliminary)33 b(De\014nitions)41 b Fw(F)-7 b(ormally)g(,)27 b(w)n(e)g(will)h(use)f(the)h(follo)n(wing)f (de\014nitions:)523 2510 y Fl(De\014nition)k(1.)41 b Fm(A)32 b(distribution)h Fn(G)f Fm(is)h Fl(w)m(ell-represen)m(ted)e Fm(by)i(a)g(distribution)g Fn(F)44 b Fm(if)33 b Fn(F)523 2610 y Fm(and)d Fn(G)g Fm(agr)l(e)l(e)g(on)g(their)h(\014rst)e(thr)l(e) l(e)g(moments.)523 2793 y Fw(The)g(normalized)f(momen)n(ts,)h(in)n(tro) r(duced)g(in)g([18],)g(help)g(pro)n(vide)f(a)h(simple)g(represen)n(ta-) 523 2892 y(tion)f(and)f(analysis)f(of)i(our)f(closed-form)f(solution.)h (These)g(are)g(de\014ned)h(as)f(follo)n(ws:)523 3083 y Fl(De\014nition)k(2.)41 b Fm(L)l(et)22 b Fn(\026)1271 3053 y Fh(F)1271 3107 y(k)1349 3083 y Fm(b)l(e)h(the)f Fn(k)s Fm(-th)h(moment)f(of)h(a)g(distribution)h Fn(F)34 b Fm(for)24 b Fn(k)i Fw(=)c(1)p Fn(;)14 b Fw(2)p Fn(;)g Fw(3)p Fm(.)22 b(The)523 3211 y Fl(normalized)32 b Fn(k)s Fl(-th)i(momen)m(t)29 b Fn(m)1660 3181 y Fh(F)1660 3234 y(k)1746 3211 y Fm(of)k Fn(F)43 b Fm(for)32 b Fn(k)d Fw(=)d(2)p Fn(;)14 b Fw(3)30 b Fm(is)i(de\014ne)l(d)g(to)f(b)l(e)h Fn(m)3041 3181 y Fh(F)3041 3232 y Fv(2)3122 3211 y Fw(=)3265 3169 y Fh(\026)3305 3144 y Fg(F)3305 3185 y Ff(2)p 3223 3192 173 4 v 3223 3244 a Fv(\()p Fh(\026)3289 3224 y Fg(F)3289 3264 y Ff(1)3337 3244 y Fv(\))3363 3227 y Ff(2)523 3368 y Fm(and)e Fn(m)757 3338 y Fh(F)757 3388 y Fv(3)835 3368 y Fw(=)977 3325 y Fh(\026)1017 3300 y Fg(F)1017 3342 y Ff(3)p 933 3349 176 4 v 933 3400 a Fh(\026)973 3380 y Fg(F)973 3420 y Ff(1)1021 3400 y Fh(\026)1061 3380 y Fg(F)1061 3420 y Ff(2)1118 3368 y Fm(.)523 3569 y Fw(Notice)g(the)h(corresp)r(ondence)d(to)i(the)h(co)r(e\016cien)n(t)f (of)g(v)-5 b(ariabilit)n(y)29 b Fn(C)2707 3581 y Fh(F)2793 3569 y Fw(and)h(sk)n(ewness)f Fn(\015)3350 3581 y Fh(F)523 3690 y Fw(of)i Fn(F)12 b Fw(:)30 b Fn(m)812 3659 y Fh(F)812 3710 y Fv(2)895 3690 y Fw(=)e Fn(C)1053 3659 y Fv(2)1047 3712 y Fh(F)1123 3690 y Fw(+)20 b(1)30 b(and)h Fn(m)1518 3659 y Fh(F)1518 3710 y Fv(3)1601 3690 y Fw(=)c Fn(\027)1734 3702 y Fh(F)1790 3616 y Fe(p)p 1873 3616 129 4 v 74 x Fn(m)1946 3661 y Fh(F)1946 3712 y Fv(2)2001 3690 y Fw(,)j(where)g Fn(\027)2338 3702 y Fh(F)2422 3690 y Fw(=)2596 3647 y Fh(\026)2636 3622 y Fg(F)2636 3664 y Ff(3)p 2524 3670 231 4 v 2524 3722 a Fv(\()p Fh(\026)2590 3702 y Fg(F)2590 3742 y Ff(2)2638 3722 y Fv(\))2664 3706 y Ff(3)p Fg(=)p Ff(2)2765 3690 y Fw(.)h(\()p Fn(\027)2892 3702 y Fh(F)2978 3690 y Fw(and)f Fn(\015)3185 3702 y Fh(F)3271 3690 y Fw(and)523 3846 y(closely)e(related,)g(since)h Fn(\015)1339 3858 y Fh(F)1419 3846 y Fw(=)1596 3804 y Fv(\026)-38 b Fh(\026)1631 3779 y Fg(F)1631 3820 y Ff(3)p 1519 3827 V 1519 3879 a Fv(\()5 b(\026)-38 b Fh(\026)1585 3859 y Fg(F)1585 3899 y Ff(2)1633 3879 y Fv(\))1659 3862 y Ff(3)p Fg(=)p Ff(2)1760 3846 y Fw(,)29 b(where)35 b(\026)-49 b Fn(\026)2103 3816 y Fh(F)2103 3870 y(k)2187 3846 y Fw(is)29 b(the)g(cen)n(tralized)f Fn(k)s Fw(-th)h(momen)n(t)g(of)523 3969 y Fn(F)40 b Fw(for)27 b Fn(k)f Fw(=)c(2)p Fn(;)14 b Fw(3.\))523 4160 y Fl(De\014nition)31 b(3.)41 b Fk(P)7 b(H)1220 4172 y Fv(3)1294 4160 y Fm(r)l(efers)37 b(to)f(the)h(set)f(of) h(distributions)g(that)f(ar)l(e)h(wel)t(l-r)l(epr)l(esente)l(d)523 4260 y(by)30 b(a)g(PH)g(distribution.)523 4443 y Fw(It)h(is)g(kno)n(wn) f(that)h(a)g(distribution)g Fn(G)g Fw(is)g(in)g Fk(P)7 b(H)2099 4455 y Fv(3)2168 4443 y Fw(i\013)31 b(its)g(normalized)f (momen)n(ts)g(satisfy)523 4542 y Fn(m)596 4512 y Fh(G)596 4563 y Fv(3)675 4542 y Fn(>)22 b(m)835 4512 y Fh(G)835 4563 y Fv(2)914 4542 y Fn(>)h Fw(1)d([10].)h(Since)g(an)n(y)f (nonnegativ)n(e)g(distribution)h Fn(G)g Fw(satis\014es)g Fn(m)2958 4512 y Fh(G)2958 4563 y Fv(3)3036 4542 y Fk(\025)i Fn(m)3197 4512 y Fh(G)3197 4563 y Fv(2)3276 4542 y Fk(\025)f Fw(1)523 4642 y([13],)27 b(almost)g(all)g(the)h(nonnegativ)n(e)e (distributions)i(are)e(in)i Fk(P)7 b(H)2547 4654 y Fv(3)2584 4642 y Fw(.)p 523 4728 473 4 v 606 4833 a Fu(acyclic.)35 b(Ho)n(w)n(ev)n(er,)f(the)g(necessary)g(and)g(su\016cien)n(t)g(n)n(um)n (b)r(er)e(of)i(phases)h(required)e(to)h(w)n(ell-)606 4924 y(represen)n(t)26 b(a)g(giv)n(en)f(distribution)h(b)n(y)e(a)i(\(p) r(ossibly-cyclic\))h(PH)e(distribution)h(is)g(unkno)n(wn.)p eop %%Page: 4 4 4 3 bop 1380 365 a 9212070 2368143 0 0 37166694 9538355 startTexFig 1380 365 a %%BeginDocument: n-stage-coxian.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: n-stage-coxian.eps %%Creator: fig2dev Version 3.2 Patchlevel 3d %%CreationDate: Thu Jun 19 09:17:41 2003 %%For: osogami@gs57.sp.cs.cmu.edu (Takayuki Osogami) %%BoundingBox: 0 0 565 145 %%Magnification: 1.0000 %%EndComments /$F2psDict 200 dict def F2psDictbeginF2psDict begin F2psDictbeginF2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save newpath 0 145 moveto 0 0 lineto 565 0 lineto 565 145 lineto closepath clip newpath -13.5 355.7 translate 1 -1 scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /reencdict 12 dict def /ReEncode { reencdict begin /newcodesandnames exch def /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup /FID ne { dup /Encoding eq { exch dup length array copy newfont 3 1 roll put } { exch newfont 3 1 roll put } ifelse } { pop pop } ifelse } forall newfont /FontName newfontname put newcodesandnames aload pop 128 1 255 { newfont /Encoding get exch /.notdef put } for newcodesandnames length 2 idiv { newfont /Encoding get 3 1 roll put } repeat newfontname newfont definefont pop end } def /isovec [ 8#055 /minus 8#200 /grave 8#201 /acute 8#202 /circumflex 8#203 /tilde 8#204 /macron 8#205 /breve 8#206 /dotaccent 8#207 /dieresis 8#210 /ring 8#211 /cedilla 8#212 /hungarumlaut 8#213 /ogonek 8#214 /caron 8#220 /dotlessi 8#230 /oe 8#231 /OE 8#240 /space 8#241 /exclamdown 8#242 /cent 8#243 /sterling 8#244 /currency 8#245 /yen 8#246 /brokenbar 8#247 /section 8#250 /dieresis 8#251 /copyright 8#252 /ordfeminine 8#253 /guillemotleft 8#254 /logicalnot 8#255 /hyphen 8#256 /registered 8#257 /macron 8#260 /degree 8#261 /plusminus 8#262 /twosuperior 8#263 /threesuperior 8#264 /acute 8#265 /mu 8#266 /paragraph 8#267 /periodcentered 8#270 /cedilla 8#271 /onesuperior 8#272 /ordmasculine 8#273 /guillemotright 8#274 /onequarter 8#275 /onehalf 8#276 /threequarters 8#277 /questiondown 8#300 /Agrave 8#301 /Aacute 8#302 /Acircumflex 8#303 /Atilde 8#304 /Adieresis 8#305 /Aring 8#306 /AE 8#307 /Ccedilla 8#310 /Egrave 8#311 /Eacute 8#312 /Ecircumflex 8#313 /Edieresis 8#314 /Igrave 8#315 /Iacute 8#316 /Icircumflex 8#317 /Idieresis 8#320 /Eth 8#321 /Ntilde 8#322 /Ograve 8#323 /Oacute 8#324 /Ocircumflex 8#325 /Otilde 8#326 /Odieresis 8#327 /multiply 8#330 /Oslash 8#331 /Ugrave 8#332 /Uacute 8#333 /Ucircumflex 8#334 /Udieresis 8#335 /Yacute 8#336 /Thorn 8#337 /germandbls 8#340 /agrave 8#341 /aacute 8#342 /acircumflex 8#343 /atilde 8#344 /adieresis 8#345 /aring 8#346 /ae 8#347 /ccedilla 8#350 /egrave 8#351 /eacute 8#352 /ecircumflex 8#353 /edieresis 8#354 /igrave 8#355 /iacute 8#356 /icircumflex 8#357 /idieresis 8#360 /eth 8#361 /ntilde 8#362 /ograve 8#363 /oacute 8#364 /ocircumflex 8#365 /otilde 8#366 /odieresis 8#367 /divide 8#370 /oslash 8#371 /ugrave 8#372 /uacute 8#373 /ucircumflex 8#374 /udieresis 8#375 /yacute 8#376 /thorn 8#377 /ydieresis] def /Times-Roman /Times-Roman-iso isovec ReEncode /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def F2psBegin10setmiterlimit0.060000.06000scF2psBegin 10 setmiterlimit 0.06000 0.06000 sc % % Fig objects follow % 7.500 slw % Ellipse n 5737 4237 54 54 0 360 DrawEllipse gs col0 s gr % Ellipse n 6037 4237 54 54 0 360 DrawEllipse gs col0 s gr % Ellipse n 6337 4237 54 54 0 360 DrawEllipse gs col0 s gr /Symbol ff 540.00 scf sf 1575 4650 m gs 1 -1 sc (l) col0 sh gr /Times-Roman-iso ff 360.00 scf sf 1875 4725 m gs 1 -1 sc (1) col0 sh gr /Symbol ff 540.00 scf sf 3975 4650 m gs 1 -1 sc (l) col0 sh gr /Times-Roman-iso ff 360.00 scf sf 4275 4725 m gs 1 -1 sc (2) col0 sh gr /Symbol ff 540.00 scf sf 7575 4650 m gs 1 -1 sc (l) col0 sh gr /Times-Roman-iso ff 360.00 scf sf 7875 4725 m gs 1 -1 sc (n) col0 sh gr 30.000 slw % Ellipse n 4196 4171 629 629 0 360 DrawEllipse gs col0 s gr % Ellipse n 7792 4191 629 629 0 360 DrawEllipse gs col0 s gr % Ellipse n 1806 4188 629 629 0 360 DrawEllipse gs col0 s gr % Polyline gs clippath 5415 4245 m 5415 4155 l 5188 4155 l 5368 4200 l 5188 4245 l cp eoclip n 4800 4200 m 5400 4200 l gs col0 s gr gr % arrowhead 15.000 slw n 5188 4245 m 5368 4200 l 5188 4155 l 5188 4245 l cp gs 0.00 setgray ef gr col0 s % Polyline 30.000 slw gs clippath 1140 4245 m 1140 4155 l 913 4155 l 1093 4200 l 913 4245 l cp eoclip n 300 4200 m 1125 4200 l gs col0 s gr gr % arrowhead 15.000 slw n 913 4245 m 1093 4200 l 913 4155 l 913 4245 l cp gs 0.00 setgray ef gr col0 s % Polyline 30.000 slw gs clippath 3615 4245 m 3615 4155 l 3388 4155 l 3568 4200 l 3388 4245 l cp eoclip n 2400 4200 m 3600 4200 l gs col0 s gr gr % arrowhead 15.000 slw n 3388 4245 m 3568 4200 l 3388 4155 l 3388 4245 l cp gs 0.00 setgray ef gr col0 s % Polyline 30.000 slw gs clippath 9615 4245 m 9615 4155 l 9388 4155 l 9568 4200 l 9388 4245 l cp eoclip n 8400 4200 m 9600 4200 l gs col0 s gr gr % arrowhead 15.000 slw n 9388 4245 m 9568 4200 l 9388 4155 l 9388 4245 l cp gs 0.00 setgray ef gr col0 s % Polyline 30.000 slw gs clippath 7215 4245 m 7215 4155 l 6988 4155 l 7168 4200 l 6988 4245 l cp eoclip n 6600 4200 m 7200 4200 l gs col0 s gr gr % arrowhead 15.000 slw n 6988 4245 m 7168 4200 l 6988 4155 l 6988 4245 l cp gs 0.00 setgray ef gr col0 s % Polyline 30.000 slw gs clippath 9540 5895 m 9540 5805 l 9313 5805 l 9493 5850 l 9313 5895 l cp eoclip n 4800 4350 m 6300 5850 l 9525 5850 l gs col0 s gr gr % arrowhead 15.000 slw n 9313 5895 m 9493 5850 l 9313 5805 l 9313 5895 l cp gs 0.00 setgray ef gr col0 s % Polyline 30.000 slw n 300 4200 m 1950 5850 l 9075 5850 l gs col0 s gr % Polyline n 2400 4425 m 3825 5850 l 9150 5850 l gs col0 s gr /Times-Roman-iso ff 540.00 scf sf 2775 3900 m gs 1 -1 sc (p) col0 sh gr /Times-Roman-iso ff 540.00 scf sf 4950 3900 m gs 1 -1 sc (p) col0 sh gr /Times-Roman-iso ff 540.00 scf sf 6600 3900 m gs 1 -1 sc (p) col0 sh gr /Times-Roman-iso ff 360.00 scf sf 3000 4050 m gs 1 -1 sc (2) col0 sh gr /Times-Roman-iso ff 360.00 scf sf 5175 4050 m gs 1 -1 sc (3) col0 sh gr /Times-Roman-iso ff 360.00 scf sf 6825 4050 m gs 1 -1 sc (n) col0 sh gr /Times-Roman-iso ff 360.00 scf sf 750 4050 m gs 1 -1 sc (1) col0 sh gr /Times-Roman-iso ff 540.00 scf sf 525 3900 m gs 1 -1 sc (p) col0 sh gr /Times-Roman-iso ff 360.00 scf sf 975 5400 m gs 1 -1 sc (1) col0 sh gr /Times-Roman-iso ff 540.00 scf sf 225 5250 m gs 1 -1 sc (1-p) col0 sh gr /Times-Roman-iso ff 360.00 scf sf 2850 5400 m gs 1 -1 sc (2) col0 sh gr /Times-Roman-iso ff 360.00 scf sf 5250 5400 m gs 1 -1 sc (3) col0 sh gr /Times-Roman-iso ff 540.00 scf sf 2100 5250 m gs 1 -1 sc (1-p) col0 sh gr /Times-Roman-iso ff 540.00 scf sf 4500 5250 m gs 1 -1 sc (1-p) col0 sh gr /Times-Roman-iso ff 540.00 scf sf 1425 4125 m gs 1 -1 sc (Exp) col0 sh gr /Times-Roman-iso ff 540.00 scf sf 3825 4125 m gs 1 -1 sc (Exp) col0 sh gr /Times-Roman-iso ff 540.00 scf sf 7425 4125 m gs 1 -1 sc (Exp) col0 sh gr F2psBegin10setmiterlimit0.060000.06000scF2psEnd rs %%EndDocument endTexFig 523 840 a Fr(Fig.)15 b(2.)25 b Fu(An)f Fq(n)p Fu(-phase)i(Co)n(xian)g (distribution)f(is)h(a)g(particular)g Fq(n)p Fu(-phase)f(PH)h (distribution)f(whose)523 931 y(underlying)31 b(Mark)n(o)n(v)g(c)n (hain)g(is)h(of)g(the)e(form)h(in)h(the)e(\014gure,)i(where)f(0)g Ft(\024)g Fq(p)2804 939 y Fi(i)2860 931 y Ft(\024)f Fu(1)i(and)f Fq(\025)3220 939 y Fi(i)3276 931 y Fq(>)g Fu(0)523 1022 y(for)j(all)h(0)g Ft(\024)f Fq(i)g Ft(\024)g Fq(n)p Fu(.)g(An)f Fq(n)p Fu(-phase)g(Co)n(xian)1846 991 y Fj(+)1931 1022 y Fu(distribution)h(is)g(a)g(particular)g Fq(n)p Fu(-phase)f(Co)n(xian) 523 1114 y(distribution)26 b(with)g Fq(p)1156 1122 y Fj(1)1211 1114 y Fu(=)21 b(1.)523 1418 y Fl(De\014nition)31 b(4.)41 b Fn(O)r(P)12 b(T)g Fw(\()p Fn(G)p Fw(\))21 b Fm(is)h(de\014ne)l(d)f(to)h(b)l(e)f(the)g(minimum)g(numb)l(er)g(of)h (ne)l(c)l(essary)g(phases)523 1517 y(for)31 b(a)f(distribution)g Fn(G)g Fm(to)g(b)l(e)g(wel)t(l-r)l(epr)l(esente)l(d)g(by)h(an)e (acyclic)j(PH)e(distribution.)3159 1487 y Fv(3)523 1760 y Fo(Pr)-5 b(evious)40 b(Work)49 b Fw(Prior)30 b(w)n(ork)i(has)g(con)n (tributed)h(a)f(v)n(ery)f(large)h(n)n(um)n(b)r(er)g(of)h(momen)n(t)523 1860 y(matc)n(hing)f(algorithms.)e(While)j(all)f(of)g(these)g (algorithms)f(excel)g(with)i(resp)r(ect)f(to)g(some)523 1960 y(of)24 b(the)g(four)f(measures)g(men)n(tioned)h(earlier)e(\(n)n (um)n(b)r(er)i(of)f(momen)n(ts)h(matc)n(hed;)g(generalit)n(y)523 2059 y(of)h(the)g(solution;)g(computational)f(e\016ciency)h(of)g(the)g (algorithm;)f(and)h(minimalit)n(y)g(of)g(the)523 2159 y(n)n(um)n(b)r(er)35 b(of)h(phases\),)f(they)h(all)g(are)e(de\014cien)n (t)i(in)g(at)f(least)h(one)f(of)h(these)f(measures)g(as)523 2259 y(explained)27 b(b)r(elo)n(w.)648 2367 y(In)h(cases)f(where)g (matc)n(hing)g(only)h(t)n(w)n(o)f(momen)n(ts)h(su\016ces,)g(it)g(is)g (p)r(ossible)f(to)h(ac)n(hiev)n(e)523 2467 y(solutions)d(whic)n(h)h(p)r (erform)f(v)n(ery)g(w)n(ell)h(along)e(all)i(the)g(other)g(three)f (measures.)g(Sauer)g(and)523 2566 y(Chandy)h([21])f(pro)n(vide)f(a)i (closed-form)e(solution)h(for)h(matc)n(hing)f(t)n(w)n(o)g(momen)n(ts)h (of)f(a)h(gen-)523 2666 y(eral)f(distribution)h(in)g Fk(P)7 b(H)1365 2678 y Fv(3)1402 2666 y Fw(.)26 b(They)g(use)g(a)f(t)n (w)n(o-branc)n(h)f(h)n(yp)r(er-exp)r(onen)n(tial)h(distribution)523 2765 y(for)31 b(matc)n(hing)g(distributions)g(with)g(squared)f(co)r (e\016cien)n(t)i(of)f(v)-5 b(ariabilit)n(y)30 b Fn(C)2965 2735 y Fv(2)3032 2765 y Fn(>)e Fw(1)j(and)g(a)523 2865 y(generalized)g(Erlang)f(distribution)j(for)e(matc)n(hing)h (distributions)g(with)h Fn(C)2935 2835 y Fv(2)3003 2865 y Fn(<)d Fw(1.)i(Marie)523 2965 y([15])26 b(pro)n(vides)g(a)h (closed-form)e(solution)i(for)f(matc)n(hing)h(t)n(w)n(o)f(momen)n(ts)h (of)g(a)g(general)e(dis-)523 3064 y(tribution)32 b(in)g Fk(P)7 b(H)1114 3076 y Fv(3)1151 3064 y Fw(.)32 b(He)g(uses)f(a)g(t)n (w)n(o-phase)f(Co)n(xian)2237 3034 y Fv(+)2322 3064 y Fw(distribution)2747 3034 y Fv(4)2816 3064 y Fw(for)h(distributions)523 3164 y(with)23 b Fn(C)772 3134 y Fv(2)833 3164 y Fn(>)f Fw(1)g(and)h(a)f(generalized)f(Erlang)g(distribution)i(for)f (distributions)g(with)h Fn(C)3192 3134 y Fv(2)3253 3164 y Fn(<)f Fw(1.)648 3272 y(If)39 b(one)g(is)g(willing)f(to)h(matc)n(h)g (only)g(a)f(subset)h(of)g(distributions,)g(then)h(again)e(it)h(is)523 3372 y(p)r(ossible)27 b(to)g(ac)n(hiev)n(e)e(solutions)i(whic)n(h)g(p)r (erform)f(v)n(ery)g(w)n(ell)h(along)f(the)h(remaining)f(three)523 3472 y(measures.)h(Whitt)i([26])f(and)g(Altiok)g([2])g(fo)r(cus)g(on)g (the)h(set)f(of)g(distributions)g(with)h Fn(C)3279 3441 y Fv(2)3340 3472 y Fn(>)523 3571 y Fw(1)37 b(and)f(su\016cien)n(tly)h (high)g(third)g(momen)n(t.)g(They)g(obtain)g(a)f(closed-form)g (solution)h(for)523 3671 y(matc)n(hing)h(three)g(momen)n(ts)g(of)g(an)n (y)f(distribution)i(in)f(this)h(set.)f(Whitt)i(matc)n(hes)d(to)h(a)523 3770 y(t)n(w)n(o-branc)n(h)32 b(h)n(yp)r(er-exp)r(onen)n(tial)h (distribution)h(and)g(Altiok)g(matc)n(hes)g(to)g(a)g(t)n(w)n(o-phase) 523 3870 y(Co)n(xian)778 3840 y Fv(+)853 3870 y Fw(distribution.)21 b(T)-7 b(elek)21 b(and)g(Heindl)h([25])f(fo)r(cus)g(on)g(the)g(set)g (of)g(distributions)g(with)523 3970 y Fn(C)588 3940 y Fv(2)649 3970 y Fk(\025)746 3937 y Fv(1)p 746 3951 34 4 v 746 3998 a(2)813 3970 y Fw(and)i(v)-5 b(arious)22 b(constrain)n(ts)g(on)h(the)h(third)g(momen)n(t.)f(They)g(obtain)g(a)g (closed-form)523 4069 y(solution)29 b(for)g(matc)n(hing)g(three)g (momen)n(ts)g(of)g(an)n(y)g(distribution)g(in)h(this)g(set,)f(b)n(y)g (using)g(a)523 4169 y(t)n(w)n(o-phase)d(Co)n(xian)1166 4139 y Fv(+)1247 4169 y Fw(distribution.)648 4277 y(Johnson)f(and)i(T) -7 b(aa\013e)26 b([10,)13 b(9])26 b(come)h(closest)f(to)g(ac)n(hieving) g(all)g(four)h(measures.)e(They)523 4377 y(pro)n(vide)i(a)i (closed-form)e(solution)h(for)g(matc)n(hing)g(the)h(\014rst)f(three)h (momen)n(ts)f(of)h(an)n(y)e(dis-)523 4477 y(tribution)i Fn(G)24 b Fk(2)h(P)7 b(H)1180 4489 y Fv(3)1217 4477 y Fw(.)29 b(They)f(use)g(a)g(mixed)h(Erlang)e(distribution)h(with)h (common)f(order.)p 523 4565 473 4 v 546 4619 a Fj(3)606 4650 y Fu(The)22 b(n)n(um)n(b)r(er)e(of)j(necessary)f(phases)h(in)f (general)h(PH)e(distributions)i(is)f(not)g(kno)n(wn.)g(As)g(sho)n(wn) 606 4742 y(in)g(the)h(next)e(section,)j(all)f(the)f(previous)h(w)n(ork) f(on)h(computationally)f(e\016cien)n(t)h(algorithms)g(for)606 4833 y(mapping)i(general)i(distributions)f(concen)n(trates)g(on)f(a)h (subset)g(of)g(acyclic)h(PH)e(distributions.)546 4893 y Fj(4)606 4924 y Fu(Co)n(xian)840 4893 y Fj(+)911 4924 y Fu(and)20 b(Co)n(xian)h(distributions)f(are)g(particular)h(PH)f (distributions)g(sho)n(wn)h(in)f(Figure)g(2.)p eop %%Page: 5 5 5 4 bop 523 448 a Fw(Unfortunately)-7 b(,)26 b(this)g(mixed)h(Erlang)d (distribution)i(do)r(es)g(not)f(pro)r(duce)h(a)g(minimal)g(solu-)523 548 y(tion.)i(Their)f(solution)g(requires)f(2)p Fn(O)r(P)12 b(T)g Fw(\()p Fn(G)p Fw(\))19 b(+)f(2)27 b(phases)g(in)h(the)g(w)n (orst)e(case.)648 648 y(In)h(complemen)n(tary)f(w)n(ork,)f(Johnson)h (and)h(T)-7 b(aa\013e)27 b([12,)13 b(11])26 b(again)g(lo)r(ok)h(at)f (the)i(prob-)523 747 y(lem)d(of)h(matc)n(hing)e(the)i(\014rst)f(three)g (momen)n(ts)f(of)i(an)n(y)e(distribution)h Fn(G)f Fk(2)f(P)7 b(H)2998 759 y Fv(3)3036 747 y Fw(,)25 b(this)g(time)523 847 y(using)30 b(three)g(t)n(yp)r(es)g(of)g(PH)h(distributions:)f(a)g (mixture)g(of)g(t)n(w)n(o)g(Erlang)e(distributions,)i(a)523 946 y(Co)n(xian)778 916 y Fv(+)868 946 y Fw(distribution,)36 b(and)g(a)f(general)g(PH)g(distribution.)h(Their)g(solution)f(is)h (nearly)523 1046 y(minimal)f(in)g(that)g(it)g(requires)e(at)i(most)f Fn(O)r(P)12 b(T)g Fw(\()p Fn(G)p Fw(\))23 b(+)g(2)34 b(phases.)g(Unfortunately)-7 b(,)35 b(their)523 1146 y(algorithm)g(requires)f(solving)h(a)g(nonlinear)g(programing)e (problem)i(and)h(hence)g(is)f(v)n(ery)523 1245 y(computationally)27 b(ine\016cien)n(t.)648 1345 y(Ab)r(o)n(v)n(e)d(w)n(e)h(ha)n(v)n(e)e (describ)r(ed)i(the)h(prior)e(w)n(ork)f(fo)r(cusing)i(on)g(momen)n (t-matc)n(hing)f(algo-)523 1445 y(rithms)31 b(\(three)f(momen)n(ts\),)h (whic)n(h)f(is)h(the)g(fo)r(cus)f(of)h(this)f(pap)r(er.)g(There)g(is)h (also)e(a)h(large)523 1544 y(b)r(o)r(dy)g(of)g(w)n(ork)e(fo)r(cusing)i (on)g(\014tting)g(the)g Fm(shap)l(e)h Fw(of)f(an)g(input)g (distribution)g(using)g(a)f(PH)523 1644 y(distribution.)g(Of)f (particular)f(recen)n(t)h(in)n(terest)g(has)g(b)r(een)h(w)n(ork)d(on)j (\014tting)f(hea)n(vy-tailed)523 1743 y(distributions)e(to)g(PH)g (distributions,)g(see)g(for)g(example)f(the)i(w)n(ork)e(of)h([3,)13 b(6,)h(7,)f(14,)g(20,)g(24].)523 1843 y(There)26 b(is)g(also)g(w)n(ork) f(whic)n(h)h(com)n(bines)g(the)h(goals)e(of)h(momen)n(t)h(matc)n(hing)f (with)g(the)h(goal)523 1943 y(of)i(\014tting)h(the)f(shap)r(e)g(of)g (the)g(distribution,)g(see)g(for)g(example)f(the)i(w)n(ork)d(of)i([8,) 14 b(22].)28 b(The)523 2042 y(w)n(ork)d(ab)r(o)n(v)n(e)g(is)i(clearly)e (broader)g(in)h(its)h(goals)e(than)h(simply)h(matc)n(hing)f(three)g (momen)n(ts.)523 2142 y(Unfortunately)c(there's)f(a)g(tradeo\013:)g (obtaining)g(a)g(more)g(precise)g(\014t)h(requires)e(man)n(y)h(more)523 2242 y(phases.)i(Additionally)g(it)g(can)g(sometimes)g(b)r(e)h(v)n(ery) e(computationally)g(ine\016cien)n(t)h([8,)14 b(22].)523 2441 y Fo(The)45 b(Ide)-5 b(a)46 b(Behind)e(the)h(EC)g(Distribution)52 b Fw(In)38 b(all)f(the)h(prior)e(w)n(ork)g(on)h(compu-)523 2541 y(tationally)30 b(e\016cien)n(t)h(momen)n(t-matc)n(hing)f (algorithms,)f(the)i(approac)n(h)e(w)n(as)g(to)i(matc)n(h)f(a)523 2640 y(general)g(input)i(distribution)g Fn(G)g Fw(to)f(some)g(subset)g Fk(S)38 b Fw(of)32 b(the)f(PH)h(distributions.)f(In)h(this)523 2740 y(pap)r(er,)h(w)n(e)f(sho)n(w)h(that)g(b)n(y)g(using)g(the)g(set)g (of)g(EC)g(distributions)g(as)f(our)h(subset)g Fk(S)6 b Fw(,)33 b(w)n(e)523 2839 y(ac)n(hiev)n(e)26 b(a)g(solution)h(whic)n (h)g(excels)f(in)h(all)g(four)g(desirable)f(measures)f(men)n(tioned)i (earlier.)523 2939 y(W)-7 b(e)28 b(de\014ne)g(the)g(EC)f(distributions) g(as)g(follo)n(ws:)523 3113 y Fl(De\014nition)k(5.)41 b Fm(A)n(n)29 b Fn(n)p Fm(-phase)i(EC)g(\(Erlang-Coxian\))h (distribution)f(is)g(a)g(p)l(articular)g(PH)523 3213 y(distribution)g(whose)f(underlying)h(Markov)g(chain)g(is)f(of)h(the)f (form)g(in)g(Figur)l(e)g(3.)1092 3418 y 13758847 3315386 0 0 32561971 7828029 startTexFig 1092 3418 a %%BeginDocument: ECn.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: ECn.eps %%Creator: fig2dev Version 3.2 Patchlevel 3d %%CreationDate: Thu Jun 19 09:34:47 2003 %%For: osogami@gs57.sp.cs.cmu.edu (Takayuki Osogami) %%BoundingBox: 0 0 495 119 %%Magnification: 1.0000 %%EndComments /$F2psDict 200 dict def F2psDictbeginF2psDict begin F2psDictbeginF2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save newpath 0 119 moveto 0 0 lineto 495 0 lineto 495 119 lineto closepath clip newpath -16.0 324.2 translate 1 -1 scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /reencdict 12 dict def /ReEncode { reencdict begin /newcodesandnames exch def /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup /FID ne { dup /Encoding eq { exch dup length array copy newfont 3 1 roll put } { exch newfont 3 1 roll put } ifelse } { pop pop } ifelse } forall newfont /FontName newfontname put newcodesandnames aload pop 128 1 255 { newfont /Encoding get exch /.notdef put } for newcodesandnames length 2 idiv { newfont /Encoding get 3 1 roll put } repeat newfontname newfont definefont pop end } def /isovec [ 8#055 /minus 8#200 /grave 8#201 /acute 8#202 /circumflex 8#203 /tilde 8#204 /macron 8#205 /breve 8#206 /dotaccent 8#207 /dieresis 8#210 /ring 8#211 /cedilla 8#212 /hungarumlaut 8#213 /ogonek 8#214 /caron 8#220 /dotlessi 8#230 /oe 8#231 /OE 8#240 /space 8#241 /exclamdown 8#242 /cent 8#243 /sterling 8#244 /currency 8#245 /yen 8#246 /brokenbar 8#247 /section 8#250 /dieresis 8#251 /copyright 8#252 /ordfeminine 8#253 /guillemotleft 8#254 /logicalnot 8#255 /hyphen 8#256 /registered 8#257 /macron 8#260 /degree 8#261 /plusminus 8#262 /twosuperior 8#263 /threesuperior 8#264 /acute 8#265 /mu 8#266 /paragraph 8#267 /periodcentered 8#270 /cedilla 8#271 /onesuperior 8#272 /ordmasculine 8#273 /guillemotright 8#274 /onequarter 8#275 /onehalf 8#276 /threequarters 8#277 /questiondown 8#300 /Agrave 8#301 /Aacute 8#302 /Acircumflex 8#303 /Atilde 8#304 /Adieresis 8#305 /Aring 8#306 /AE 8#307 /Ccedilla 8#310 /Egrave 8#311 /Eacute 8#312 /Ecircumflex 8#313 /Edieresis 8#314 /Igrave 8#315 /Iacute 8#316 /Icircumflex 8#317 /Idieresis 8#320 /Eth 8#321 /Ntilde 8#322 /Ograve 8#323 /Oacute 8#324 /Ocircumflex 8#325 /Otilde 8#326 /Odieresis 8#327 /multiply 8#330 /Oslash 8#331 /Ugrave 8#332 /Uacute 8#333 /Ucircumflex 8#334 /Udieresis 8#335 /Yacute 8#336 /Thorn 8#337 /germandbls 8#340 /agrave 8#341 /aacute 8#342 /acircumflex 8#343 /atilde 8#344 /adieresis 8#345 /aring 8#346 /ae 8#347 /ccedilla 8#350 /egrave 8#351 /eacute 8#352 /ecircumflex 8#353 /edieresis 8#354 /igrave 8#355 /iacute 8#356 /icircumflex 8#357 /idieresis 8#360 /eth 8#361 /ntilde 8#362 /ograve 8#363 /oacute 8#364 /ocircumflex 8#365 /otilde 8#366 /odieresis 8#367 /divide 8#370 /oslash 8#371 /ugrave 8#372 /uacute 8#373 /ucircumflex 8#374 /udieresis 8#375 /yacute 8#376 /thorn 8#377 /ydieresis] def /Times-Roman /Times-Roman-iso isovec ReEncode /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def F2psBegin10setmiterlimit0.060000.06000scF2psBegin 10 setmiterlimit 0.06000 0.06000 sc % % Fig objects follow % 7.500 slw % Ellipse n 3637 4162 54 54 0 360 DrawEllipse gs col0 s gr % Ellipse n 3937 4162 54 54 0 360 DrawEllipse gs col0 s gr % Ellipse n 4237 4162 54 54 0 360 DrawEllipse gs col0 s gr /Times-Roman-iso ff 270.00 scf sf 3300 3675 m gs 1 -1 sc (N) col0 sh gr /Times-Roman-iso ff 360.00 scf sf 3075 3675 m gs 1 -1 sc (E) col0 sh gr /Times-Roman-iso ff 360.00 scf sf 6525 3675 m gs 1 -1 sc (COX) col0 sh gr /Times-Roman-iso ff 270.00 scf sf 7350 3675 m gs 1 -1 sc (2) col0 sh gr 30.000 slw % Ellipse n 5206 4184 338 338 0 360 DrawEllipse gs col0 s gr % Ellipse n 6423 4199 338 338 0 360 DrawEllipse gs col0 s gr % Ellipse n 7617 4180 338 338 0 360 DrawEllipse gs col0 s gr % Ellipse n 2656 4184 338 338 0 360 DrawEllipse gs col0 s gr % Ellipse n 1462 4200 338 338 0 360 DrawEllipse gs col0 s gr % Polyline gs clippath 1140 4245 m 1140 4155 l 913 4155 l 1093 4200 l 913 4245 l cp eoclip n 300 4200 m 1125 4200 l gs col0 s gr gr % arrowhead 15.000 slw n 913 4245 m 1093 4200 l 913 4155 l 913 4245 l cp gs 0.00 setgray ef gr col0 s % Polyline 30.000 slw gs clippath 8490 5370 m 8490 5280 l 8263 5280 l 8443 5325 l 8263 5370 l cp eoclip n 300 4200 m 1425 5325 l 8475 5325 l gs col0 s gr gr % arrowhead 15.000 slw n 8263 5370 m 8443 5325 l 8263 5280 l 8263 5370 l cp gs 0.00 setgray ef gr col0 s % Polyline 30.000 slw gs clippath 2340 4245 m 2340 4155 l 2113 4155 l 2293 4200 l 2113 4245 l cp eoclip n 1800 4200 m 2325 4200 l gs col0 s gr gr % arrowhead 15.000 slw n 2113 4245 m 2293 4200 l 2113 4155 l 2113 4245 l cp gs 0.00 setgray ef gr col0 s % Polyline 30.000 slw gs clippath 3540 4245 m 3540 4155 l 3313 4155 l 3493 4200 l 3313 4245 l cp eoclip n 3000 4200 m 3525 4200 l gs col0 s gr gr % arrowhead 15.000 slw n 3313 4245 m 3493 4200 l 3313 4155 l 3313 4245 l cp gs 0.00 setgray ef gr col0 s % Polyline 30.000 slw gs clippath 4890 4245 m 4890 4155 l 4663 4155 l 4843 4200 l 4663 4245 l cp eoclip n 4350 4200 m 4875 4200 l gs col0 s gr gr % arrowhead 15.000 slw n 4663 4245 m 4843 4200 l 4663 4155 l 4663 4245 l cp gs 0.00 setgray ef gr col0 s % Polyline 30.000 slw gs clippath 6090 4245 m 6090 4155 l 5863 4155 l 6043 4200 l 5863 4245 l cp eoclip n 5550 4200 m 6075 4200 l gs col0 s gr gr % arrowhead 15.000 slw n 5863 4245 m 6043 4200 l 5863 4155 l 5863 4245 l cp gs 0.00 setgray ef gr col0 s % Polyline 30.000 slw gs clippath 7290 4245 m 7290 4155 l 7063 4155 l 7243 4200 l 7063 4245 l cp eoclip n 6750 4200 m 7275 4200 l gs col0 s gr gr % arrowhead 15.000 slw n 7063 4245 m 7243 4200 l 7063 4155 l 7063 4245 l cp gs 0.00 setgray ef gr col0 s % Polyline 30.000 slw gs clippath 8490 4245 m 8490 4155 l 8263 4155 l 8443 4200 l 8263 4245 l cp eoclip n 7950 4200 m 8475 4200 l gs col0 s gr gr % arrowhead 15.000 slw n 8263 4245 m 8443 4200 l 8263 4155 l 8263 4245 l cp gs 0.00 setgray ef gr col0 s % Polyline 30.000 slw gs clippath 8490 5370 m 8490 5280 l 8263 5280 l 8443 5325 l 8263 5370 l cp eoclip n 6750 4200 m 7875 5325 l 8475 5325 l gs col0 s gr gr % arrowhead 15.000 slw n 8263 5370 m 8443 5325 l 8263 5280 l 8263 5370 l cp gs 0.00 setgray ef gr col0 s % Polyline 7.500 slw [60] 0 sd n 1080 3750 m 975 3750 975 4545 105 arcto 4 {pop} repeat 975 4650 5595 4650 105 arcto 4 {pop} repeat 5700 4650 5700 3855 105 arcto 4 {pop} repeat 5700 3750 1080 3750 105 arcto 4 {pop} repeat cp gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 6030 3750 m 5925 3750 5925 4545 105 arcto 4 {pop} repeat 5925 4650 7995 4650 105 arcto 4 {pop} repeat 8100 4650 8100 3855 105 arcto 4 {pop} repeat 8100 3750 6030 3750 105 arcto 4 {pop} repeat cp gs col0 s gr [] 0 sd /Times-Roman-iso ff 360.00 scf sf 600 4050 m gs 1 -1 sc (p) col0 sh gr /Times-Roman-iso ff 360.00 scf sf 375 5100 m gs 1 -1 sc (1-p) col0 sh gr /Times-Roman-iso ff 360.00 scf sf 6600 5025 m gs 1 -1 sc (1-p) col0 sh gr /Times-Roman-iso ff 360.00 scf sf 6900 3975 m gs 1 -1 sc (p) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 7050 4050 m gs 1 -1 sc (X) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 7125 5100 m gs 1 -1 sc (X) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 1500 4425 m gs 1 -1 sc (Y) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 2700 4425 m gs 1 -1 sc (Y) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 5250 4425 m gs 1 -1 sc (Y) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 6450 4425 m gs 1 -1 sc (X1) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 7575 4425 m gs 1 -1 sc (X2) col0 sh gr /Times-Roman-iso ff 300.00 scf sf 1275 4125 m gs 1 -1 sc (Exp) col0 sh gr /Times-Roman-iso ff 300.00 scf sf 2475 4125 m gs 1 -1 sc (Exp) col0 sh gr /Times-Roman-iso ff 300.00 scf sf 6225 4125 m gs 1 -1 sc (Exp) col0 sh gr /Times-Roman-iso ff 300.00 scf sf 7425 4125 m gs 1 -1 sc (Exp) col0 sh gr /Times-Roman-iso ff 300.00 scf sf 5025 4125 m gs 1 -1 sc (Exp) col0 sh gr /Symbol ff 330.00 scf sf 1350 4425 m gs 1 -1 sc (l) col0 sh gr /Symbol ff 330.00 scf sf 2550 4425 m gs 1 -1 sc (l) col0 sh gr /Symbol ff 330.00 scf sf 5100 4425 m gs 1 -1 sc (l) col0 sh gr /Symbol ff 330.00 scf sf 6300 4425 m gs 1 -1 sc (l) col0 sh gr /Symbol ff 330.00 scf sf 7425 4425 m gs 1 -1 sc (l) col0 sh gr F2psBegin10setmiterlimit0.060000.06000scF2psEnd rs %%EndDocument endTexFig 523 4013 a Fr(Fig.)15 b(3.)21 b Fu(The)g(Mark)n(o)n(v)g(c)n(hain)g (underlying)g(an)g(EC)h(distribution,)f(where)h(the)f(\014rst)g(b)r(o)n (x)f(ab)r(o)n(v)n(e)i(de-)523 4104 y(picts)i(the)e(underlying)h(con)n (tin)n(uous)g(time)g(Mark)n(o)n(v)g(c)n(hain)g(in)g(an)h Fq(N)8 b Fu(-phase)23 b(Erlang)h(distribution,)523 4195 y(where)i Fq(N)k Fu(=)21 b Fq(n)c Ft(\000)f Fu(2,)26 b(and)g(the)f(second)g(b)r(o)n(x)g(depicts)h(the)f(underlying)g(con)n (tin)n(uous)g(time)g(Mark)n(o)n(v)523 4287 y(c)n(hain)30 b(in)g(a)g(t)n(w)n(o-phase)g(Co)n(xian)1489 4255 y Fj(+)1570 4287 y Fu(distribution.)g(Notice)g(that)f(the)h(rates)g(in)g(the)f (\014rst)h(b)r(o)n(x)f(are)523 4378 y(the)c(same)h(for)g(all)h(states.) 648 4725 y Fw(W)-7 b(e)31 b(no)n(w)f(pro)n(vide)f(some)h(in)n(tuition)h (b)r(ehind)h(the)f(creation)e(of)i(the)g(EC)f(distribution.)523 4825 y(Recall)20 b(that)g(a)g(Co)n(xian)f(distribution)h(is)g(v)n(ery)f (go)r(o)r(d)g(for)g(appro)n(ximating)g(an)n(y)g(distribution)523 4924 y(with)26 b(high)f(v)-5 b(ariabilit)n(y)e(.)25 b(In)g(particular,) g(a)g(t)n(w)n(o-phase)e(Co)n(xian)i(distribution)g(is)g(kno)n(wn)g(to)p eop %%Page: 6 6 6 5 bop 523 448 a Fw(w)n(ell-represen)n(t)34 b(an)n(y)h(distribution)h (that)g(has)f(high)g(second)h(and)f(third)h(momen)n(ts)f(\(an)n(y)523 548 y(distribution)c Fn(G)h Fw(that)g(satis\014es)e Fn(m)1642 518 y Fh(G)1642 569 y Fv(2)1727 548 y Fn(>)f Fw(2)i(and)g Fn(m)2132 518 y Fh(G)2132 569 y Fv(3)2217 548 y Fn(>)2321 515 y Fv(3)p 2321 529 34 4 v 2321 577 a(2)2364 548 y Fn(m)2437 518 y Fh(G)2437 569 y Fv(2)2493 548 y Fw(\))g([18].)g(Ho)n(w) n(ev)n(er)e(a)i(Co)n(xian)523 648 y(distribution)39 b(requires)f(man)n (y)g(more)h(phases)f(for)g(appro)n(ximating)g(distributions)g(with)523 747 y(lo)n(w)n(er)20 b(second)h(and)g(third)h(momen)n(ts.)f(\(F)-7 b(or)21 b(example,)g(a)g(Co)n(xian)f(distribution)i(requires)e(at)523 847 y(least)h Fn(n)h Fw(phases)f(to)h(w)n(ell-represen)n(t)e(a)h (distribution)h Fn(G)g Fw(with)g Fn(m)2511 817 y Fh(G)2511 867 y Fv(2)2590 847 y Fk(\024)2688 814 y Fh(n)p Fv(+1)p 2688 828 126 4 v 2730 875 a Fh(n)2845 847 y Fw(for)f(in)n(tegers)f Fn(n)j Fk(\025)523 946 y Fw(1\))31 b([18].)f(The)h(large)f(n)n(um)n(b)r (er)g(of)h(phases)f(needed)h(implies)g(that)h(man)n(y)e(free)h (parameters)523 1046 y(m)n(ust)21 b(b)r(e)h(determined)f(whic)n(h)g (implies)h(that)f(an)n(y)g(algorithm)f(that)h(tries)g(to)g(w)n (ell-represen)n(t)523 1146 y(an)e(arbitrary)e(distribution)j(using)f(a) g(minimal)h(n)n(um)n(b)r(er)f(of)g(phases)g(is)g(lik)n(ely)g(to)g (su\013er)g(from)523 1245 y(computational)27 b(ine\016ciency)-7 b(.)648 1345 y(By)20 b(con)n(trast,)g(an)h Fn(n)p Fw(-phase)f(Erlang)f (distribution)i(has)f(only)h(t)n(w)n(o)f(free)h(parameters)e(and)523 1445 y(is)i(also)g(kno)n(wn)f(to)i(ha)n(v)n(e)e(the)h(least)g (normalized)g(second)g(momen)n(t)g(among)f(all)h(the)h Fn(n)p Fw(-phase)523 1544 y(PH)j(distributions)f([1].)h(Ho)n(w)n(ev)n (er)d(the)j(Erlang)e(distribution)i(is)g(ob)n(viously)e(limited)j(in)f (the)523 1644 y(set)j(of)f(distributions)h(whic)n(h)f(it)h(can)f(w)n (ell-represen)n(t.)648 1743 y(Our)h(approac)n(h)f(is)i(therefore)f(to)h (com)n(bine)g(the)g(Erlang)f(distribution)h(with)h(the)f(t)n(w)n(o-)523 1843 y(phase)c(Co)n(xian)g(distribution,)h(allo)n(wing)e(us)h(to)h (represen)n(t)e(distributions)i(with)g(all)g(ranges)523 1943 y(of)k(v)-5 b(ariabilit)n(y)e(,)29 b(while)i(using)e(only)h(a)g (small)f(n)n(um)n(b)r(er)h(of)g(phases.)f(F)-7 b(urthermore)29 b(the)i(fact)523 2042 y(that)23 b(the)g(EC)f(distribution)g(has)g(v)n (ery)g(few)h(free)f(parameters)f(allo)n(ws)g(us)h(to)h(obtain)f (closed-)523 2142 y(from)28 b(expressions)f(for)i(the)g(parameters)e (\()p Fn(n)p Fw(,)i Fn(p)p Fw(,)f Fn(\025)2133 2154 y Fh(Y)2191 2142 y Fw(,)h Fn(\025)2291 2154 y Fh(X)5 b Fv(1)2388 2142 y Fw(,)29 b Fn(\025)2488 2154 y Fh(X)5 b Fv(2)2584 2142 y Fw(,)29 b Fn(p)2678 2154 y Fh(X)2741 2142 y Fw(\))g(of)g(the)g(EC)f(distri-)523 2242 y(bution)g(that)g(w)n (ell-represen)n(ts)d(an)n(y)i(giv)n(en)g(distribution)h(in)f Fk(P)7 b(H)2584 2254 y Fv(3)2621 2242 y Fw(.)523 2416 y Fo(Outline)29 b(of)i(Pap)-5 b(er)38 b Fw(W)-7 b(e)25 b(b)r(egin)e(in)h(Section)g(2)g(b)n(y)f(c)n(haracterizing)e(the)k(EC)e (distribution)523 2516 y(in)f(terms)f(of)g(normalized)g(momen)n(ts.)g (W)-7 b(e)22 b(\014nd)g(that)g(for)f(the)h(purp)r(ose)f(of)g(momen)n(t) h(matc)n(h-)523 2616 y(ing)31 b(it)g(su\016ces)g(to)g(narro)n(w)d(do)n (wn)j(the)g(set)g(of)g(EC)f(distributions)h(further)g(from)g(six)f (free)523 2715 y(parameters)c(to)h(\014v)n(e)h(free)f(parameters,)f(b)n (y)h(optimally)g(\014xing)h(one)f(of)h(the)g(parameters.)648 2815 y(W)-7 b(e)34 b(next)h(presen)n(t)e(three)h(v)-5 b(arian)n(ts)33 b(for)h(closed-form)f(solutions)h(for)f(the)i (remaining)523 2914 y(free)28 b(parameters)e(of)i(the)g(EC)g (distribution,)g(eac)n(h)f(of)h(whic)n(h)g(ac)n(hiev)n(es)e(sligh)n (tly)i(di\013eren)n(t)523 3014 y(goals.)k(The)h(\014rst)g(closed-form)f (solution)g(pro)n(vided,)h(whic)n(h)g(w)n(e)g(refer)f(to)h(as)g Fm(the)i(simple)523 3114 y(solution)p Fw(,)23 b(\(see)f(Section)g(3\))g (has)g(the)h(adv)-5 b(an)n(tage)20 b(of)j(simplicit)n(y)f(and)g (readabilit)n(y;)f(ho)n(w)n(ev)n(er)523 3213 y(it)28 b(do)r(es)f(not)h(w)n(ork)e(for)h(all)h(distributions)f(in)h Fk(P)7 b(H)2105 3225 y Fv(3)2170 3213 y Fw(\(although)27 b(it)h(w)n(orks)e(for)h(almost)g(all\).)523 3313 y(This)40 b(solution)g(requires)f(at)h(most)g Fn(O)r(P)12 b(T)g Fw(\()p Fn(G)p Fw(\))28 b(+)e(2)40 b(phases.)f(The)i(second)e (closed-form)523 3413 y(solution)29 b(pro)n(vided,)g(whic)n(h)h(w)n(e)f (refer)h(to)f(as)g Fm(the)j(impr)l(ove)l(d)i(solution)p Fw(,)c(\(see)g(Section)g(4.1\))523 3512 y(is)f(de\014ned)g(for)f(all)h (the)g(input)g(distributions)g(in)g Fk(P)7 b(H)2227 3524 y Fv(3)2293 3512 y Fw(and)28 b(uses)h(at)f(most)h Fn(O)r(P)12 b(T)g Fw(\()p Fn(G)p Fw(\))20 b(+)e(1)523 3612 y(phases.)36 b(This)h(solution)g(is)g(only)f(lac)n(king)g(in)i(n)n(umerical)e (stabilit)n(y)-7 b(.)37 b(The)g(third)g(closed-)523 3711 y(form)24 b(solution)f(pro)n(vided,)g(whic)n(h)h(w)n(e)g(refer)f(to)h (as)f Fm(the)k(numeric)l(al)t(ly)h(stable)f(solution)p Fw(,)d(\(see)523 3811 y(Section)29 b(4.2\))g(again)f(is)h(de\014ned)g (for)g(all)g(input)h(distributions)f(in)g Fk(P)7 b(H)2767 3823 y Fv(3)2805 3811 y Fw(.)29 b(It)g(uses)g(at)g(most)523 3911 y Fn(O)r(P)12 b(T)g Fw(\()p Fn(G)p Fw(\))22 b(+)g(2)32 b(phases)g(and)g(is)h(n)n(umerically)e(stable)i(in)g(that)f(the)i (momen)n(ts)e(of)g(the)h(EC)523 4010 y(distribution)28 b(are)e(insensitiv)n(e)i(to)f(a)g(small)g(p)r(erturbation)g(in)h(its)g (parameters.)523 4251 y Fp(2)112 b(EC)37 b(Distribution:)e(Motiv)-6 b(ation)36 b(and)i(Prop)s(erties)523 4426 y Fw(The)29 b(purp)r(ose)f(of)g(this)h(section)g(is)f(t)n(w)n(ofold:)g(to)h(pro)n (vide)e(a)h(detailed)h(c)n(haracterization)d(of)523 4526 y(the)34 b(EC)g(distribution,)g(and)g(to)f(discuss)h(a)f(narro)n(w)n (ed-do)n(wn)e(subset)j(of)g(the)g(EC)f(distri-)523 4625 y(butions)j(with)g(only)f(\014v)n(e)h(free)f(parameters)f(\()p Fn(\025)2060 4637 y Fh(Y)2154 4625 y Fw(is)i(\014xed\))g(whic)n(h)g(w)n (e)f(will)h(use)g(in)g(our)523 4725 y(momen)n(t-matc)n(hing)25 b(metho)r(d.)i(Both)e(of)h(these)g(results)g(are)f(summarized)g(in)h (Theorem)f(1.)648 4825 y(T)-7 b(o)20 b(motiv)-5 b(ate)21 b(the)g(theorem)f(in)h(this)g(section,)g(consider)e(the)i(follo)n(wing) f(story)-7 b(.)20 b(Supp)r(ose)523 4924 y(one)34 b(is)g(trying)f(to)h (matc)n(h)g(the)g(\014rst)g(three)g(momen)n(ts)g(of)g(a)g(giv)n(en)f (distribution)h Fn(G)g Fw(to)g(a)p eop %%Page: 7 7 7 6 bop 523 448 a Fw(distribution)27 b Fn(P)39 b Fw(whic)n(h)27 b(consists)g(of)g(a)f(generalized)g(Erlang)f(distribution)i(\(in)h(a)f (general-)523 548 y(ized)f(Erlang)d(distribution)j(the)g(rates)e(of)i (the)g(exp)r(onen)n(tial)f(phases)f(ma)n(y)h(di\013er\))h(follo)n(w)n (ed)523 648 y(b)n(y)i(a)f(t)n(w)n(o-phase)g(Co)n(xian)1352 617 y Fv(+)1434 648 y Fw(distribution.)h(If)h(the)f(distribution)g Fn(G)h Fw(has)e(su\016cien)n(tly)h(high)523 747 y(second)23 b(and)h(third)g(momen)n(ts,)f(then)h(a)g(t)n(w)n(o-phase)e(Co)n(xian) 2411 717 y Fv(+)2489 747 y Fw(distribution)h(alone)g(su\016ces)523 847 y(and)h(w)n(e)g(need)h(zero)e(phases)g(of)i(the)f(generalized)f (Erlang)g(distribution.)i(If)f(the)h(v)-5 b(ariabilit)n(y)523 946 y(of)32 b Fn(G)g Fw(is)g(lo)n(w)n(er,)f(ho)n(w)n(ev)n(er,)e(w)n(e)j (migh)n(t)g(try)g(app)r(ending)g(a)f(single-phase)g(generalized)f(Er-) 523 1046 y(lang)e(distribution)g(to)h(the)f(t)n(w)n(o-phase)f(Co)n (xian)2047 1016 y Fv(+)2129 1046 y Fw(distribution.)i(If)g(that)f(do)r (esn't)h(su\016ce,)523 1146 y(w)n(e)g(migh)n(t)g(app)r(end)h(a)f(t)n(w) n(o-phase)e(generalized)h(Erlang)g(distribution)h(to)g(the)h(t)n(w)n (o-phase)523 1245 y(Co)n(xian)778 1215 y Fv(+)867 1245 y Fw(distribution.)36 b(If)g(our)e(distribution)i Fn(G)g Fw(has)f(v)n(ery)f(lo)n(w)h(v)-5 b(ariabilit)n(y)34 b(w)n(e)h(migh)n(t) 523 1345 y(b)r(e)c(forced)g(to)g(use)f(man)n(y)h(phases)f(of)h(the)g (generalized)f(Erlang)f(distribution)i(to)g(get)g(the)523 1445 y(v)-5 b(ariabilit)n(y)26 b(of)h Fn(P)39 b Fw(to)28 b(b)r(e)f(lo)n(w)g(enough.)f(Therefore,)g(to)h(minimize)h(the)g(n)n(um) n(b)r(er)f(of)g(phases)523 1544 y(in)c Fn(P)12 b Fw(,)22 b(it)h(seems)f(desirable)f(to)h(c)n(ho)r(ose)f(the)i(rates)e(of)i(the)f (generalized)f(Erlang)g(distribution)523 1644 y(so)27 b(that)h(the)g(o)n(v)n(erall)d(v)-5 b(ariabilit)n(y)27 b(of)g Fn(P)40 b Fw(is)27 b(minimized.)648 1743 y(Con)n(tin)n(uing)18 b(with)i(our)f(story)-7 b(,)18 b(one)h(could)g(express)g(the)g(app)r (ending)h(of)f(eac)n(h)g(additional)523 1843 y(phase)36 b(of)g(the)h(generalized)e(Erlang)g(distribution)h(as)g(a)g (\\function")g(whose)g(goal)f(is)h(to)523 1943 y(reduce)27 b(the)h(v)-5 b(ariabilit)n(y)27 b(of)g Fn(P)40 b Fw(y)n(et)27 b(further.)h(W)-7 b(e)28 b(call)f(this)h(\\function)g Fn(\036)p Fw(.")523 2079 y Fl(De\014nition)j(6.)41 b Fm(L)l(et)24 b Fn(X)32 b Fm(b)l(e)25 b(an)g(arbitr)l(ary)h (distribution.)h Fl(F)-8 b(unction)26 b Fn(\036)f Fm(maps)h Fn(X)31 b Fm(to)26 b Fn(\036)p Fw(\()p Fn(X)7 b Fw(\))523 2179 y Fm(such)34 b(that)h Fn(\036)p Fw(\()p Fn(X)7 b Fw(\))32 b(=)f Fn(Y)40 b Fk(\003)21 b Fn(X)7 b Fm(,)35 b(wher)l(e)g Fn(Y)53 b Fm(is)34 b(an)h(exp)l(onential)g(distribution)g (with)g(r)l(ate)f Fn(\025)3347 2191 y Fh(Y)523 2278 y Fm(indep)l(endent)25 b(of)g Fn(X)7 b Fm(,)24 b Fn(Y)h Fk(\003)6 b Fn(X)30 b Fm(is)24 b(the)g(c)l(onvolution)h(of)g Fn(Y)43 b Fm(and)25 b Fn(X)7 b Fm(,)23 b(and)i Fn(\025)2733 2290 y Fh(Y)2815 2278 y Fm(is)f(chosen)h(so)g(that)523 2378 y(the)g(normalize)l(d)i(se)l(c)l(ond)e(moment)g(of)h Fn(\036)p Fw(\()p Fn(X)7 b Fw(\))25 b Fm(is)g(minimize)l(d.)i(A)n(lso,) f Fn(\036)2715 2348 y Fh(l)2741 2378 y Fw(\()p Fn(X)7 b Fw(\))22 b(=)h Fn(\036)p Fw(\()p Fn(\036)3121 2348 y Fh(l)p Fd(\000)p Fv(1)3233 2378 y Fw(\()p Fn(X)7 b Fw(\)\))523 2477 y Fm(r)l(efers)27 b(to)f(the)g(distribution)h(obtaine) l(d)g(by)f(applying)j(function)d Fn(\036)g Fm(to)g Fn(\036)2720 2447 y Fh(l)p Fd(\000)p Fv(1)2831 2477 y Fw(\()p Fn(X)7 b Fw(\))26 b Fm(for)h(inte)l(gers)523 2577 y Fn(l)e Fk(\025)d Fw(1)p Fm(,)30 b(wher)l(e)g Fn(\036)1040 2547 y Fv(0)1078 2577 y Fw(\()p Fn(X)7 b Fw(\))23 b(=)g Fn(X)7 b Fm(.)648 2713 y Fw(Observ)n(e)24 b(that,)j(when)g Fn(X)33 b Fw(is)26 b(a)g Fn(k)s Fw(-phase)g(PH)g(distribution,)h Fn(\036)p Fw(\()p Fn(X)7 b Fw(\))26 b(is)h(a)f(\()p Fn(k)19 b Fw(+)d(1\)-phase) 523 2813 y(PH)28 b(distribution)g(whose)g(underlying)f(Mark)n(o)n(v)f (c)n(hain)i(can)g(b)r(e)g(obtained)g(b)n(y)g(app)r(ending)523 2913 y(a)g(state)g(with)h(rate)f Fn(\025)1207 2925 y Fh(Y)1294 2913 y Fw(to)g(the)h(Mark)n(o)n(v)d(c)n(hain)i(underlying)g Fn(X)7 b Fw(,)28 b(where)g Fn(\025)2893 2925 y Fh(Y)2980 2913 y Fw(is)g(c)n(hosen)g(so)523 3024 y(that)21 b Fn(m)769 2981 y Fh(\036)p Fv(\()p Fh(X)5 b Fv(\))769 3046 y(2)944 3024 y Fw(is)20 b(minimized.)h(In)g(theory)-7 b(,)19 b(function)i Fn(\036)g Fw(allo)n(ws)e(eac)n(h)h(successiv)n(e)f(exp)r (onen)n(tial)523 3123 y(distribution)31 b(whic)n(h)f(is)h(app)r(ended)g (to)g(ha)n(v)n(e)e(a)h(di\013eren)n(t)h(\014rst)g(momen)n(t.)f(The)h (follo)n(wing)523 3223 y(theorem)20 b(sho)n(ws)e(that)j(if)f(the)h(exp) r(onen)n(tial)e(distribution)h Fn(Y)39 b Fw(b)r(eing)20 b(app)r(ended)h(b)n(y)e(function)523 3323 y Fn(\036)h Fw(is)f(c)n(hosen)g(so)g(as)g(to)g(minimize)h(the)g(normalized)e (second)h(momen)n(t)g(of)h Fn(\036)p Fw(\()p Fn(X)7 b Fw(\))20 b(\(as)f(sp)r(eci\014ed)523 3422 y(b)n(y)k(the)i (de\014nition\),)f(then)g(the)g(\014rst)g(momen)n(t)f(of)h(eac)n(h)f (successiv)n(e)f Fn(Y)43 b Fw(is)23 b(alw)n(a)n(ys)f Fm(the)k(same)523 3522 y Fw(and)31 b(is)g(de\014ned)g(b)n(y)f(the)i (simple)f(form)n(ula)f(sho)n(wn)g(in)h(\(1\).)g(The)g(theorem)g(b)r (elo)n(w)f(further)523 3622 y(c)n(haracterizes)25 b(the)j(normalized)f (momen)n(ts)g(of)h Fn(\036)2080 3591 y Fh(l)2106 3622 y Fw(\()p Fn(X)7 b Fw(\).)523 3749 y Fl(Theorem)30 b(1.)41 b Fm(L)l(et)26 b Fn(\036)1233 3719 y Fh(l)1259 3749 y Fw(\()p Fn(X)7 b Fw(\))23 b(=)f Fn(Y)1557 3761 y Fh(l)1593 3749 y Fk(\003)10 b Fn(\036)1694 3719 y Fh(l)p Fd(\000)p Fv(1)1804 3749 y Fw(\()p Fn(X)d Fw(\))26 b Fm(and)h(let)e Fn(\025)2288 3761 y Fh(Y)2327 3770 y Fg(l)2379 3749 y Fw(=)2512 3717 y Fv(1)p 2477 3731 103 4 v 2477 3797 a Fh(\026)2517 3762 y Fg(Y)2551 3777 y(l)2517 3817 y Ff(1)2615 3749 y Fm(for)i Fn(l)e Fw(=)d(1)p Fn(;)14 b(:::;)g(N)9 b Fm(.)26 b(Then,)1637 3962 y Fq(\025)1682 3970 y Fi(Y)1718 3982 y Fg(l)1768 3962 y Fu(=)2051 3913 y(1)p 1859 3944 422 4 v 1859 4014 a(\()p Fq(m)1957 3988 y Fi(X)1957 4034 y Fj(2)2031 4014 y Ft(\000)17 b Fu(1\))p Fq(\026)2222 3988 y Fi(X)2222 4034 y Fj(1)3307 3962 y Fu(\(1\))553 4140 y Fm(for)30 b Fn(l)25 b Fw(=)d(1)p Fn(;)14 b(:::;)g(N)9 b Fm(.)648 4231 y(The)30 b(normalize)l(d)h(moments)f(of)g Fn(Z)1747 4243 y Fh(N)1833 4231 y Fw(=)23 b Fn(\036)1970 4201 y Fh(N)2033 4231 y Fw(\()p Fn(X)7 b Fw(\))30 b Fm(ar)l(e:)626 4415 y Fq(m)694 4378 y Fi(Z)736 4389 y Fg(N)694 4435 y Fj(2)816 4415 y Fu(=)909 4366 y(\()p Fq(m)1007 4334 y Fi(X)1007 4380 y Fj(2)1082 4366 y Ft(\000)17 b Fu(1\)\()p Fq(N)25 b Fu(+)17 b(1\))g(+)g(1)p 909 4398 712 4 v 1005 4468 a(\()p Fq(m)1103 4442 y Fi(X)1103 4488 y Fj(2)1178 4468 y Ft(\000)g Fu(1\))p Fq(N)25 b Fu(+)17 b(1)1631 4415 y(;)1655 b(\(2\))626 4635 y Fq(m)694 4598 y Fi(Z)736 4609 y Fg(N)694 4654 y Fj(3)816 4635 y Fu(=)1482 4587 y Fq(m)1550 4555 y Fi(X)1550 4600 y Fj(2)1608 4587 y Fq(m)1676 4555 y Fi(X)1676 4600 y Fj(3)p 909 4618 1398 4 v 909 4699 a Fu(\(\()p Fq(m)1037 4673 y Fi(X)1037 4719 y Fj(2)1112 4699 y Ft(\000)16 b Fu(1\)\()p Fq(N)26 b Fu(+)17 b(1\))g(+)g(1\))c(\(\()p Fq(m)1821 4673 y Fi(X)1821 4719 y Fj(2)1895 4699 y Ft(\000)k Fu(1\))p Fq(N)26 b Fu(+)17 b(1\))2272 4660 y Fj(2)828 4882 y Fu(+)898 4820 y(\()p Fq(m)996 4789 y Fi(X)996 4834 y Fj(2)1071 4820 y Ft(\000)f Fu(1\))p Fq(N)1298 4755 y Fe(\000)1336 4820 y Fu(3)p Fq(m)1442 4789 y Fi(X)1442 4834 y Fj(2)1517 4820 y Fu(+)h(\()p Fq(m)1692 4789 y Fi(X)1692 4834 y Fj(2)1766 4820 y Ft(\000)g Fu(1\)\()p Fq(m)2009 4789 y Fi(X)2009 4834 y Fj(2)2084 4820 y Fu(+)g(2\)\()p Fq(N)25 b Fu(+)17 b(1\))g(+)g(\()p Fq(m)2682 4789 y Fi(X)2682 4834 y Fj(2)2757 4820 y Ft(\000)f Fu(1\))2901 4789 y Fj(2)2936 4820 y Fu(\()p Fq(N)26 b Fu(+)16 b(1\))3197 4789 y Fj(2)3232 4755 y Fe(\001)p 898 4865 2373 4 v 1386 4946 a Fu(\()o(\()p Fq(m)1513 4920 y Fi(X)1513 4966 y Fj(2)1588 4946 y Ft(\000)h Fu(1\)\()p Fq(N)25 b Fu(+)17 b(1\))h(+)e(1\))d(\(\()p Fq(m)2297 4920 y Fi(X)2297 4966 y Fj(2)2372 4946 y Ft(\000)k Fu(1\))p Fq(N)25 b Fu(+)17 b(1\))2748 4907 y Fj(2)3280 4882 y Fq(:)6 b Fu(\(3\))p eop %%Page: 8 8 8 7 bop 648 586 a Fw(Observ)n(e)18 b(that,)j(when)g Fn(X)26 b Fw(is)21 b(a)f Fn(k)s Fw(-phase)f(PH)h(distribution,)h Fn(\036)2553 556 y Fh(N)2616 586 y Fw(\()p Fn(X)7 b Fw(\))21 b(is)f(a)g(\()p Fn(k)7 b Fw(+)t Fn(N)i Fw(\)-phase)523 685 y(PH)28 b(distribution)g(whose)g(underlying)f(Mark)n(o)n(v)f(c)n (hain)i(can)g(b)r(e)g(obtained)g(b)n(y)g(app)r(ending)523 785 y Fn(N)40 b Fw(states)31 b(with)g(rate)g Fn(\025)1285 797 y Fh(Y)1374 785 y Fw(to)g(the)g(Mark)n(o)n(v)e(c)n(hain)h (underlying)h Fn(X)7 b Fw(,)30 b(where)h Fn(\025)2991 797 y Fh(Y)3080 785 y Fw(is)g(c)n(hosen)523 896 y(so)d(that)g Fn(m)879 853 y Fh(\036)p Fv(\()p Fh(X)5 b Fv(\))879 918 y(2)1062 896 y Fw(is)28 b(minimized.)h(The)g(remainder)e(of)h(this)h (section)f(will)g(pro)n(v)n(e)f(the)h(ab)r(o)n(v)n(e)523 996 y(theorem)f(and)h(a)f(corollary)-7 b(.)523 1133 y Fm(Pr)l(o)l(of)43 b(\(The)l(or)l(em)30 b(1\).)648 1233 y Fw(W)-7 b(e)28 b(\014rst)f(c)n(haracterize)e Fn(Z)k Fw(=)23 b Fn(\036)p Fw(\()p Fn(X)7 b Fw(\))23 b(=)g Fn(Y)37 b Fk(\003)18 b Fn(X)p 648 1267 1468 4 v 6 w Fw(,)33 b(where)g Fn(X)40 b Fw(is)33 b(an)g(arbitrary)e(distribu-)523 1332 y(tion)e(with)h(a)e(\014nite)i(third)f(momen)n(t)g(and)g Fn(Y)48 b Fw(is)29 b(an)f(exp)r(onen)n(tial)h(distribution.)g(The)g (nor-)523 1453 y(malized)e(second)g(momen)n(t)g(of)h Fn(Z)33 b Fw(is)27 b Fn(m)1756 1422 y Fh(Z)1756 1473 y Fv(2)1833 1453 y Fw(=)1930 1410 y Fh(m)1989 1385 y Fg(X)1989 1427 y Ff(2)2043 1410 y Fv(+2)p Fh(y)r Fv(+2)p Fh(y)2283 1385 y Ff(2)p 1930 1433 385 4 v 2020 1481 a Fv(\(1+)p Fh(y)r Fv(\))2192 1465 y Ff(2)2325 1453 y Fw(,)g(where)g Fn(y)f Fw(=)2781 1410 y Fh(\026)2821 1385 y Fg(Y)2821 1427 y Ff(1)p 2780 1433 94 4 v 2780 1485 a Fh(\026)2820 1465 y Fg(X)2820 1505 y Ff(1)2883 1453 y Fw(.)i(Observ)n(e)e(that)523 1583 y Fn(m)596 1553 y Fh(Z)596 1604 y Fv(2)677 1583 y Fw(is)i(minimized)g(when)g Fn(y)d Fw(=)e Fn(m)1606 1553 y Fh(X)1606 1604 y Fv(2)1687 1583 y Fk(\000)18 b Fw(1,)28 b(namely)-7 b(,)1615 1729 y Fn(\026)1665 1695 y Fh(Y)1665 1750 y Fv(1)1746 1729 y Fw(=)22 b(\()p Fn(m)1938 1695 y Fh(X)1938 1750 y Fv(2)2020 1729 y Fk(\000)c Fw(1\))p Fn(\026)2227 1695 y Fh(X)2227 1750 y Fv(1)2290 1729 y Fn(:)986 b Fw(\(4\))523 1866 y(Observ)n(e)30 b(that)i(when)g(equation)f (\(4\))h(is)f(satis\014ed,)h(the)g(normalized)e(second)h(momen)n(t)h (of)523 1958 y Fn(Z)h Fw(satis\014es:)1705 2062 y Fq(m)1773 2027 y Fi(Z)1773 2076 y Fj(2)1843 2062 y Fu(=)21 b(2)d Ft(\000)2110 2014 y Fu(1)p 2066 2045 126 4 v 2066 2115 a Fq(m)2134 2089 y Fi(X)2134 2135 y Fj(2)2202 2062 y Fq(;)1084 b Fu(\(5\))551 2219 y Fw(and)27 b(the)h(normalized)f(third)g (momen)n(t)h(of)g Fn(Z)33 b Fw(satis\014es:)1295 2383 y Fq(m)1363 2347 y Fi(Z)1363 2396 y Fj(3)1434 2383 y Fu(=)1746 2335 y(1)p 1525 2366 482 4 v 1525 2436 a Fq(m)1593 2410 y Fi(X)1593 2456 y Fj(2)1650 2436 y Fu(\(2)p Fq(m)1786 2410 y Fi(X)1786 2456 y Fj(2)1861 2436 y Ft(\000)17 b Fu(1\))2016 2383 y Fq(m)2084 2347 y Fi(X)2084 2396 y Fj(3)2159 2383 y Fu(+)2245 2334 y(3\()p Fq(m)2381 2303 y Fi(X)2381 2348 y Fj(2)2456 2334 y Ft(\000)g Fu(1\))p 2245 2366 356 4 v 2361 2436 a Fq(m)2429 2410 y Fi(X)2429 2456 y Fj(2)2611 2383 y Fq(:)675 b Fu(\(6\))648 2624 y Fw(W)-7 b(e)28 b(next)f(c)n(haracterize)f Fn(Z)1494 2636 y Fh(l)1542 2624 y Fw(=)d Fn(\036)1679 2593 y Fh(l)1705 2624 y Fw(\()p Fn(X)7 b Fw(\))22 b(=)h Fn(Y)2003 2636 y Fh(l)2047 2624 y Fk(\003)18 b Fn(\036)2156 2593 y Fh(l)p Fd(\000)p Fv(1)2267 2624 y Fw(\()p Fn(X)7 b Fw(\))28 b(for)f(2)22 b Fk(\024)h Fn(l)h Fk(\024)f Fn(N)p 648 2658 2280 4 v 9 w Fw(:)37 b(By)h(\(5\))f(and)523 2723 y(\(6\),)31 b(\(2\))g(and)f(\(3\))h(follo)n(w)f(from)g(solving)f(the)i (follo)n(wing)f(recursiv)n(e)f(form)n(ulas)g(\(where)h(w)n(e)523 2849 y(use)d Fn(b)702 2861 y Fh(l)755 2849 y Fw(to)h(denote)f Fn(m)1197 2806 y Fh(\036)1237 2781 y Fg(l)1261 2806 y Fv(\()p Fh(X)5 b Fv(\))1197 2871 y(2)1404 2849 y Fw(and)27 b Fn(B)1628 2861 y Fh(l)1681 2849 y Fw(to)h(denote)f Fn(m)2123 2806 y Fh(\036)2163 2781 y Fg(l)2187 2806 y Fv(\()p Fh(X)5 b Fv(\))2123 2871 y(3)2302 2849 y Fw(\):)1427 3035 y Fn(b)1463 3047 y Fh(l)p Fv(+1)1595 3035 y Fw(=)23 b(2)18 b Fk(\000)1846 2979 y Fw(1)p 1836 3016 62 4 v 1836 3092 a Fn(b)1872 3104 y Fh(l)1907 3035 y Fw(;)1369 b(\(7\))1399 3256 y Fn(B)1462 3268 y Fh(l)p Fv(+1)1595 3256 y Fw(=)1834 3200 y Fn(B)1897 3212 y Fh(l)p 1693 3237 372 4 v 1693 3313 a Fn(b)1729 3325 y Fh(l)1754 3313 y Fw(\(2)p Fn(b)1864 3325 y Fh(l)1907 3313 y Fk(\000)18 b Fw(1\))2093 3256 y(+)2185 3200 y(3\()p Fn(b)2295 3212 y Fh(l)2339 3200 y Fk(\000)g Fw(1\))p 2185 3237 311 4 v 2310 3313 a Fn(b)2346 3325 y Fh(l)2506 3256 y Fn(:)770 b Fw(\(8\))523 3451 y(The)28 b(solution)f(for)g(\(7\))h(is)f(giv)n(en)g (b)n(y)1570 3626 y Fq(b)1603 3635 y Fi(l)1649 3626 y Fu(=)1836 3577 y(\()p Fq(b)1899 3585 y Fj(1)1950 3577 y Ft(\000)17 b Fu(1\))p Fq(l)h Fu(+)f(1)p 1740 3609 608 4 v 1740 3676 a(\()p Fq(b)1803 3684 y Fj(1)1854 3676 y Ft(\000)g Fu(1\)\()p Fq(l)h Ft(\000)f Fu(1\))g(+)g(1)3307 3626 y(\(9\))551 3807 y Fw(for)27 b(all)g Fn(l)e Fk(\025)d Fw(1,)27 b(and)h(the)g(solution)f(for)g(\(8\))h(is)f(giv)n(en)g(b)n(y) 858 4007 y Fq(B)916 4016 y Fi(l)961 4007 y Fu(=)1052 3945 y Fq(b)1085 3953 y Fj(1)1120 3945 y Fq(B)1178 3953 y Fj(1)1229 3945 y Fu(+)17 b(\()p Fq(b)1369 3953 y Fj(1)1421 3945 y Ft(\000)g Fu(1\)\()p Fq(l)h Ft(\000)f Fu(1\))1795 3880 y Fe(\000)1833 3945 y Fu(3)p Fq(b)1904 3953 y Fj(1)1956 3945 y Fu(+)g(\()p Fq(b)2096 3953 y Fj(1)2147 3945 y Ft(\000)g Fu(1\)\()p Fq(b)2355 3953 y Fj(1)2406 3945 y Fu(+)g(2\))p Fq(l)i Fu(+)e(\()p Fq(b)2733 3953 y Fj(1)2784 3945 y Ft(\000)g Fu(1\))2929 3914 y Fj(2)2964 3945 y Fq(l)2988 3914 y Fj(2)3022 3880 y Fe(\001)p 1052 3990 2009 4 v 1461 4067 a Fu(\(\()p Fq(b)1554 4075 y Fj(1)1605 4067 y Ft(\000)g Fu(1\))p Fq(l)i Fu(+)d(1\))d(\(\()p Fq(b)2042 4075 y Fj(1)2094 4067 y Ft(\000)k Fu(1\)\()p Fq(l)h Ft(\000)f Fu(1\))g(+)g(1\))2617 4032 y Fj(2)3268 4007 y Fu(\(10\))552 4198 y Fw(for)28 b(all)g Fn(l)e Fk(\025)f Fw(1.)j(Equations)f(\(9\))i(and)g(\(10\))f(can)g(b)r(e)h (easily)f(v)n(eri\014ed)g(b)n(y)g(substitution)h(in)n(to)523 4297 y(\(7\))f(and)f(\(8\),)h(resp)r(ectiv)n(ely)-7 b(.)27 b(This)g(completes)h(the)g(pro)r(of)f(of)g(\(2\))h(and)f(\(3\).)648 4388 y(The)33 b(pro)r(of)g(of)h(\(1\))f(pro)r(ceeds)g(b)n(y)g (induction.)h(When)g Fn(l)h Fw(=)d(1,)h(\(1\))h(follo)n(ws)f(from)g (\(4\).)523 4480 y(Assume)24 b(that)h(\(1\))f(holds)g(when)g Fn(l)h Fw(=)d(1)p Fn(;)14 b(:::;)g(t)p Fw(.)24 b(Let)h Fn(Z)2163 4492 y Fh(t)2215 4480 y Fw(=)d Fn(\036)2351 4450 y Fh(t)2381 4480 y Fw(\()p Fn(X)7 b Fw(\).)24 b(By)g(\(2\),)g (whic)n(h)g(is)h(pro)n(v)n(ed)523 4605 y(ab)r(o)n(v)n(e,)h Fn(m)854 4568 y Fh(Z)899 4576 y Fg(t)854 4627 y Fv(2)954 4605 y Fw(=)1052 4562 y Fv(\()p Fh(m)1137 4537 y Fg(X)1137 4579 y Ff(2)1190 4562 y Fd(\000)p Fv(1\)\()p Fh(t)p Fv(+1\)+1)p 1052 4586 495 4 v 1120 4637 a(\()p Fh(m)1205 4617 y Fg(X)1205 4657 y Ff(2)1258 4637 y Fd(\000)p Fv(1\))p Fh(t)p Fv(+1)1556 4605 y Fw(.)i(Th)n(us,)f(b)n(y)h(\(4\))1321 4784 y Fq(\026)1367 4741 y Fi(Y)1403 4752 y Fg(t)p Ff(+1)1367 4803 y Fj(1)1528 4784 y Fu(=)21 b(\()p Fq(m)1707 4748 y Fi(Z)1749 4756 y Fg(t)1707 4802 y Fj(2)1797 4784 y Ft(\000)16 b Fu(1\))p Fq(\026)1987 4748 y Fi(Z)2029 4756 y Fg(t)1987 4802 y Fj(1)2083 4784 y Fu(=)21 b(\()p Fq(m)2262 4748 y Fi(X)2262 4797 y Fj(2)2336 4784 y Ft(\000)c Fu(1\))p Fq(\026)2527 4748 y Fi(X)2527 4797 y Fj(1)2585 4784 y Fq(:)3350 4924 y Fk(u)-55 b(t)p eop %%Page: 9 9 9 8 bop 523 448 a Fl(Corollary)32 b(1.)41 b Fm(L)l(et)29 b Fn(Z)1264 460 y Fh(N)1350 448 y Fw(=)22 b Fn(\036)1486 418 y Fh(N)1550 448 y Fw(\()p Fn(X)7 b Fw(\))p Fm(.)30 b(If)g Fn(X)f Fk(2)2009 381 y Fe(\010)2057 448 y Fn(F)35 b Fk(j)53 b Fw(2)22 b Fn(<)h(m)2446 418 y Fh(F)2446 469 y Fv(2)2501 381 y Fe(\011)2550 448 y Fm(,)30 b(then)523 581 y Fn(Z)580 593 y Fh(N)666 581 y Fk(2)744 489 y Fe(n)800 581 y Fn(F)865 486 y Fe(\014)865 535 y(\014)865 585 y(\014)902 548 y Fh(N)6 b Fv(+2)p 902 562 143 4 v 902 610 a Fh(N)g Fv(+1)1078 581 y Fn(<)23 b(m)1239 551 y Fh(F)1239 602 y Fv(2)1317 581 y Fn(<)1414 548 y Fh(N)6 b Fv(+1)p 1414 562 V 1456 610 a Fh(N)1567 489 y Fe(o)1623 581 y Fm(.)523 775 y Fw(Corollary)25 b(1)h(suggests)g(the)h(n)n(um)n(b)r(er)g Fn(N)35 b Fw(of)27 b(times)h(that)f(function)g Fn(\036)h Fw(m)n(ust)f(b)r(e)g(applied)g(to)523 874 y Fn(X)40 b Fw(to)34 b(bring)f Fn(m)1036 837 y Fh(Z)1081 845 y Fg(N)1036 896 y Fv(2)1173 874 y Fw(in)n(to)g(the)i(desired)e(range,)g(giv)n(en)g (the)h(v)-5 b(alue)34 b(of)f Fn(m)2807 844 y Fh(X)2807 895 y Fv(2)2870 874 y Fw(.)i(Observ)n(e)d(that)523 974 y(an)n(y)27 b(Co)n(xian)935 944 y Fv(+)1016 974 y Fw(distribution)h(is) g(in)f Fk(f)p Fn(F)35 b Fk(j)23 b Fw(2)g Fn(<)f(m)2050 944 y Fh(F)2050 995 y Fv(2)2105 974 y Fk(g)523 1147 y Fm(Pr)l(o)l(of)43 b(\(Cor)l(ol)t(lary)31 b(1\).)f Fw(By)d(\(2\),)g Fn(m)1650 1110 y Fh(Z)1695 1118 y Fg(N)1650 1169 y Fv(2)1780 1147 y Fw(is)g(a)g(con)n(tin)n(uous)f(and)h(monotonically)e(increasing) 523 1247 y(function)38 b(of)f Fn(m)1035 1217 y Fh(X)1035 1267 y Fv(2)1098 1247 y Fw(.)g(Th)n(us,)g(the)h(in\014m)n(um)f(and)g (the)h(suprem)n(um)f(of)g Fn(m)2803 1210 y Fh(Z)2848 1218 y Fg(N)2803 1269 y Fv(2)2943 1247 y Fw(are)f(giv)n(en)g(b)n(y)523 1346 y(ev)-5 b(aluating)28 b Fn(m)996 1309 y Fh(Z)1041 1317 y Fg(N)996 1369 y Fv(2)1128 1346 y Fw(at)g(the)i(in\014m)n(um)f (and)g(the)g(suprem)n(um,)g(resp)r(ectiv)n(ely)-7 b(,)28 b(of)g Fn(m)3075 1316 y Fh(X)3075 1367 y Fv(2)3138 1346 y Fw(.)h(When)523 1446 y Fn(m)596 1416 y Fh(X)596 1467 y Fv(2)682 1446 y Fk(!)23 b Fw(2,)k Fn(m)953 1409 y Fh(Z)998 1417 y Fg(N)953 1468 y Fv(2)1079 1446 y Fk(!)1195 1413 y Fh(N)6 b Fv(+2)p 1195 1427 V 1195 1475 a Fh(N)g Fv(+1)1348 1446 y Fw(.)28 b(When)g Fn(m)1714 1416 y Fh(X)1714 1467 y Fv(2)1800 1446 y Fk(!)23 b(1)p Fw(,)28 b Fn(m)2113 1409 y Fh(Z)2158 1417 y Fg(N)2113 1468 y Fv(2)2238 1446 y Fk(!)2355 1413 y Fh(N)6 b Fv(+1)p 2355 1427 V 2397 1475 a Fh(N)2507 1446 y Fw(.)820 b Fk(u)-55 b(t)523 1711 y Fp(3)112 b(A)37 b(Simple)f(Closed-F)-9 b(orm)37 b(Solution)523 1901 y Fw(Theorem)31 b(1)g(implies)h(that)g(the)g(parameter)f Fn(\025)2018 1913 y Fh(Y)2107 1901 y Fw(of)h(the)g(EC)g(distribution)f (can)h(b)r(e)g(\014xed)523 1992 y(without)c(excluding)f(the)h (distributions)f(of)g(lo)n(w)n(est)g(v)-5 b(ariabilit)n(y)26 b(from)h(the)h(set)f(of)g(EC)g(dis-)523 2083 y(tributions.)h(In)f(the)h (rest)g(of)f(the)h(pap)r(er,)f(w)n(e)g(constrain)g Fn(\025)2363 2095 y Fh(Y)2449 2083 y Fw(as)f(follo)n(ws:)1609 2274 y Fq(\025)1654 2282 y Fi(Y)1728 2274 y Fu(=)2011 2226 y(1)p 1819 2257 422 4 v 1819 2327 a(\()p Fq(m)1917 2301 y Fi(X)1917 2346 y Fj(2)1992 2327 y Ft(\000)17 b Fu(1\))p Fq(\026)2183 2301 y Fi(X)2183 2346 y Fj(1)2251 2274 y Fq(;)996 b Fu(\(11\))562 2487 y Fw(and)38 b(deriv)n(e)g(closed-form)g (represen)n(tations)e(of)j(the)g(remaining)f(free)h(parameters)e(\()p Fn(n)p Fw(,)523 2586 y Fn(p)p Fw(,)d Fn(\025)670 2598 y Fh(X)5 b Fv(1)767 2586 y Fw(,)35 b Fn(\025)873 2598 y Fh(X)5 b Fv(2)969 2586 y Fw(,)35 b Fn(p)1069 2598 y Fh(X)1132 2586 y Fw(\),)g(where)f(these)g(free)g(parameters)f(will)i (determine)f Fn(m)2922 2556 y Fh(X)2922 2607 y Fv(2)3020 2586 y Fw(and)g Fn(\026)3238 2556 y Fh(X)3238 2607 y Fv(1)3336 2586 y Fw(in)523 2686 y(\(11\).)29 b(Ob)n(viously)-7 b(,)29 b(at)g(least)g(three)g(degrees)g(of)g(freedom)g(are)g(necessary) e(to)j(matc)n(h)f(three)523 2786 y(momen)n(ts.)j(As)g(w)n(e)f(will)h (see,)g(the)g(additional)g(degrees)e(of)i(freedom)g(allo)n(w)f(us)h(to) f(accept)523 2885 y(all)39 b(input)g(distributions)g(in)g Fk(P)7 b(H)1622 2897 y Fv(3)1660 2885 y Fw(,)38 b(use)h(a)f(smaller)g (n)n(um)n(b)r(er)h(of)g(phases,)f(and)g(ac)n(hiev)n(e)523 2985 y(n)n(umerical)27 b(stabilit)n(y)-7 b(.)648 3084 y(W)g(e)31 b(in)n(tro)r(duce)f(the)h(follo)n(wing)e(sets)i(of)f (distributions)h(to)f(describ)r(e)g(the)h(closed-form)523 3184 y(solutions)c(compactly:)523 3333 y Fl(De\014nition)k(7.)41 b Fm(L)l(et)27 b Fk(U)1278 3345 y Fh(i)1305 3333 y Fm(,)h Fk(M)1458 3345 y Fh(i)1486 3333 y Fm(,)g(and)g Fk(L)g Fm(b)l(e)g(the)f(sets)h(of)g(distributions)g(de\014ne)l(d)g(as)g(fol)t (lows:)1086 3522 y Ft(U)1134 3530 y Fj(0)1192 3522 y Fu(=)1274 3432 y Fe(n)1330 3522 y Fq(F)1390 3429 y Fe(\014)1390 3479 y(\014)1390 3528 y(\014)1417 3522 y Fq(m)1485 3486 y Fi(F)1485 3535 y Fj(2)1557 3522 y Fq(>)21 b Fu(2)28 b Fc(and)g Fq(m)1921 3486 y Fi(F)1921 3535 y Fj(3)1993 3522 y Fq(>)21 b Fu(2)p Fq(m)2180 3486 y Fi(F)2180 3535 y Fj(2)2248 3522 y Ft(\000)c Fu(1)2363 3432 y Fe(o)2431 3522 y Fq(;)1094 3709 y Ft(U)1142 3717 y Fi(i)1192 3709 y Fu(=)1274 3619 y Fe(n)1330 3709 y Fq(F)1390 3616 y Fe(\014)1390 3665 y(\014)1390 3715 y(\014)1427 3661 y Fq(i)h Fu(+)e(2)p 1427 3692 159 4 v 1427 3760 a Fq(i)i Fu(+)e(1)1617 3709 y Fq(<)21 b(m)1766 3673 y Fi(F)1766 3722 y Fj(2)1838 3709 y Fq(<)1929 3661 y(i)c Fu(+)g(1)p 1929 3692 V 1995 3760 a Fq(i)2125 3709 y Fc(and)28 b Fq(m)2342 3673 y Fi(F)2342 3722 y Fj(3)2414 3709 y Fq(>)21 b Fu(2)p Fq(m)2601 3673 y Fi(F)2601 3722 y Fj(2)2669 3709 y Ft(\000)16 b Fu(1)2783 3619 y Fe(o)2852 3709 y Fq(;)1042 3895 y Ft(M)1134 3903 y Fj(0)1192 3895 y Fu(=)1274 3805 y Fe(n)1330 3895 y Fq(F)1390 3802 y Fe(\014)1390 3852 y(\014)1390 3902 y(\014)1417 3895 y Fq(m)1485 3859 y Fi(F)1485 3909 y Fj(2)1557 3895 y Fq(>)21 b Fu(2)28 b Fc(and)g Fq(m)1921 3859 y Fi(F)1921 3909 y Fj(3)1993 3895 y Fu(=)21 b(2)p Fq(m)2180 3859 y Fi(F)2180 3909 y Fj(2)2248 3895 y Ft(\000)c Fu(1)2363 3805 y Fe(o)2431 3895 y Fq(;)1050 4082 y Ft(M)1142 4090 y Fi(i)1192 4082 y Fu(=)1274 3992 y Fe(n)1330 4082 y Fq(F)1390 3989 y Fe(\014)1390 4039 y(\014)1390 4088 y(\014)1427 4034 y Fq(i)h Fu(+)e(2)p 1427 4065 V 1427 4133 a Fq(i)i Fu(+)e(1)1617 4082 y Fq(<)21 b(m)1766 4046 y Fi(F)1766 4095 y Fj(2)1838 4082 y Fq(<)1929 4034 y(i)c Fu(+)g(1)p 1929 4065 V 1995 4133 a Fq(i)2125 4082 y Fc(and)28 b Fq(m)2342 4046 y Fi(F)2342 4095 y Fj(3)2414 4082 y Fu(=)21 b(2)p Fq(m)2601 4046 y Fi(F)2601 4095 y Fj(2)2669 4082 y Ft(\000)16 b Fu(1)2783 3992 y Fe(o)2852 4082 y Fq(;)1115 4269 y Ft(L)24 b Fu(=)1274 4178 y Fe(n)1330 4269 y Fq(F)1390 4175 y Fe(\014)1390 4225 y(\014)1390 4275 y(\014)1417 4269 y Fq(m)1485 4233 y Fi(F)1485 4282 y Fj(2)1557 4269 y Fq(>)d Fu(1)28 b Fc(and)g Fq(m)1921 4233 y Fi(F)1921 4282 y Fj(2)1993 4269 y Fq(<)21 b(m)2142 4233 y Fi(F)2142 4282 y Fj(3)2214 4269 y Fq(<)g Fu(2)p Fq(m)2401 4233 y Fi(F)2401 4282 y Fj(2)2469 4269 y Ft(\000)16 b Fu(1)2583 4178 y Fe(o)2652 4269 y Fq(;)546 4469 y Fm(for)23 b(nonne)l(gative)g(inte)l (gers)g Fn(i)p Fm(.)g(A)n(lso,)g(let)g Fk(U)1874 4439 y Fv(+)1952 4469 y Fw(=)g Fk([)2095 4439 y Fd(1)2095 4490 y Fh(i)p Fv(=1)2207 4469 y Fk(U)2259 4481 y Fh(i)2286 4469 y Fm(,)g Fk(M)2434 4439 y Fv(+)2512 4469 y Fw(=)g Fk([)2655 4439 y Fd(1)2655 4490 y Fh(i)p Fv(=1)2767 4469 y Fk(M)2867 4481 y Fh(i)2894 4469 y Fm(,)g Fk(U)32 b Fw(=)22 b Fk(U)3165 4481 y Fv(0)3206 4469 y Fk([)s(U)3324 4439 y Fv(+)3379 4469 y Fm(,)523 4568 y(and)30 b Fk(M)23 b Fw(=)g Fk(M)995 4580 y Fv(0)1050 4568 y Fk([)c(M)1224 4538 y Fv(+)1279 4568 y Fm(.)523 4725 y Fw(These)30 b(sets)g(are)f (illustrated)g(in)i(Figure)e(4.)h(The)g(next)g(theorem)g(pro)n(vides)e (the)j(in)n(tuition)523 4825 y(b)r(ehind)e(the)g(sets)g Fk(U)8 b Fw(,)29 b Fk(M)p Fw(,)f(and)g Fk(L)p Fw(;)h(namely)-7 b(,)29 b(for)f(all)g(distributions)g Fn(X)7 b Fw(,)28 b(the)h(distributions)523 4924 y Fn(X)34 b Fw(and)27 b Fn(A)p Fw(\()p Fn(X)7 b Fw(\))28 b(are)f(in)h(the)g(same)f (classi\014cation)f(region)g(\(Figure)h(4\).)p eop %%Page: 10 10 10 9 bop 1455 365 a 8028005 7104430 0 0 29536010 26115358 startTexFig 1455 365 a %%BeginDocument: region2.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: region2.eps %%Creator: fig2dev Version 3.2 Patchlevel 3d %%CreationDate: Thu Jun 19 09:36:16 2003 %%For: osogami@gs57.sp.cs.cmu.edu (Takayuki Osogami) %%BoundingBox: 0 0 449 397 %%Magnification: 1.0000 %%EndComments /MyAppDict 100 dict dup begin def /$F2psDict 200 dict def F2psDictbeginF2psDict begin F2psDictbeginF2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save newpath 0 397 moveto 0 0 lineto 449 0 lineto 449 397 lineto closepath clip newpath -31.5 466.2 translate 1 -1 scale % This junk string is used by the show operators /PATsstr 1 string def /PATawidthshow { % cx cy cchar rx ry string % Loop over each character in the string { % cx cy cchar rx ry char % Show the character dup % cx cy cchar rx ry char char PATsstr dup 0 4 -1 roll put % cx cy cchar rx ry char (char) false charpath % cx cy cchar rx ry char /clip load PATdraw % Move past the character (charpath modified the % current point) currentpoint % cx cy cchar rx ry char x y newpath moveto % cx cy cchar rx ry char % Reposition by cx,cy if the character in the string is cchar 3 index eq { % cx cy cchar rx ry 4 index 4 index rmoveto } if % Reposition all characters by rx ry 2 copy rmoveto % cx cy cchar rx ry } forall pop pop pop pop pop % - currentpoint newpath moveto } bind def /PATcg { 7 dict dup begin /lw currentlinewidth def /lc currentlinecap def /lj currentlinejoin def /ml currentmiterlimit def /ds [ currentdash ] def /cc [ currentrgbcolor ] def /cm matrix currentmatrix def end } bind def % PATdraw - calculates the boundaries of the object and % fills it with the current pattern /PATdraw { % proc save exch PATpcalc % proc nw nh px py 5 -1 roll exec % nw nh px py newpath PATfill % - restore } bind def % PATfill - performs the tiling for the shape /PATfill { % nw nh px py PATfill - PATDict /CurrentPattern get dup begin setfont % Set the coordinate system to Pattern Space PatternGState PATsg % Set the color for uncolored pattezns PaintType 2 eq { PATDict /PColor get PATsc } if % Create the string for showing 3 index string % nw nh px py str % Loop for each of the pattern sources 0 1 Multi 1 sub { % nw nh px py str source % Move to the starting location 3 index 3 index % nw nh px py str source px py moveto % nw nh px py str source % For multiple sources, set the appropriate color Multi 1 ne { dup PC exch get PATsc } if % Set the appropriate string for the source 0 1 7 index 1 sub { 2 index exch 2 index put } for pop % Loop over the number of vertical cells 3 index % nw nh px py str nh { % nw nh px py str currentpoint % nw nh px py str cx cy 2 index oldshow % nw nh px py str cx cy YStep add moveto % nw nh px py str } repeat % nw nh px py str } for 5 { pop } repeat end } bind def % PATkshow - kshow with the current pattezn /PATkshow { % proc string exch bind % string proc 1 index 0 get % string proc char % Loop over all but the last character in the string 0 1 4 index length 2 sub { % string proc char idx % Find the n+1th character in the string 3 index exch 1 add get % string proe char char+1 exch 2 copy % strinq proc char+1 char char+1 char % Now show the nth character PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr) false charpath % string proc char+1 char char+1 /clip load PATdraw % Move past the character (charpath modified the current point) currentpoint newpath moveto % Execute the user proc (should consume char and char+1) mark 3 1 roll % string proc char+1 mark char char+1 4 index exec % string proc char+1 mark... cleartomark % string proc char+1 } for % Now display the last character PATsstr dup 0 4 -1 roll put % string proc (char+1) false charpath % string proc /clip load PATdraw neewath pop pop % - } bind def % PATmp - the makepattern equivalent /PATmp { % patdict patmtx PATmp patinstance exch dup length 7 add % We will add 6 new entries plus 1 FID dict copy % Create a new dictionary begin % Matrix to install when painting the pattern TilingType PATtcalc /PatternGState PATcg def PatternGState /cm 3 -1 roll put % Check for multi pattern sources (Level 1 fast color patterns) currentdict /Multi known not { /Multi 1 def } if % Font dictionary definitions /FontType 3 def % Create a dummy encoding vector /Encoding 256 array def 3 string 0 1 255 { Encoding exch dup 3 index cvs cvn put } for pop /FontMatrix matrix def /FontBBox BBox def /BuildChar { mark 3 1 roll % mark dict char exch begin Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata] PaintType 2 eq Multi 1 ne or { XStep 0 FontBBox aload pop setcachedevice } { XStep 0 setcharwidth } ifelse currentdict % mark [paintdata] dict /PaintProc load % mark [paintdata] dict paintproc end gsave false PATredef exec true PATredef grestore cleartomark % - } bind def currentdict end % newdict /foo exch % /foo newlict definefont % newfont } bind def % PATpcalc - calculates the starting point and width/height % of the tile fill for the shape /PATpcalc { % - PATpcalc nw nh px py PATDict /CurrentPattern get begin gsave % Set up the coordinate system to Pattern Space % and lock down pattern PatternGState /cm get setmatrix BBox aload pop pop pop translate % Determine the bounding box of the shape pathbbox % llx lly urx ury grestore % Determine (nw, nh) the # of cells to paint width and height PatHeight div ceiling % llx lly urx qh 4 1 roll % qh llx lly urx PatWidth div ceiling % qh llx lly qw 4 1 roll % qw qh llx lly PatHeight div floor % qw qh llx ph 4 1 roll % ph qw qh llx PatWidth div floor % ph qw qh pw 4 1 roll % pw ph qw qh 2 index sub cvi abs % pw ph qs qh-ph exch 3 index sub cvi abs exch % pw ph nw=qw-pw nh=qh-ph % Determine the starting point of the pattern fill %(px, py) 4 2 roll % nw nh pw ph PatHeight mul % nw nh pw py exch % nw nh py pw PatWidth mul exch % nw nh px py end } bind def % Save the original routines so that we can use them later on /oldfill /fill load def /oldeofill /eofill load def /oldstroke /stroke load def /oldshow /show load def /oldashow /ashow load def /oldwidthshow /widthshow load def /oldawidthshow /awidthshow load def /oldkshow /kshow load def % These defs are necessary so that subsequent procs don't bind in % the originals /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def /PATredef { MyAppDict begin { /fill { /clip load PATdraw newpath } bind def /eofill { /eoclip load PATdraw newpath } bind def /stroke { PATstroke } bind def /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def /ashow { 0 0 null 6 3 roll PATawidthshow } bind def /widthshow { 0 0 3 -1 roll PATawidthshow } bind def /awidthshow { PATawidthshow } bind def /kshow { PATkshow } bind def } { /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def } ifelse end } bind def false PATredef % Conditionally define setcmykcolor if not available /setcmykcolor where { pop } { /setcmykcolor { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor - pop } bind def } ifelse /PATsc { % colorarray aload length % c1 ... cn length dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /PATsg { % dict begin lw setlinewidth lc setlinecap lj setlinejoin ml setmiterlimit ds aload pop setdash cc aload pop setrgbcolor cm setmatrix end } bind def /PATDict 3 dict def /PATsp { true PATredef PATDict begin /CurrentPattern exch def % If it's an uncolored pattern, save the color CurrentPattern /PaintType get 2 eq { /PColor exch def } if /CColor [ currentrgbcolor ] def end } bind def % PATstroke - stroke with the current pattern /PATstroke { countdictstack save mark { currentpoint strokepath moveto PATpcalc % proc nw nh px py clip newpath PATfill } stopped { (*** PATstroke Warning: Path is too complex, stroking with gray) = cleartomark restore countdictstack exch sub dup 0 gt { { end } repeat } { pop } ifelse gsave 0.5 setgray oldstroke grestore } { pop restore pop } ifelse newpath } bind def /PATtcalc { % modmtx tilingtype PATtcalc tilematrix % Note: tiling types 2 and 3 are not supported gsave exch concat % tilingtype matrix currentmatrix exch % cmtx tilingtype % Tiling type 1 and 3: constant spacing 2 ne { % Distort the pattern so that it occupies % an integral number of device pixels dup 4 get exch dup 5 get exch % tx ty cmtx XStep 0 dtransform round exch round exch % tx ty cmtx dx.x dx.y XStep div exch XStep div exch % tx ty cmtx a b 0 YStep dtransform round exch round exch % tx ty cmtx a b dy.x dy.y YStep div exch YStep div exch % tx ty cmtx a b c d 7 -3 roll astore % { a b c d tx ty } } if grestore } bind def /PATusp { false PATredef PATDict begin CColor PATsc end } bind def % left45 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 32 32 true [ 32 0 0 -32 0 32 ] {<808080804040404020202020101010100808080804040404 020202020101010180808080404040402020202010101010 080808080404040402020202010101018080808040404040 202020201010101008080808040404040202020201010101 808080804040404020202020101010100808080804040404 0202020201010101>} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P4 exch def % horizontal sawtooth lines 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 32 16 true [ 32 0 0 -16 0 16 ] {<000000000000000000000000000000000000000000000000 000000000100010002800280044004400820082010101010 20082008400440048002800200010001>} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P21 exch def % vertical sawtooth lines 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 16 32 true [ 16 0 0 -32 0 32 ] {<400020001000080004000200010000800100020004000800 100020004000800040002000100008000400020001000080 01000200040008001000200040008000>} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P22 exch def /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def F2psBegin10setmiterlimit0.060000.06000scF2psBegin 10 setmiterlimit 0.06000 0.06000 sc % % Fig objects follow % % Polyline n 7200 1200 m 1200 7200 l 7200 4200 l cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P21 [16 0 0 -8 80.00 80.00] PATmp PATsp ef gr PATusp % Polyline n 1200 1200 m 1200 7200 l 4200 4200 l 4200 1200 l cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P22 [8 0 0 -16 80.00 80.00] PATmp PATsp ef gr PATusp % Polyline n 4200 1200 m 4200 4200 l 7200 1200 l 4200 1200 l cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P4 [16 0 0 -16 280.00 80.00] PATmp PATsp ef gr PATusp % Polyline 30.000 slw gs clippath 1290 1185 m 1110 1185 l 1110 1626 l 1200 1266 l 1290 1626 l cp eoclip n 1200 7200 m 1200 1200 l gs col0 s gr gr % arrowhead n 1290 1626 m 1200 1266 l 1110 1626 l col0 s % Polyline gs clippath 7215 7290 m 7215 7110 l 6774 7110 l 7134 7200 l 6774 7290 l cp eoclip n 1200 7200 m 7200 7200 l gs col0 s gr gr % arrowhead n 6774 7290 m 7134 7200 l 6774 7110 l col0 s % Polyline n 4200 1200 m 4200 4200 l gs col0 s gr % Polyline n 1200 7200 m 7200 1200 l gs col0 s gr % Polyline n 4200 7125 m 4200 7275 l gs col0 s gr % Polyline n 1125 4200 m 1275 4200 l gs col0 s gr % Polyline 60.000 slw [15 90] 90 sd n 1230 7230 m 1230 1830 l gs col0 s gr [] 0 sd % Polyline [15 90] 90 sd n 1200 7200 m 7200 4200 l gs col0 s gr [] 0 sd % Polyline 30.000 slw n 2175 1275 m 2175 6225 l gs col0 s gr % Polyline n 2700 1275 m 2700 5700 l gs col0 s gr % Polyline n 4200 4200 m 7200 2175 l gs col0 s gr % Polyline n 2700 5700 m 7200 2775 l gs col0 s gr % Polyline n 2169 6217 m 7200 3150 l gs col0 s gr % Polyline 2 slj 15.000 slw gs clippath 5436 2953 m 5522 2925 l 5452 2708 l 5465 2894 l 5367 2736 l cp eoclip n 5700 2325 m 5698 2326 l 5695 2330 l 5689 2335 l 5680 2343 l 5668 2354 l 5655 2368 l 5639 2383 l 5623 2400 l 5606 2418 l 5588 2438 l 5571 2459 l 5554 2481 l 5537 2505 l 5520 2532 l 5504 2561 l 5489 2592 l 5475 2625 l 5463 2662 l 5454 2697 l 5449 2728 l 5446 2757 l 5446 2783 l 5447 2807 l 5450 2829 l 5454 2850 l 5458 2869 l 5463 2886 l 5467 2900 l 5475 2925 l gs col0 s gr gr % arrowhead 0 slj n 5367 2736 m 5465 2894 l 5452 2708 l 5367 2736 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj gs clippath 3336 5053 m 3422 5025 l 3352 4808 l 3365 4994 l 3267 4836 l cp eoclip n 3600 4425 m 3598 4426 l 3595 4430 l 3589 4435 l 3580 4443 l 3568 4454 l 3555 4468 l 3539 4483 l 3523 4500 l 3506 4518 l 3488 4538 l 3471 4559 l 3454 4581 l 3437 4605 l 3420 4632 l 3404 4661 l 3389 4692 l 3375 4725 l 3363 4762 l 3354 4797 l 3349 4828 l 3346 4857 l 3346 4883 l 3347 4907 l 3350 4929 l 3354 4950 l 3358 4969 l 3363 4986 l 3367 5000 l 3375 5025 l gs col0 s gr gr % arrowhead 0 slj n 3267 4836 m 3365 4994 l 3352 4808 l 3267 4836 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj gs clippath 2361 5953 m 2447 5925 l 2377 5708 l 2390 5894 l 2292 5736 l cp eoclip n 2625 5325 m 2623 5326 l 2620 5330 l 2614 5335 l 2605 5343 l 2593 5354 l 2580 5368 l 2564 5383 l 2548 5400 l 2531 5418 l 2513 5438 l 2496 5459 l 2479 5481 l 2462 5505 l 2445 5532 l 2429 5561 l 2414 5592 l 2400 5625 l 2388 5662 l 2379 5697 l 2374 5728 l 2371 5757 l 2371 5783 l 2372 5807 l 2375 5829 l 2379 5850 l 2383 5869 l 2388 5886 l 2392 5900 l 2400 5925 l gs col0 s gr gr % arrowhead 0 slj n 2292 5736 m 2390 5894 l 2377 5708 l 2292 5736 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj gs clippath 6967 3132 m 7056 3143 l 7084 2917 l 7018 3091 l 6994 2906 l cp eoclip n 7425 2775 m 7423 2776 l 7419 2777 l 7412 2779 l 7401 2783 l 7388 2788 l 7371 2793 l 7352 2800 l 7332 2808 l 7311 2816 l 7289 2826 l 7267 2836 l 7244 2847 l 7221 2859 l 7197 2873 l 7173 2889 l 7149 2906 l 7125 2925 l 7098 2950 l 7076 2974 l 7059 2996 l 7046 3016 l 7037 3035 l 7029 3052 l 7024 3069 l 7020 3084 l 7017 3098 l 7014 3123 l gs col0 s gr gr % arrowhead 0 slj n 6994 2906 m 7018 3091 l 7084 2917 l 6994 2906 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj gs clippath 6952 2652 m 7041 2663 l 7069 2437 l 7003 2611 l 6979 2426 l cp eoclip n 7410 2295 m 7408 2296 l 7404 2297 l 7397 2299 l 7386 2303 l 7373 2308 l 7356 2313 l 7337 2320 l 7317 2328 l 7296 2336 l 7274 2346 l 7252 2356 l 7229 2367 l 7206 2379 l 7182 2393 l 7158 2409 l 7134 2426 l 7110 2445 l 7083 2470 l 7061 2494 l 7044 2516 l 7031 2536 l 7022 2555 l 7014 2572 l 7009 2589 l 7005 2604 l 7002 2618 l 6999 2643 l gs col0 s gr gr % arrowhead 0 slj n 6979 2426 m 7003 2611 l 7069 2437 l 6979 2426 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj gs clippath 6967 1932 m 7056 1943 l 7084 1717 l 7018 1891 l 6994 1706 l cp eoclip n 7425 1575 m 7423 1576 l 7419 1577 l 7412 1579 l 7401 1583 l 7388 1588 l 7371 1593 l 7352 1600 l 7332 1608 l 7311 1616 l 7289 1626 l 7267 1636 l 7244 1647 l 7221 1659 l 7197 1673 l 7173 1689 l 7149 1706 l 7125 1725 l 7098 1750 l 7076 1774 l 7059 1796 l 7046 1816 l 7037 1835 l 7029 1852 l 7024 1869 l 7020 1884 l 7017 1898 l 7014 1923 l gs col0 s gr gr % arrowhead 0 slj n 6994 1706 m 7018 1891 l 7084 1717 l 6994 1706 l cp gs 0.00 setgray ef gr col0 s /Times-Roman ff 420.00 scf sf 7020 7770 m gs 1 -1 sc (2) col0 sh gr /Times-Roman ff 420.00 scf sf 945 1665 m gs 1 -1 sc (3) col0 sh gr /Times-Roman ff 540.00 scf sf 3225 2550 m gs 1 -1 sc (U) col0 sh gr /Times-Roman ff 390.00 scf sf 3600 2625 m gs 1 -1 sc (1) col0 sh gr /Times-Roman ff 390.00 scf sf 2550 3000 m gs 1 -1 sc (2) col0 sh gr /Times-Roman ff 540.00 scf sf 4500 2175 m gs 1 -1 sc (U) col0 sh gr /Times-Roman ff 390.00 scf sf 4875 2250 m gs 1 -1 sc (0) col0 sh gr /Times-Roman ff 540.00 scf sf 5475 2250 m gs 1 -1 sc (M) col0 sh gr /Times-Roman ff 390.00 scf sf 5925 2325 m gs 1 -1 sc (0) col0 sh gr /Times-Roman ff 540.00 scf sf 3450 4350 m gs 1 -1 sc (M) col0 sh gr /Times-Roman ff 390.00 scf sf 3900 4350 m gs 1 -1 sc (1) col0 sh gr /Times-Roman ff 540.00 scf sf 2400 5250 m gs 1 -1 sc (M) col0 sh gr /Times-Roman ff 390.00 scf sf 2850 5325 m gs 1 -1 sc (2) col0 sh gr /Times-Roman ff 540.00 scf sf 7425 1800 m gs 1 -1 sc (L) col0 sh gr /Times-Roman ff 390.00 scf sf 7800 1875 m gs 1 -1 sc (0) col0 sh gr /Times-Roman ff 540.00 scf sf 7425 2475 m gs 1 -1 sc (L) col0 sh gr /Times-Roman ff 390.00 scf sf 7800 2550 m gs 1 -1 sc (1) col0 sh gr /Times-Roman ff 540.00 scf sf 7425 3000 m gs 1 -1 sc (L) col0 sh gr /Times-Roman ff 390.00 scf sf 7800 3075 m gs 1 -1 sc (2) col0 sh gr /Times-Roman ff 540.00 scf sf 2175 2925 m gs 1 -1 sc (U) col0 sh gr /Times-Roman ff 540.00 scf sf 6600 7650 m gs 1 -1 sc (m) col0 sh gr /Times-Roman ff 540.00 scf sf 525 1500 m gs 1 -1 sc (m) col0 sh gr /Times-Roman ff 540.00 scf sf 1200 7725 m gs 1 -1 sc (1) col0 sh gr /Times-Roman ff 540.00 scf sf 4125 7725 m gs 1 -1 sc (2) col0 sh gr /Times-Roman ff 540.00 scf sf 825 7275 m gs 1 -1 sc (1) col0 sh gr /Times-Roman ff 540.00 scf sf 825 4350 m gs 1 -1 sc (3) col0 sh gr F2psBegin10setmiterlimit0.060000.06000scF2psEnd rs end %%EndDocument endTexFig 523 1440 a Fr(Fig.)15 b(4.)28 b Fu(A)g(classi\014cation)j(of)e (distributions.)g(The)g(dotted)f(lines)h(delineate)h(the)e(set)h(of)g (all)h(non-)523 1531 y(negativ)n(e)c(distributions)g Fq(G)g Fu(\()p Fq(m)1459 1499 y Fi(G)1459 1545 y Fj(3)1531 1531 y Ft(\025)21 b Fq(m)1680 1499 y Fi(G)1680 1545 y Fj(2)1753 1531 y Ft(\025)g Fu(1\).)523 1802 y Fl(Lemma)30 b(1.)40 b Fm(L)l(et)i Fn(Z)1190 1814 y Fh(N)1298 1802 y Fw(=)i Fn(A)1469 1772 y Fh(N)1532 1802 y Fw(\()p Fn(X)7 b Fw(\))42 b Fm(for)h(inte)l(gers)f Fn(N)54 b Fk(\025)44 b Fw(1)p Fm(.)e(If)g Fn(X)52 b Fk(2)45 b(U)50 b Fm(\(r)l(esp)l(e)l (ctively,)523 1902 y Fn(X)29 b Fk(2)24 b(M)p Fm(,)k Fn(X)i Fk(2)23 b(L)p Fm(\),)29 b(then)f Fn(Z)1415 1914 y Fh(N)1501 1902 y Fk(2)23 b(U)37 b Fm(\(r)l(esp)l(e)l(ctively,)30 b Fn(Z)2217 1914 y Fh(N)2302 1902 y Fk(2)24 b(M)p Fm(,)k Fn(Z)2591 1914 y Fh(N)2677 1902 y Fk(2)23 b(L)p Fm(\))29 b(for)g(al)t(l)g Fn(N)j Fk(\025)23 b Fw(1)p Fm(.)523 2043 y(Pr)l(o)l(of.)43 b Fw(W)-7 b(e)29 b(pro)n(v)n(e)d(the)i(case)f (when)i Fn(N)j Fw(=)23 b(1.)28 b(The)g(theorem)f(then)i(follo)n(ws)e(b) n(y)h(induction.)523 2134 y(Let)g Fn(Z)g Fw(=)23 b Fn(A)p Fw(\()p Fn(X)7 b Fw(\).)28 b(By)f(\(2\),)h Fn(m)1458 2104 y Fh(X)1458 2155 y Fv(2)1544 2134 y Fw(=)1720 2102 y Fv(1)p 1642 2116 191 4 v 1642 2167 a(2)p Fd(\000)p Fh(m)1786 2147 y Fg(Z)1786 2187 y Ff(2)1842 2134 y Fw(,)f(and)935 2360 y Fq(m)1003 2324 y Fi(Z)1003 2373 y Fj(3)1075 2360 y Fu(=)c(\(resp)r(ectiv)n(ely)q Fq(;)f(<;)39 b Fu(and)46 b Fq(>)p Fu(\))2184 2312 y(2)p Fq(m)2290 2280 y Fi(X)2290 2326 y Fj(2)2365 2312 y Ft(\000)17 b Fu(1)p 2092 2343 482 4 v 2092 2413 a Fq(m)2160 2387 y Fi(X)2160 2433 y Fj(2)2217 2413 y Fu(\(2)p Fq(m)2353 2387 y Fi(X)2353 2433 y Fj(2)2428 2413 y Ft(\000)g Fu(1\))2600 2360 y(+)g(3)2725 2312 y Fq(m)2793 2280 y Fi(X)2793 2326 y Fj(2)2868 2312 y Ft(\000)g Fu(1)p 2725 2343 258 4 v 2791 2413 a Fq(m)2859 2387 y Fi(X)2859 2433 y Fj(2)1075 2536 y Fu(=)23 b(\(resp)r(ectiv)n (ely)q Fq(;)f(<;)39 b Fu(and)46 b Fq(>)p Fu(\))77 b(2)p Fq(m)2188 2500 y Fi(Z)2188 2549 y Fj(2)2254 2536 y Ft(\000)17 b Fu(1)p Fq(;)551 2686 y Fw(where)27 b(the)h(last)f(equalit)n(y)g (follo)n(ws)g(from)g Fn(m)1949 2656 y Fh(X)1949 2707 y Fv(2)2035 2686 y Fw(=)2211 2654 y Fv(1)p 2133 2668 191 4 v 2133 2719 a(2)p Fd(\000)p Fh(m)2277 2699 y Fg(Z)2277 2739 y Ff(2)2332 2686 y Fw(.)995 b Fk(u)-55 b(t)648 2855 y Fw(By)27 b(Corollary)e(1)i(and)h(Lemma)f(1,)g(it)h(follo)n(ws)f (that:)523 2993 y Fl(Corollary)32 b(2.)41 b Fm(L)l(et)29 b Fn(Z)1264 3005 y Fh(N)1351 2993 y Fw(=)23 b Fn(A)1501 2963 y Fh(N)1564 2993 y Fw(\()p Fn(X)7 b Fw(\))30 b Fm(for)h Fn(N)i Fk(\025)23 b Fw(0)p Fm(.)30 b(If)g Fn(X)g Fk(2)25 b(U)2470 3005 y Fv(0)2537 2993 y Fm(\(r)l(esp)l(e)l(ctively,)31 b Fn(X)f Fk(2)25 b(M)3309 3005 y Fv(0)3346 2993 y Fm(\),)523 3093 y(then)30 b Fn(Z)765 3105 y Fh(N)850 3093 y Fk(2)24 b(U)981 3105 y Fh(N)1073 3093 y Fm(\(r)l(esp)l(e)l(ctively,)32 b Fn(Z)1624 3105 y Fh(N)1709 3093 y Fk(2)24 b(M)1888 3105 y Fh(N)1950 3093 y Fm(\).)648 3231 y Fw(The)k(corollary)e(implies) i(that)h(for)f(all)g Fn(G)c Fk(2)h(U)2092 3243 y Fh(N)2174 3231 y Fk([)19 b(M)2348 3243 y Fh(N)2411 3231 y Fw(,)28 b Fn(G)h Fw(can)f(b)r(e)h(w)n(ell-represen)n(ted)523 3330 y(b)n(y)g(an)f(\()p Fn(N)h Fw(+)19 b(2\)-phase)28 b(EC)g(distribution)h(with)g(no)g(mass)f(probabilit)n(y)g(at)h(zero)f (\()p Fn(p)d Fw(=)g(1\),)523 3430 y(since,)34 b(for)g(all)g Fn(F)46 b Fk(2)35 b(U)1253 3442 y Fv(0)1313 3430 y Fk([)23 b(M)1491 3442 y Fv(0)1528 3430 y Fw(,)34 b Fn(F)47 b Fw(can)33 b(b)r(e)i(w)n(ell-represen)n(ted)e(b)n(y)h(t)n(w)n(o-phase)e (Co)n(xian)3351 3400 y Fv(+)523 3530 y Fw(distribution,)i(and)f Fn(Z)1229 3542 y Fh(N)1324 3530 y Fw(=)f Fn(A)1483 3499 y Fh(N)1546 3530 y Fw(\()p Fn(X)7 b Fw(\))34 b(can)f(b)r(e)g(w)n (ell-represen)n(ted)f(b)n(y)h(\(2)22 b(+)g Fn(N)9 b Fw(\)-phase)32 b(EC)523 3629 y(distribution.)j(It)f(can)g(also)g(b)r(e)g(easily)g(sho) n(wn)f(that)i(for)f(all)g Fn(G)h Fk(2)f(L)2737 3641 y Fh(N)2801 3629 y Fw(,)g Fn(G)h Fw(can)f(b)r(e)h(w)n(ell-)523 3729 y(represen)n(ted)24 b(b)n(y)h(an)h(\()p Fn(N)d Fw(+)13 b(2\)-phase)25 b(EC)g(distribution)g(with)h(nonzero)e(mass)h (probabilit)n(y)523 3828 y(at)j(zero)e(\()p Fn(p)d(<)g Fw(1\).)648 3928 y(F)-7 b(rom)35 b(these)i(prop)r(erties)e(of)h Fn(A)1660 3898 y Fh(N)1723 3928 y Fw(\()p Fn(X)7 b Fw(\),)36 b(it)h(is)f(relativ)n(ely)f(easy)g(to)h(pro)n(vide)f(a)h(closed-)523 4028 y(form)29 b(solution)f(for)h(the)h(parameters)d(\()p Fn(n)p Fw(,)i Fn(p)p Fw(,)g Fn(\025)2017 4040 y Fh(X)5 b Fv(1)2114 4028 y Fw(,)29 b Fn(\025)2214 4040 y Fh(X)5 b Fv(2)2311 4028 y Fw(,)29 b Fn(p)2405 4040 y Fh(X)2468 4028 y Fw(\))h(of)f(an)f(EC)h(distribution)g Fn(Z)523 4127 y Fw(so)19 b(that)h(a)g(giv)n(en)e(distribution)i Fn(G)g Fw(is)g(w)n(ell-represen)n(ted)e(b)n(y)h Fn(Z)6 b Fw(.)20 b(Essen)n(tially)-7 b(,)18 b(one)i(just)g(needs)523 4227 y(to)32 b(\014nd)g(an)f(appropriate)f Fn(N)41 b Fw(and)31 b(solv)n(e)g Fn(Z)k Fw(=)30 b Fn(A)2100 4197 y Fh(N)2163 4227 y Fw(\()p Fn(X)7 b Fw(\))32 b(for)f Fn(X)38 b Fw(in)32 b(terms)f(of)h(normalized)523 4327 y(momen)n(ts,)21 b(whic)n(h)g(is)h(immediate)f(since)g Fn(N)31 b Fw(is)21 b(giv)n(en)f(b)n(y)h(Corollary)e(1)i(and)g(the)h (normalized)523 4426 y(momen)n(ts)i(of)f Fn(X)30 b Fw(can)24 b(b)r(e)g(obtained)f(from)h(Theorem)f(1.)g(A)h(little)h(more)e (e\013ort)g(is)h(necessary)523 4526 y(to)k(minimize)g(the)g(n)n(um)n(b) r(er)f(of)h(phases)e(and)i(to)f(guaran)n(tee)f(n)n(umerical)h(stabilit) n(y)-7 b(.)648 4625 y(In)24 b(this)h(section,)f(w)n(e)g(giv)n(e)f(a)h (simple)h(solution,)f(whic)n(h)g(assumes)f(the)i(follo)n(wing)f(condi-) 523 4725 y(tion)j(on)f Fn(A)p Fw(:)h Fn(A)d Fk(2)f(P)7 b(H)1219 4689 y Fd(\000)1219 4747 y Fv(3)1275 4725 y Fw(,)26 b(where)h Fk(P)7 b(H)1699 4689 y Fd(\000)1699 4747 y Fv(3)1778 4725 y Fw(=)23 b Fk(U)i([)17 b(M)f([)h(L)p Fw(.)27 b(Observ)n(e)e Fk(P)7 b(H)2764 4689 y Fd(\000)2764 4747 y Fv(3)2847 4725 y Fw(includes)27 b(almost)523 4825 y(all)22 b(distributions)g(in)h Fk(P)7 b(H)1340 4837 y Fv(3)1377 4825 y Fw(.)23 b(Only)f(the)g(b)r(orders)g(b)r(et)n(w)n (een)g(the)g Fk(U)2558 4837 y Fh(i)2586 4825 y Fw('s)g(are)g(not)g (included.)h(W)-7 b(e)523 4924 y(also)27 b(analyze)f(the)i(n)n(um)n(b)r (er)f(of)h(necessary)e(phases)h(and)g(pro)n(v)n(e)f(the)i(follo)n(wing) e(theorem:)p eop %%Page: 11 11 11 10 bop 523 448 a Fl(Theorem)30 b(2.)41 b Fm(Under)34 b(the)h(simple)g(solution,)g(the)f(numb)l(er)f(of)i(phases)h(ne)l(e)l (de)l(d)e(to)g(wel)t(l-)523 548 y(r)l(epr)l(esent)40 b(any)h(distribution)g Fn(G)f Fm(by)h(an)f(EC)h(distribution)g(is)g(at) f(most)g Fn(O)r(P)12 b(T)g Fw(\()p Fn(G)p Fw(\))26 b(+)g(2)p Fm(.)523 848 y Fo(The)37 b(Close)-5 b(d-F)e(orm)37 b(Solution:)42 b Fw(The)30 b(solution)f(di\013ers)h(according)d(to)j(the)g (classi\014ca-)523 947 y(tion)37 b(of)g(the)h(input)g(distribution)f Fn(G)p Fw(.)g(When)h Fn(G)h Fk(2)h(U)2281 959 y Fv(0)2343 947 y Fk([)25 b(M)2523 959 y Fv(0)2560 947 y Fw(,)37 b(a)g(t)n(w)n(o-phase)e(Co)n(xian)3351 917 y Fv(+)523 1047 y Fw(distribution)g(su\016ces)f(to)h(matc)n(h)f(the)h(\014rst)g (three)f(momen)n(ts.)h(When)g Fn(G)g Fk(2)g(U)3070 1017 y Fv(+)3149 1047 y Fk([)23 b(M)3327 1017 y Fv(+)3382 1047 y Fw(,)523 1146 y Fn(G)37 b Fw(is)g(w)n(ell-represen)n(ted)d(b)n (y)j(an)f(EC)g(distribution)h(with)g Fn(p)h Fw(=)g(1.)e(When)h Fn(G)i Fk(2)f(L)p Fw(,)f Fn(G)g Fw(is)523 1246 y(w)n(ell-represen)n (ted)23 b(b)n(y)i(an)f(EC)h(distribution)g(with)g Fn(p)e(<)f Fw(1.)j(F)-7 b(or)24 b(all)h(cases,)e(the)j(parameters)523 1346 y(\()p Fn(n)p Fw(,)i Fn(p)p Fw(,)f Fn(\025)796 1358 y Fh(X)5 b Fv(1)893 1346 y Fw(,)28 b Fn(\025)992 1358 y Fh(X)5 b Fv(2)1088 1346 y Fw(,)28 b Fn(p)1181 1358 y Fh(X)1244 1346 y Fw(\))g(are)f(giv)n(en)f(b)n(y)i(simple)f(closed)g (form)n(ulas.)648 1437 y(\(i\))f(If)i Fn(G)23 b Fk(2)g(U)1062 1449 y Fv(0)1118 1437 y Fk([)c(M)1292 1449 y Fv(0)p 761 1463 569 4 v 1329 1437 a Fw(,)25 b(then)h(a)f(t)n(w)n(o-phase)f(Co)n (xian)2272 1407 y Fv(+)2351 1437 y Fw(distribution)i(su\016ces)f(to)g (matc)n(h)523 1528 y(the)31 b(\014rst)g(three)f(momen)n(ts,)h(i.e.,)g Fn(p)d Fw(=)f(1)k(and)f Fn(n)e Fw(=)g(2)i(\()p Fn(N)38 b Fw(=)28 b(0\).)i(The)h(parameters)e(\()p Fn(\025)3285 1540 y Fh(X)5 b Fv(1)3382 1528 y Fw(,)523 1620 y Fn(\025)571 1632 y Fh(X)g Fv(2)668 1620 y Fw(,)27 b Fn(p)760 1632 y Fh(X)823 1620 y Fw(\))h(of)g(the)g(t)n(w)n(o-phase)e(Co)n(xian)1764 1590 y Fv(+)1845 1620 y Fw(distribution)i(are)e(c)n(hosen)h(as)g(follo) n(ws)g([25,)13 b(18]:)597 1836 y Fq(\025)642 1844 y Fi(X)t Fj(1)752 1836 y Fu(=)843 1787 y Fq(u)k Fu(+)981 1726 y Ft(p)p 1045 1726 252 4 v 61 x Fq(u)1089 1766 y Fj(2)1141 1787 y Ft(\000)f Fu(4)p Fq(v)p 843 1819 454 4 v 1001 1888 a Fu(2)p Fq(\026)1085 1863 y Fi(G)1085 1908 y Fj(1)1306 1836 y Fq(;)90 b(\025)1462 1844 y Fi(X)t Fj(2)1571 1836 y Fu(=)1662 1787 y Fq(u)17 b Ft(\000)1800 1726 y(p)p 1864 1726 252 4 v 61 x Fq(u)1908 1766 y Fj(2)1960 1787 y Ft(\000)g Fu(4)p Fq(v)p 1662 1819 454 4 v 1821 1888 a Fu(2)p Fq(\026)1905 1863 y Fi(G)1905 1908 y Fj(1)2125 1836 y Fq(;)90 b Fu(and)77 b Fq(p)2476 1844 y Fi(X)2554 1836 y Fu(=)2645 1787 y Fq(\025)2690 1795 y Fi(X)t Fj(2)2779 1787 y Fq(\026)2825 1755 y Fi(G)2825 1800 y Fj(1)2876 1787 y Fu(\()p Fq(\025)2951 1795 y Fi(X)t Fj(1)3039 1787 y Fq(\026)3085 1755 y Fi(G)3085 1800 y Fj(1)3154 1787 y Ft(\000)17 b Fu(1\))p 2645 1819 655 4 v 2857 1888 a Fq(\025)2902 1896 y Fi(X)t Fj(1)2990 1888 y Fq(\026)3036 1863 y Fi(G)3036 1908 y Fj(1)3309 1836 y Fq(;)551 2083 y Fw(where)27 b Fn(u)22 b Fw(=)1013 2041 y Fv(6)p Fd(\000)p Fv(2)p Fh(m)1190 2016 y Fg(G)1190 2057 y Ff(3)p 959 2064 333 4 v 959 2116 a Fv(3)p Fh(m)1051 2096 y Fg(G)1051 2136 y Ff(2)1099 2116 y Fd(\000)p Fv(2)p Fh(m)1243 2096 y Fg(G)1243 2136 y Ff(3)1329 2083 y Fw(and)28 b Fn(v)e Fw(=)1771 2041 y Fv(12)p Fd(\000)p Fv(6)p Fh(m)1981 2016 y Fg(G)1981 2057 y Ff(2)p 1654 2064 492 4 v 1654 2116 a Fh(m)1713 2096 y Fg(G)1713 2136 y Ff(2)1762 2116 y Fv(\(3)p Fh(m)1880 2096 y Fg(G)1880 2136 y Ff(2)1928 2116 y Fd(\000)p Fv(2)p Fh(m)2072 2096 y Fg(G)2072 2136 y Ff(3)2120 2116 y Fv(\))2156 2083 y Fw(.)648 2210 y(\(ii\))i(If)g Fn(G)23 b Fk(2)h(U)1096 2180 y Fv(+)1169 2210 y Fk([)19 b(M)1343 2180 y Fv(+)p 786 2227 613 4 v 1398 2210 a Fw(,)28 b(Corollary)d(1)i(sp)r(eci\014es)h(n)n(um)n(b)r(er,)f Fn(n)p Fw(,)h(of)f(phases)g(needed:)1200 2429 y Fq(n)d Fu(=)f(min)1494 2339 y Fe(n)1549 2429 y Fq(k)1591 2335 y Fe(\014)1591 2385 y(\014)1591 2435 y(\014)1619 2429 y Fq(m)1687 2393 y Fi(G)1687 2442 y Fj(2)1760 2429 y Fq(>)1917 2380 y(k)p 1851 2412 175 4 v 1851 2479 a(k)c Ft(\000)e Fu(1)2035 2339 y Fe(o)2112 2429 y Fu(=)2193 2314 y Fe(\026)2317 2380 y Fq(m)2385 2349 y Fi(G)2385 2394 y Fj(2)p 2251 2412 252 4 v 2251 2481 a Fq(m)2319 2456 y Fi(G)2319 2501 y Fj(2)2388 2481 y Ft(\000)f Fu(1)2530 2429 y(+)h(1)2645 2314 y Fe(\027)2706 2429 y Fq(;)541 b Fu(\(12\))559 2685 y Fw(\()p Fn(N)47 b Fw(=)807 2593 y Fe(j)903 2642 y Fh(m)962 2617 y Fg(G)962 2659 y Ff(2)p 861 2666 193 4 v 861 2717 a Fh(m)920 2697 y Fg(G)920 2737 y Ff(2)968 2717 y Fd(\000)p Fv(1)1082 2685 y Fk(\000)18 b Fw(1)1207 2593 y Fe(k)1250 2685 y Fw(\).)37 b(Next,)f(w)n(e)g(\014nd) h(the)g(t)n(w)n(o-phase)d(Co)n(xian)2685 2655 y Fv(+)2775 2685 y Fw(distribution)j Fn(X)44 b Fk(2)523 2821 y(U)575 2833 y Fv(0)633 2821 y Fk([)20 b(M)808 2833 y Fv(0)876 2821 y Fw(suc)n(h)30 b(that)g Fn(G)h Fw(is)f(w)n(ell-represen)n(ted)f (b)n(y)h Fn(Z)6 b Fw(,)30 b(where)f Fn(Z)6 b Fw(\()p Fk(\001)p Fw(\))28 b(=)g Fn(Y)2856 2791 y Fv(\()p Fh(n)p Fd(\000)p Fv(2\))p Fd(\003)3072 2821 y Fw(\()p Fk(\001)p Fw(\))21 b Fk(\003)f Fn(X)7 b Fw(\()p Fk(\001)p Fw(\))523 2923 y(and)36 b Fn(Y)54 b Fw(is)36 b(an)g(exp)r(onen)n(tial)f (distribution)h(satisfying)f(\(1\),)i Fn(Y)2535 2893 y Fv(\()p Fh(n)p Fd(\000)p Fv(2\))p Fd(\003)2787 2923 y Fw(is)f(the)g(\()p Fn(n)24 b Fk(\000)g Fw(2\)-th)523 3026 y(con)n(v)n(olution)37 b(of)h Fn(Y)18 b Fw(,)39 b(and)f Fn(Y)1450 2996 y Fv(\()p Fh(n)p Fd(\000)p Fv(2\))p Fd(\003)1692 3026 y Fk(\003)25 b Fn(X)44 b Fw(is)39 b(the)f(con)n(v)n (olution)f(of)h Fn(Y)2747 2996 y Fv(\()p Fh(n)p Fd(\000)p Fv(2\))p Fd(\003)3002 3026 y Fw(and)g Fn(X)7 b Fw(.)37 b(T)-7 b(o)523 3118 y(shed)34 b(ligh)n(t)f(on)g(this)h(expression,)e (consider)h(i.i.d.)h(random)e(v)-5 b(ariables)33 b Fn(V)2868 3130 y Fv(1)2905 3118 y Fw(,)h(...)g Fn(V)3113 3130 y Fh(k)3188 3118 y Fw(whose)523 3224 y(distributions)23 b(are)g Fn(Y)42 b Fw(and)24 b(a)f(random)f(v)-5 b(ariable)23 b Fn(V)2108 3236 y Fh(k)q Fv(+1)2233 3224 y Fw(.)h(Then)g(random)e(v)-5 b(ariable)3101 3162 y Fe(P)3189 3182 y Fh(k)q Fv(+1)3189 3249 y Fh(t)p Fv(=1)3327 3224 y Fn(V)3375 3236 y Fh(t)523 3315 y Fw(has)27 b(distribution)h Fn(Z)6 b Fw(.)27 b(By)g(Theorem)g(1,) g(this)h(can)g(b)r(e)g(ac)n(hiev)n(ed)e(b)n(y)h(setting)565 3526 y Fq(m)633 3490 y Fi(X)633 3539 y Fj(2)712 3526 y Fu(=)803 3477 y(\()p Fq(n)18 b Ft(\000)e Fu(3\))p Fq(m)1109 3445 y Fi(G)1109 3490 y Fj(2)1178 3477 y Ft(\000)h Fu(\()p Fq(n)g Ft(\000)g Fu(2\))p 803 3509 690 4 v 803 3578 a(\()p Fq(n)h Ft(\000)e Fu(2\))p Fq(m)1109 3553 y Fi(G)1109 3598 y Fj(2)1178 3578 y Ft(\000)h Fu(\()p Fq(n)g Ft(\000)g Fu(1\))1503 3526 y(;)90 b Fq(m)1682 3490 y Fi(X)1682 3539 y Fj(3)1761 3526 y Fu(=)1852 3477 y Fq(\014)t(m)1967 3446 y Fi(G)1967 3491 y Fj(3)2035 3477 y Ft(\000)17 b Fq(\013)p 1852 3509 310 4 v 1944 3578 a(m)2012 3553 y Fi(X)2012 3598 y Fj(2)2171 3526 y Fu(;)90 b Fq(\026)2328 3490 y Fi(X)2328 3539 y Fj(1)2408 3526 y Fu(=)2798 3477 y Fq(\026)2844 3446 y Fi(G)2844 3491 y Fj(1)p 2499 3509 697 4 v 2499 3578 a Fu(\()p Fq(n)17 b Ft(\000)g Fu(2\))p Fq(m)2805 3553 y Fi(X)2805 3598 y Fj(2)2880 3578 y Ft(\000)f Fu(\()p Fq(n)i Ft(\000)f Fu(3\))3205 3526 y Fq(;)42 b Fu(\(13\))551 3735 y Fw(where)732 3897 y Fq(\013)23 b Fu(=)g(\()p Fq(n)17 b Ft(\000)g Fu(2\)\()p Fq(m)1223 3861 y Fi(X)1223 3910 y Fj(2)1298 3897 y Ft(\000)g Fu(1\))1456 3832 y Fe(\000)1494 3897 y Fq(n)p Fu(\()p Fq(n)g Ft(\000)g Fu(1\)\()p Fq(m)1876 3861 y Fi(X)1876 3910 y Fj(2)1934 3897 y Fu(\))1964 3861 y Fj(2)2015 3897 y Ft(\000)g Fq(n)p Fu(\(2)p Fq(n)h Ft(\000)f Fu(5\))p Fq(m)2483 3861 y Fi(X)2483 3910 y Fj(2)2558 3897 y Fu(+)g(\()p Fq(n)g Ft(\000)g Fu(1\)\()p Fq(n)g Ft(\000)g Fu(3\))3111 3832 y Fe(\001)3162 3897 y Fq(;)734 4043 y(\014)27 b Fu(=)887 3978 y Fe(\000)925 4043 y Fu(\()p Fq(n)17 b Ft(\000)g Fu(1\))p Fq(m)1231 4007 y Fi(X)1231 4056 y Fj(2)1306 4043 y Ft(\000)g Fu(\()p Fq(n)g Ft(\000)g Fu(2\))1621 3978 y Fe(\001)c(\000)1710 4043 y Fu(\()p Fq(n)k Ft(\000)g Fu(2\))p Fq(m)2016 4007 y Fi(X)2016 4056 y Fj(2)2091 4043 y Ft(\000)g Fu(\()p Fq(n)g Ft(\000)g Fu(3\))2406 3978 y Fe(\001)2444 3994 y Fj(2)2492 4043 y Fq(:)550 4214 y Fw(Th)n(us,)26 b(w)n(e)h(set)f Fn(p)d Fw(=)g(1,)j(and)h(the)g(parameters)e(\()p Fn(\025)2088 4226 y Fh(X)5 b Fv(1)2185 4214 y Fw(,)27 b Fn(\025)2283 4226 y Fh(X)5 b Fv(2)2379 4214 y Fw(,)27 b Fn(p)2471 4226 y Fh(X)2534 4214 y Fw(\))g(of)g Fn(X)33 b Fw(are)26 b(giv)n(en)f(b)n(y)i(case)523 4313 y(\(i\),)h(using)e(the)h(\014rst)g (momen)n(t)g(and)g(the)g(normalized)f(momen)n(ts)g(of)h Fn(X)34 b Fw(sp)r(eci\014ed)27 b(b)n(y)f(\(13\).)648 4405 y(\(iii\))i(If)g Fn(G)23 b Fk(2)h(L)p 809 4421 308 4 v Fw(,)k(then)g(let)855 4619 y Fq(p)21 b Fu(=)1172 4571 y(1)p 1006 4602 371 4 v 1006 4672 a(2)p Fq(m)1112 4646 y Fi(G)1112 4691 y Fj(2)1181 4672 y Ft(\000)c Fq(m)1326 4646 y Fi(G)1326 4691 y Fj(3)1387 4619 y Fq(;)90 b(m)1566 4583 y Fi(W)1566 4632 y Fj(2)1657 4619 y Fu(=)21 b Fq(pm)1845 4583 y Fi(G)1845 4632 y Fj(2)1895 4619 y Fq(;)90 b(m)2074 4583 y Fi(W)2074 4632 y Fj(3)2165 4619 y Fu(=)21 b Fq(pm)2353 4583 y Fi(G)2353 4632 y Fj(3)2404 4619 y Fq(;)90 b Fu(and)76 b Fq(\026)2761 4583 y Fi(W)2761 4632 y Fj(1)2852 4619 y Fu(=)2943 4571 y Fq(\026)2989 4539 y Fi(G)2989 4584 y Fj(1)p 2943 4602 98 4 v 2973 4670 a Fq(p)3051 4619 y(:)196 b Fu(\(14\))543 4833 y Fn(G)21 b Fw(is)e(then)i(w)n (ell-represen)n(ted)d(b)n(y)i(distribution)g Fn(Z)6 b Fw(,)20 b(where)f Fn(Z)6 b Fw(\()p Fk(\001)p Fw(\))24 b(=)e Fn(W)12 b Fw(\()p Fk(\001)p Fw(\))p Fn(p)s Fw(+)s(1)s Fk(\000)s Fn(p)p Fw(.)20 b(T)-7 b(o)20 b(shed)523 4924 y(ligh)n(t)j(on)f(this)h(expression,)e(consider)h(a)g(random)g(v)-5 b(ariables)21 b Fn(V)2465 4936 y Fv(1)2526 4924 y Fw(whose)h (distribution)g(is)h Fn(W)12 b Fw(,)p eop %%Page: 12 12 12 11 bop 523 448 a Fw(where)22 b Fn(W)35 b Fw(is)23 b(an)f(EC)g(distribution)h(whose)f(\014rst)h(momen)n(t)f(and)h (normalized)f(momen)n(ts)g(are)523 540 y(sp)r(eci\014ed)28 b(b)n(y)f(\(14\).)h(Then,)1394 763 y Fq(V)1439 771 y Fj(2)1495 763 y Fu(=)1576 648 y Fe(\032)1650 716 y Fq(V)1695 724 y Fj(1)1752 716 y Fu(with)e(probabilit)n(y)g Fq(p)1650 808 y Fu(0)64 b(with)26 b(probabilit)n(y)g(1)18 b Ft(\000)e Fq(p:)551 990 y Fw(has)27 b(distribution)g Fn(Z)6 b Fw(,)28 b(since)f(Pr)o(\()p Fn(V)1637 1002 y Fv(2)1698 990 y Fn(<)c(t)p Fw(\))g(=)g Fn(p)14 b Fw(Pr)n(\()p Fn(V)2183 1002 y Fv(1)2245 990 y Fn(<)22 b(t)p Fw(\))d(+)f(\(1)g Fk(\000)g Fn(p)p Fw(\).)648 1091 y(Observ)n(e)32 b(that)i Fn(p)g Fw(satis\014es)f(0)g Fk(\024)g Fn(p)h(<)f Fw(1)h(and)f Fn(W)46 b Fw(satis\014es)34 b Fn(W)45 b Fk(2)34 b(M)p Fw(.)g(If)g Fn(W)46 b Fk(2)34 b(M)3345 1103 y Fv(0)3382 1091 y Fw(,)523 1190 y(the)g(parameters)f(of)h Fn(W)46 b Fw(are)33 b(pro)n(vided)g(b)n(y)g(case)h(\(i\),)g(using)g(the)g (normalized)f(momen)n(ts)523 1290 y(sp)r(eci\014ed)f(b)n(y)f(\(14\).)g (If)h Fn(W)42 b Fk(2)29 b(M)1574 1260 y Fv(+)1629 1290 y Fw(,)j(the)g(parameters)e(of)h Fn(W)43 b Fw(are)31 b(pro)n(vided)f(b)n(y)i(case)e(\(ii\),)523 1390 y(using)d(the)h (normalized)f(momen)n(ts)g(sp)r(eci\014ed)h(b)n(y)f(\(14\).)648 1491 y(Figure)g(5)g(sho)n(ws)f(a)h(graphical)f(represen)n(tation)g(of)i (the)g(simple)g(solution.)556 1699 y 6763386 6630772 0 0 26575831 25918013 startTexFig 556 1699 a %%BeginDocument: outline.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: outline.eps %%Creator: fig2dev Version 3.2 Patchlevel 3d %%CreationDate: Thu Jun 19 09:38:26 2003 %%For: osogami@gs57.sp.cs.cmu.edu (Takayuki Osogami) %%BoundingBox: 0 0 404 394 %%Magnification: 1.0000 %%EndComments /MyAppDict 100 dict dup begin def /$F2psDict 200 dict def F2psDictbeginF2psDict begin F2psDictbeginF2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save newpath 0 394 moveto 0 0 lineto 404 0 lineto 404 394 lineto closepath clip newpath -31.5 463.5 translate 1 -1 scale % This junk string is used by the show operators /PATsstr 1 string def /PATawidthshow { % cx cy cchar rx ry string % Loop over each character in the string { % cx cy cchar rx ry char % Show the character dup % cx cy cchar rx ry char char PATsstr dup 0 4 -1 roll put % cx cy cchar rx ry char (char) false charpath % cx cy cchar rx ry char /clip load PATdraw % Move past the character (charpath modified the % current point) currentpoint % cx cy cchar rx ry char x y newpath moveto % cx cy cchar rx ry char % Reposition by cx,cy if the character in the string is cchar 3 index eq { % cx cy cchar rx ry 4 index 4 index rmoveto } if % Reposition all characters by rx ry 2 copy rmoveto % cx cy cchar rx ry } forall pop pop pop pop pop % - currentpoint newpath moveto } bind def /PATcg { 7 dict dup begin /lw currentlinewidth def /lc currentlinecap def /lj currentlinejoin def /ml currentmiterlimit def /ds [ currentdash ] def /cc [ currentrgbcolor ] def /cm matrix currentmatrix def end } bind def % PATdraw - calculates the boundaries of the object and % fills it with the current pattern /PATdraw { % proc save exch PATpcalc % proc nw nh px py 5 -1 roll exec % nw nh px py newpath PATfill % - restore } bind def % PATfill - performs the tiling for the shape /PATfill { % nw nh px py PATfill - PATDict /CurrentPattern get dup begin setfont % Set the coordinate system to Pattern Space PatternGState PATsg % Set the color for uncolored pattezns PaintType 2 eq { PATDict /PColor get PATsc } if % Create the string for showing 3 index string % nw nh px py str % Loop for each of the pattern sources 0 1 Multi 1 sub { % nw nh px py str source % Move to the starting location 3 index 3 index % nw nh px py str source px py moveto % nw nh px py str source % For multiple sources, set the appropriate color Multi 1 ne { dup PC exch get PATsc } if % Set the appropriate string for the source 0 1 7 index 1 sub { 2 index exch 2 index put } for pop % Loop over the number of vertical cells 3 index % nw nh px py str nh { % nw nh px py str currentpoint % nw nh px py str cx cy 2 index oldshow % nw nh px py str cx cy YStep add moveto % nw nh px py str } repeat % nw nh px py str } for 5 { pop } repeat end } bind def % PATkshow - kshow with the current pattezn /PATkshow { % proc string exch bind % string proc 1 index 0 get % string proc char % Loop over all but the last character in the string 0 1 4 index length 2 sub { % string proc char idx % Find the n+1th character in the string 3 index exch 1 add get % string proe char char+1 exch 2 copy % strinq proc char+1 char char+1 char % Now show the nth character PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr) false charpath % string proc char+1 char char+1 /clip load PATdraw % Move past the character (charpath modified the current point) currentpoint newpath moveto % Execute the user proc (should consume char and char+1) mark 3 1 roll % string proc char+1 mark char char+1 4 index exec % string proc char+1 mark... cleartomark % string proc char+1 } for % Now display the last character PATsstr dup 0 4 -1 roll put % string proc (char+1) false charpath % string proc /clip load PATdraw neewath pop pop % - } bind def % PATmp - the makepattern equivalent /PATmp { % patdict patmtx PATmp patinstance exch dup length 7 add % We will add 6 new entries plus 1 FID dict copy % Create a new dictionary begin % Matrix to install when painting the pattern TilingType PATtcalc /PatternGState PATcg def PatternGState /cm 3 -1 roll put % Check for multi pattern sources (Level 1 fast color patterns) currentdict /Multi known not { /Multi 1 def } if % Font dictionary definitions /FontType 3 def % Create a dummy encoding vector /Encoding 256 array def 3 string 0 1 255 { Encoding exch dup 3 index cvs cvn put } for pop /FontMatrix matrix def /FontBBox BBox def /BuildChar { mark 3 1 roll % mark dict char exch begin Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata] PaintType 2 eq Multi 1 ne or { XStep 0 FontBBox aload pop setcachedevice } { XStep 0 setcharwidth } ifelse currentdict % mark [paintdata] dict /PaintProc load % mark [paintdata] dict paintproc end gsave false PATredef exec true PATredef grestore cleartomark % - } bind def currentdict end % newdict /foo exch % /foo newlict definefont % newfont } bind def % PATpcalc - calculates the starting point and width/height % of the tile fill for the shape /PATpcalc { % - PATpcalc nw nh px py PATDict /CurrentPattern get begin gsave % Set up the coordinate system to Pattern Space % and lock down pattern PatternGState /cm get setmatrix BBox aload pop pop pop translate % Determine the bounding box of the shape pathbbox % llx lly urx ury grestore % Determine (nw, nh) the # of cells to paint width and height PatHeight div ceiling % llx lly urx qh 4 1 roll % qh llx lly urx PatWidth div ceiling % qh llx lly qw 4 1 roll % qw qh llx lly PatHeight div floor % qw qh llx ph 4 1 roll % ph qw qh llx PatWidth div floor % ph qw qh pw 4 1 roll % pw ph qw qh 2 index sub cvi abs % pw ph qs qh-ph exch 3 index sub cvi abs exch % pw ph nw=qw-pw nh=qh-ph % Determine the starting point of the pattern fill %(px, py) 4 2 roll % nw nh pw ph PatHeight mul % nw nh pw py exch % nw nh py pw PatWidth mul exch % nw nh px py end } bind def % Save the original routines so that we can use them later on /oldfill /fill load def /oldeofill /eofill load def /oldstroke /stroke load def /oldshow /show load def /oldashow /ashow load def /oldwidthshow /widthshow load def /oldawidthshow /awidthshow load def /oldkshow /kshow load def % These defs are necessary so that subsequent procs don't bind in % the originals /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def /PATredef { MyAppDict begin { /fill { /clip load PATdraw newpath } bind def /eofill { /eoclip load PATdraw newpath } bind def /stroke { PATstroke } bind def /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def /ashow { 0 0 null 6 3 roll PATawidthshow } bind def /widthshow { 0 0 3 -1 roll PATawidthshow } bind def /awidthshow { PATawidthshow } bind def /kshow { PATkshow } bind def } { /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def } ifelse end } bind def false PATredef % Conditionally define setcmykcolor if not available /setcmykcolor where { pop } { /setcmykcolor { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor - pop } bind def } ifelse /PATsc { % colorarray aload length % c1 ... cn length dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /PATsg { % dict begin lw setlinewidth lc setlinecap lj setlinejoin ml setmiterlimit ds aload pop setdash cc aload pop setrgbcolor cm setmatrix end } bind def /PATDict 3 dict def /PATsp { true PATredef PATDict begin /CurrentPattern exch def % If it's an uncolored pattern, save the color CurrentPattern /PaintType get 2 eq { /PColor exch def } if /CColor [ currentrgbcolor ] def end } bind def % PATstroke - stroke with the current pattern /PATstroke { countdictstack save mark { currentpoint strokepath moveto PATpcalc % proc nw nh px py clip newpath PATfill } stopped { (*** PATstroke Warning: Path is too complex, stroking with gray) = cleartomark restore countdictstack exch sub dup 0 gt { { end } repeat } { pop } ifelse gsave 0.5 setgray oldstroke grestore } { pop restore pop } ifelse newpath } bind def /PATtcalc { % modmtx tilingtype PATtcalc tilematrix % Note: tiling types 2 and 3 are not supported gsave exch concat % tilingtype matrix currentmatrix exch % cmtx tilingtype % Tiling type 1 and 3: constant spacing 2 ne { % Distort the pattern so that it occupies % an integral number of device pixels dup 4 get exch dup 5 get exch % tx ty cmtx XStep 0 dtransform round exch round exch % tx ty cmtx dx.x dx.y XStep div exch XStep div exch % tx ty cmtx a b 0 YStep dtransform round exch round exch % tx ty cmtx a b dy.x dy.y YStep div exch YStep div exch % tx ty cmtx a b c d 7 -3 roll astore % { a b c d tx ty } } if grestore } bind def /PATusp { false PATredef PATDict begin CColor PATsc end } bind def % left45 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 32 32 true [ 32 0 0 -32 0 32 ] {<808080804040404020202020101010100808080804040404 020202020101010180808080404040402020202010101010 080808080404040402020202010101018080808040404040 202020201010101008080808040404040202020201010101 808080804040404020202020101010100808080804040404 0202020201010101>} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P4 exch def % horizontal sawtooth lines 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 32 16 true [ 32 0 0 -16 0 16 ] {<000000000000000000000000000000000000000000000000 000000000100010002800280044004400820082010101010 20082008400440048002800200010001>} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P21 exch def % vertical sawtooth lines 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 16 32 true [ 16 0 0 -32 0 32 ] {<400020001000080004000200010000800100020004000800 100020004000800040002000100008000400020001000080 01000200040008001000200040008000>} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P22 exch def /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def F2psBegin10setmiterlimit0.060000.06000scF2psBegin 10 setmiterlimit 0.06000 0.06000 sc % % Fig objects follow % % Polyline n 7200 1200 m 1200 7200 l 7200 4200 l cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P21 [16 0 0 -8 80.00 80.00] PATmp PATsp ef gr PATusp % Polyline n 4200 1200 m 4200 4200 l 7200 1200 l 4200 1200 l cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P4 [16 0 0 -16 280.00 80.00] PATmp PATsp ef gr PATusp % Polyline n 1200 1200 m 1200 7200 l 4200 4200 l 4200 1200 l cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P22 [8 0 0 -16 80.00 80.00] PATmp PATsp ef gr PATusp % Polyline 30.000 slw gs clippath 1290 1185 m 1110 1185 l 1110 1626 l 1200 1266 l 1290 1626 l cp eoclip n 1200 7200 m 1200 1200 l gs col0 s gr gr % arrowhead n 1290 1626 m 1200 1266 l 1110 1626 l col0 s % Polyline gs clippath 7215 7290 m 7215 7110 l 6774 7110 l 7134 7200 l 6774 7290 l cp eoclip n 1200 7200 m 7200 7200 l gs col0 s gr gr % arrowhead n 6774 7290 m 7134 7200 l 6774 7110 l col0 s % Polyline n 4200 1200 m 4200 4200 l gs col0 s gr % Polyline n 1200 7200 m 7200 1200 l gs col0 s gr % Polyline n 4200 7125 m 4200 7275 l gs col0 s gr % Polyline n 1125 4200 m 1275 4200 l gs col0 s gr % Polyline 60.000 slw [15 90] 90 sd n 1230 7230 m 1230 1830 l gs col0 s gr [] 0 sd % Polyline [15 90] 90 sd n 1200 7200 m 7200 4200 l gs col0 s gr [] 0 sd % Polyline 30.000 slw n 5475 1950 m 5625 2100 l gs col0 s gr % Polyline n 5625 1950 m 5475 2100 l gs col0 s gr /Times-Roman ff 540.00 scf sf 900 7275 m gs 1 -1 sc (1) col0 sh gr /Times-Roman ff 540.00 scf sf 5250 1875 m gs 1 -1 sc (G) col0 sh gr /Times-Roman ff 540.00 scf sf 4125 7725 m gs 1 -1 sc (2) col0 sh gr /Times-Roman ff 540.00 scf sf 1125 7650 m gs 1 -1 sc (1) col0 sh gr /Times-Roman ff 540.00 scf sf 825 4425 m gs 1 -1 sc (3) col0 sh gr /Times-Roman ff 540.00 scf sf 525 1500 m gs 1 -1 sc (m) col0 sh gr /Times-Roman ff 420.00 scf sf 900 1575 m gs 1 -1 sc (3) col0 sh gr /Times-Roman ff 540.00 scf sf 6525 7650 m gs 1 -1 sc (m) col0 sh gr /Times-Roman ff 420.00 scf sf 6900 7725 m gs 1 -1 sc (2) col0 sh gr F2psBegin10setmiterlimit0.060000.06000scF2psEnd rs end %%EndDocument endTexFig 944 2631 a Fu(\(i\))1508 1699 y 6697079 6630772 0 0 26575831 26115358 startTexFig 1508 1699 a %%BeginDocument: outline2.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: outline2.eps %%Creator: fig2dev Version 3.2 Patchlevel 3d %%CreationDate: Thu Jun 19 16:07:12 2003 %%For: osogami@gs57.sp.cs.cmu.edu (Takayuki Osogami) %%BoundingBox: 0 0 404 397 %%Magnification: 1.0000 %%EndComments /MyAppDict 100 dict dup begin def /$F2psDict 200 dict def F2psDictbeginF2psDict begin F2psDictbeginF2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save newpath 0 397 moveto 0 0 lineto 404 0 lineto 404 397 lineto closepath clip newpath -31.5 466.2 translate 1 -1 scale % This junk string is used by the show operators /PATsstr 1 string def /PATawidthshow { % cx cy cchar rx ry string % Loop over each character in the string { % cx cy cchar rx ry char % Show the character dup % cx cy cchar rx ry char char PATsstr dup 0 4 -1 roll put % cx cy cchar rx ry char (char) false charpath % cx cy cchar rx ry char /clip load PATdraw % Move past the character (charpath modified the % current point) currentpoint % cx cy cchar rx ry char x y newpath moveto % cx cy cchar rx ry char % Reposition by cx,cy if the character in the string is cchar 3 index eq { % cx cy cchar rx ry 4 index 4 index rmoveto } if % Reposition all characters by rx ry 2 copy rmoveto % cx cy cchar rx ry } forall pop pop pop pop pop % - currentpoint newpath moveto } bind def /PATcg { 7 dict dup begin /lw currentlinewidth def /lc currentlinecap def /lj currentlinejoin def /ml currentmiterlimit def /ds [ currentdash ] def /cc [ currentrgbcolor ] def /cm matrix currentmatrix def end } bind def % PATdraw - calculates the boundaries of the object and % fills it with the current pattern /PATdraw { % proc save exch PATpcalc % proc nw nh px py 5 -1 roll exec % nw nh px py newpath PATfill % - restore } bind def % PATfill - performs the tiling for the shape /PATfill { % nw nh px py PATfill - PATDict /CurrentPattern get dup begin setfont % Set the coordinate system to Pattern Space PatternGState PATsg % Set the color for uncolored pattezns PaintType 2 eq { PATDict /PColor get PATsc } if % Create the string for showing 3 index string % nw nh px py str % Loop for each of the pattern sources 0 1 Multi 1 sub { % nw nh px py str source % Move to the starting location 3 index 3 index % nw nh px py str source px py moveto % nw nh px py str source % For multiple sources, set the appropriate color Multi 1 ne { dup PC exch get PATsc } if % Set the appropriate string for the source 0 1 7 index 1 sub { 2 index exch 2 index put } for pop % Loop over the number of vertical cells 3 index % nw nh px py str nh { % nw nh px py str currentpoint % nw nh px py str cx cy 2 index oldshow % nw nh px py str cx cy YStep add moveto % nw nh px py str } repeat % nw nh px py str } for 5 { pop } repeat end } bind def % PATkshow - kshow with the current pattezn /PATkshow { % proc string exch bind % string proc 1 index 0 get % string proc char % Loop over all but the last character in the string 0 1 4 index length 2 sub { % string proc char idx % Find the n+1th character in the string 3 index exch 1 add get % string proe char char+1 exch 2 copy % strinq proc char+1 char char+1 char % Now show the nth character PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr) false charpath % string proc char+1 char char+1 /clip load PATdraw % Move past the character (charpath modified the current point) currentpoint newpath moveto % Execute the user proc (should consume char and char+1) mark 3 1 roll % string proc char+1 mark char char+1 4 index exec % string proc char+1 mark... cleartomark % string proc char+1 } for % Now display the last character PATsstr dup 0 4 -1 roll put % string proc (char+1) false charpath % string proc /clip load PATdraw neewath pop pop % - } bind def % PATmp - the makepattern equivalent /PATmp { % patdict patmtx PATmp patinstance exch dup length 7 add % We will add 6 new entries plus 1 FID dict copy % Create a new dictionary begin % Matrix to install when painting the pattern TilingType PATtcalc /PatternGState PATcg def PatternGState /cm 3 -1 roll put % Check for multi pattern sources (Level 1 fast color patterns) currentdict /Multi known not { /Multi 1 def } if % Font dictionary definitions /FontType 3 def % Create a dummy encoding vector /Encoding 256 array def 3 string 0 1 255 { Encoding exch dup 3 index cvs cvn put } for pop /FontMatrix matrix def /FontBBox BBox def /BuildChar { mark 3 1 roll % mark dict char exch begin Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata] PaintType 2 eq Multi 1 ne or { XStep 0 FontBBox aload pop setcachedevice } { XStep 0 setcharwidth } ifelse currentdict % mark [paintdata] dict /PaintProc load % mark [paintdata] dict paintproc end gsave false PATredef exec true PATredef grestore cleartomark % - } bind def currentdict end % newdict /foo exch % /foo newlict definefont % newfont } bind def % PATpcalc - calculates the starting point and width/height % of the tile fill for the shape /PATpcalc { % - PATpcalc nw nh px py PATDict /CurrentPattern get begin gsave % Set up the coordinate system to Pattern Space % and lock down pattern PatternGState /cm get setmatrix BBox aload pop pop pop translate % Determine the bounding box of the shape pathbbox % llx lly urx ury grestore % Determine (nw, nh) the # of cells to paint width and height PatHeight div ceiling % llx lly urx qh 4 1 roll % qh llx lly urx PatWidth div ceiling % qh llx lly qw 4 1 roll % qw qh llx lly PatHeight div floor % qw qh llx ph 4 1 roll % ph qw qh llx PatWidth div floor % ph qw qh pw 4 1 roll % pw ph qw qh 2 index sub cvi abs % pw ph qs qh-ph exch 3 index sub cvi abs exch % pw ph nw=qw-pw nh=qh-ph % Determine the starting point of the pattern fill %(px, py) 4 2 roll % nw nh pw ph PatHeight mul % nw nh pw py exch % nw nh py pw PatWidth mul exch % nw nh px py end } bind def % Save the original routines so that we can use them later on /oldfill /fill load def /oldeofill /eofill load def /oldstroke /stroke load def /oldshow /show load def /oldashow /ashow load def /oldwidthshow /widthshow load def /oldawidthshow /awidthshow load def /oldkshow /kshow load def % These defs are necessary so that subsequent procs don't bind in % the originals /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def /PATredef { MyAppDict begin { /fill { /clip load PATdraw newpath } bind def /eofill { /eoclip load PATdraw newpath } bind def /stroke { PATstroke } bind def /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def /ashow { 0 0 null 6 3 roll PATawidthshow } bind def /widthshow { 0 0 3 -1 roll PATawidthshow } bind def /awidthshow { PATawidthshow } bind def /kshow { PATkshow } bind def } { /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def } ifelse end } bind def false PATredef % Conditionally define setcmykcolor if not available /setcmykcolor where { pop } { /setcmykcolor { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor - pop } bind def } ifelse /PATsc { % colorarray aload length % c1 ... cn length dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /PATsg { % dict begin lw setlinewidth lc setlinecap lj setlinejoin ml setmiterlimit ds aload pop setdash cc aload pop setrgbcolor cm setmatrix end } bind def /PATDict 3 dict def /PATsp { true PATredef PATDict begin /CurrentPattern exch def % If it's an uncolored pattern, save the color CurrentPattern /PaintType get 2 eq { /PColor exch def } if /CColor [ currentrgbcolor ] def end } bind def % PATstroke - stroke with the current pattern /PATstroke { countdictstack save mark { currentpoint strokepath moveto PATpcalc % proc nw nh px py clip newpath PATfill } stopped { (*** PATstroke Warning: Path is too complex, stroking with gray) = cleartomark restore countdictstack exch sub dup 0 gt { { end } repeat } { pop } ifelse gsave 0.5 setgray oldstroke grestore } { pop restore pop } ifelse newpath } bind def /PATtcalc { % modmtx tilingtype PATtcalc tilematrix % Note: tiling types 2 and 3 are not supported gsave exch concat % tilingtype matrix currentmatrix exch % cmtx tilingtype % Tiling type 1 and 3: constant spacing 2 ne { % Distort the pattern so that it occupies % an integral number of device pixels dup 4 get exch dup 5 get exch % tx ty cmtx XStep 0 dtransform round exch round exch % tx ty cmtx dx.x dx.y XStep div exch XStep div exch % tx ty cmtx a b 0 YStep dtransform round exch round exch % tx ty cmtx a b dy.x dy.y YStep div exch YStep div exch % tx ty cmtx a b c d 7 -3 roll astore % { a b c d tx ty } } if grestore } bind def /PATusp { false PATredef PATDict begin CColor PATsc end } bind def % left45 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 32 32 true [ 32 0 0 -32 0 32 ] {<808080804040404020202020101010100808080804040404 020202020101010180808080404040402020202010101010 080808080404040402020202010101018080808040404040 202020201010101008080808040404040202020201010101 808080804040404020202020101010100808080804040404 0202020201010101>} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P4 exch def % horizontal sawtooth lines 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 32 16 true [ 32 0 0 -16 0 16 ] {<000000000000000000000000000000000000000000000000 000000000100010002800280044004400820082010101010 20082008400440048002800200010001>} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P21 exch def % vertical sawtooth lines 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 16 32 true [ 16 0 0 -32 0 32 ] {<400020001000080004000200010000800100020004000800 100020004000800040002000100008000400020001000080 01000200040008001000200040008000>} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P22 exch def /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def F2psBegin10setmiterlimit0.060000.06000scF2psBegin 10 setmiterlimit 0.06000 0.06000 sc % % Fig objects follow % % Polyline n 7200 1200 m 1200 7200 l 7200 4200 l cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P21 [16 0 0 -8 80.00 80.00] PATmp PATsp ef gr PATusp % Polyline n 4200 1200 m 4200 4200 l 7200 1200 l 4200 1200 l cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P4 [16 0 0 -16 280.00 80.00] PATmp PATsp ef gr PATusp % Polyline n 1200 1200 m 1200 7200 l 4200 4200 l 4200 1200 l cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P22 [8 0 0 -16 80.00 80.00] PATmp PATsp ef gr PATusp /Times-Roman ff 540.00 scf sf 3375 2850 m gs 1 -1 sc (A) col0 sh gr /Times-Roman ff 540.00 scf sf 3675 2550 m gs 1 -1 sc (N) col0 sh gr % Arc 15.000 slw gs clippath 2953 4583 m 3042 4596 l 3075 4370 l 3005 4542 l 2986 4357 l cp eoclip n 5850.0 4875.0 2865.7 -174.0 -96.0 arc gs col0 s gr gr % arrowhead n 2986 4357 m 3005 4542 l 3075 4370 l col0 s % Polyline 30.000 slw gs clippath 1290 1185 m 1110 1185 l 1110 1626 l 1200 1266 l 1290 1626 l cp eoclip n 1200 7200 m 1200 1200 l gs col0 s gr gr % arrowhead n 1290 1626 m 1200 1266 l 1110 1626 l col0 s % Polyline gs clippath 7215 7290 m 7215 7110 l 6774 7110 l 7134 7200 l 6774 7290 l cp eoclip n 1200 7200 m 7200 7200 l gs col0 s gr gr % arrowhead n 6774 7290 m 7134 7200 l 6774 7110 l col0 s % Polyline n 4200 1200 m 4200 4200 l gs col0 s gr % Polyline n 1200 7200 m 7200 1200 l gs col0 s gr % Polyline n 4200 7125 m 4200 7275 l gs col0 s gr % Polyline n 1125 4200 m 1275 4200 l gs col0 s gr % Polyline 60.000 slw [15 90] 90 sd n 1230 7230 m 1230 1830 l gs col0 s gr [] 0 sd % Polyline [15 90] 90 sd n 1200 7200 m 7200 4200 l gs col0 s gr [] 0 sd % Polyline 30.000 slw n 2925 4500 m 3075 4650 l gs col0 s gr % Polyline n 5475 1950 m 5625 2100 l gs col0 s gr % Polyline n 5625 1950 m 5475 2100 l gs col0 s gr % Polyline n 3075 4500 m 2925 4650 l gs col0 s gr /Times-Roman ff 540.00 scf sf 900 7275 m gs 1 -1 sc (1) col0 sh gr /Times-Roman ff 420.00 scf sf 7020 7770 m gs 1 -1 sc (2) col0 sh gr /Times-Roman ff 420.00 scf sf 945 1665 m gs 1 -1 sc (3) col0 sh gr /Times-Roman ff 540.00 scf sf 5250 1875 m gs 1 -1 sc (X) col0 sh gr /Times-Roman ff 540.00 scf sf 2475 4725 m gs 1 -1 sc (G) col0 sh gr /Times-Roman ff 540.00 scf sf 1125 7650 m gs 1 -1 sc (1) col0 sh gr /Times-Roman ff 540.00 scf sf 4050 7725 m gs 1 -1 sc (2) col0 sh gr /Times-Roman ff 540.00 scf sf 6600 7650 m gs 1 -1 sc (m) col0 sh gr /Times-Roman ff 540.00 scf sf 825 4425 m gs 1 -1 sc (3) col0 sh gr /Times-Roman ff 540.00 scf sf 525 1575 m gs 1 -1 sc (m) col0 sh gr F2psBegin10setmiterlimit0.060000.06000scF2psEnd rs end %%EndDocument endTexFig 1881 2631 a Fu(\(ii\))2456 1699 y 6697079 6630772 0 0 26575831 26115358 startTexFig 2456 1699 a %%BeginDocument: outline3.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: outline3.eps %%Creator: fig2dev Version 3.2 Patchlevel 3d %%CreationDate: Thu Jun 19 16:07:26 2003 %%For: osogami@gs57.sp.cs.cmu.edu (Takayuki Osogami) %%BoundingBox: 0 0 404 397 %%Magnification: 1.0000 %%EndComments /MyAppDict 100 dict dup begin def /$F2psDict 200 dict def F2psDictbeginF2psDict begin F2psDictbeginF2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save newpath 0 397 moveto 0 0 lineto 404 0 lineto 404 397 lineto closepath clip newpath -31.5 466.2 translate 1 -1 scale % This junk string is used by the show operators /PATsstr 1 string def /PATawidthshow { % cx cy cchar rx ry string % Loop over each character in the string { % cx cy cchar rx ry char % Show the character dup % cx cy cchar rx ry char char PATsstr dup 0 4 -1 roll put % cx cy cchar rx ry char (char) false charpath % cx cy cchar rx ry char /clip load PATdraw % Move past the character (charpath modified the % current point) currentpoint % cx cy cchar rx ry char x y newpath moveto % cx cy cchar rx ry char % Reposition by cx,cy if the character in the string is cchar 3 index eq { % cx cy cchar rx ry 4 index 4 index rmoveto } if % Reposition all characters by rx ry 2 copy rmoveto % cx cy cchar rx ry } forall pop pop pop pop pop % - currentpoint newpath moveto } bind def /PATcg { 7 dict dup begin /lw currentlinewidth def /lc currentlinecap def /lj currentlinejoin def /ml currentmiterlimit def /ds [ currentdash ] def /cc [ currentrgbcolor ] def /cm matrix currentmatrix def end } bind def % PATdraw - calculates the boundaries of the object and % fills it with the current pattern /PATdraw { % proc save exch PATpcalc % proc nw nh px py 5 -1 roll exec % nw nh px py newpath PATfill % - restore } bind def % PATfill - performs the tiling for the shape /PATfill { % nw nh px py PATfill - PATDict /CurrentPattern get dup begin setfont % Set the coordinate system to Pattern Space PatternGState PATsg % Set the color for uncolored pattezns PaintType 2 eq { PATDict /PColor get PATsc } if % Create the string for showing 3 index string % nw nh px py str % Loop for each of the pattern sources 0 1 Multi 1 sub { % nw nh px py str source % Move to the starting location 3 index 3 index % nw nh px py str source px py moveto % nw nh px py str source % For multiple sources, set the appropriate color Multi 1 ne { dup PC exch get PATsc } if % Set the appropriate string for the source 0 1 7 index 1 sub { 2 index exch 2 index put } for pop % Loop over the number of vertical cells 3 index % nw nh px py str nh { % nw nh px py str currentpoint % nw nh px py str cx cy 2 index oldshow % nw nh px py str cx cy YStep add moveto % nw nh px py str } repeat % nw nh px py str } for 5 { pop } repeat end } bind def % PATkshow - kshow with the current pattezn /PATkshow { % proc string exch bind % string proc 1 index 0 get % string proc char % Loop over all but the last character in the string 0 1 4 index length 2 sub { % string proc char idx % Find the n+1th character in the string 3 index exch 1 add get % string proe char char+1 exch 2 copy % strinq proc char+1 char char+1 char % Now show the nth character PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr) false charpath % string proc char+1 char char+1 /clip load PATdraw % Move past the character (charpath modified the current point) currentpoint newpath moveto % Execute the user proc (should consume char and char+1) mark 3 1 roll % string proc char+1 mark char char+1 4 index exec % string proc char+1 mark... cleartomark % string proc char+1 } for % Now display the last character PATsstr dup 0 4 -1 roll put % string proc (char+1) false charpath % string proc /clip load PATdraw neewath pop pop % - } bind def % PATmp - the makepattern equivalent /PATmp { % patdict patmtx PATmp patinstance exch dup length 7 add % We will add 6 new entries plus 1 FID dict copy % Create a new dictionary begin % Matrix to install when painting the pattern TilingType PATtcalc /PatternGState PATcg def PatternGState /cm 3 -1 roll put % Check for multi pattern sources (Level 1 fast color patterns) currentdict /Multi known not { /Multi 1 def } if % Font dictionary definitions /FontType 3 def % Create a dummy encoding vector /Encoding 256 array def 3 string 0 1 255 { Encoding exch dup 3 index cvs cvn put } for pop /FontMatrix matrix def /FontBBox BBox def /BuildChar { mark 3 1 roll % mark dict char exch begin Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata] PaintType 2 eq Multi 1 ne or { XStep 0 FontBBox aload pop setcachedevice } { XStep 0 setcharwidth } ifelse currentdict % mark [paintdata] dict /PaintProc load % mark [paintdata] dict paintproc end gsave false PATredef exec true PATredef grestore cleartomark % - } bind def currentdict end % newdict /foo exch % /foo newlict definefont % newfont } bind def % PATpcalc - calculates the starting point and width/height % of the tile fill for the shape /PATpcalc { % - PATpcalc nw nh px py PATDict /CurrentPattern get begin gsave % Set up the coordinate system to Pattern Space % and lock down pattern PatternGState /cm get setmatrix BBox aload pop pop pop translate % Determine the bounding box of the shape pathbbox % llx lly urx ury grestore % Determine (nw, nh) the # of cells to paint width and height PatHeight div ceiling % llx lly urx qh 4 1 roll % qh llx lly urx PatWidth div ceiling % qh llx lly qw 4 1 roll % qw qh llx lly PatHeight div floor % qw qh llx ph 4 1 roll % ph qw qh llx PatWidth div floor % ph qw qh pw 4 1 roll % pw ph qw qh 2 index sub cvi abs % pw ph qs qh-ph exch 3 index sub cvi abs exch % pw ph nw=qw-pw nh=qh-ph % Determine the starting point of the pattern fill %(px, py) 4 2 roll % nw nh pw ph PatHeight mul % nw nh pw py exch % nw nh py pw PatWidth mul exch % nw nh px py end } bind def % Save the original routines so that we can use them later on /oldfill /fill load def /oldeofill /eofill load def /oldstroke /stroke load def /oldshow /show load def /oldashow /ashow load def /oldwidthshow /widthshow load def /oldawidthshow /awidthshow load def /oldkshow /kshow load def % These defs are necessary so that subsequent procs don't bind in % the originals /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def /PATredef { MyAppDict begin { /fill { /clip load PATdraw newpath } bind def /eofill { /eoclip load PATdraw newpath } bind def /stroke { PATstroke } bind def /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def /ashow { 0 0 null 6 3 roll PATawidthshow } bind def /widthshow { 0 0 3 -1 roll PATawidthshow } bind def /awidthshow { PATawidthshow } bind def /kshow { PATkshow } bind def } { /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def } ifelse end } bind def false PATredef % Conditionally define setcmykcolor if not available /setcmykcolor where { pop } { /setcmykcolor { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor - pop } bind def } ifelse /PATsc { % colorarray aload length % c1 ... cn length dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /PATsg { % dict begin lw setlinewidth lc setlinecap lj setlinejoin ml setmiterlimit ds aload pop setdash cc aload pop setrgbcolor cm setmatrix end } bind def /PATDict 3 dict def /PATsp { true PATredef PATDict begin /CurrentPattern exch def % If it's an uncolored pattern, save the color CurrentPattern /PaintType get 2 eq { /PColor exch def } if /CColor [ currentrgbcolor ] def end } bind def % PATstroke - stroke with the current pattern /PATstroke { countdictstack save mark { currentpoint strokepath moveto PATpcalc % proc nw nh px py clip newpath PATfill } stopped { (*** PATstroke Warning: Path is too complex, stroking with gray) = cleartomark restore countdictstack exch sub dup 0 gt { { end } repeat } { pop } ifelse gsave 0.5 setgray oldstroke grestore } { pop restore pop } ifelse newpath } bind def /PATtcalc { % modmtx tilingtype PATtcalc tilematrix % Note: tiling types 2 and 3 are not supported gsave exch concat % tilingtype matrix currentmatrix exch % cmtx tilingtype % Tiling type 1 and 3: constant spacing 2 ne { % Distort the pattern so that it occupies % an integral number of device pixels dup 4 get exch dup 5 get exch % tx ty cmtx XStep 0 dtransform round exch round exch % tx ty cmtx dx.x dx.y XStep div exch XStep div exch % tx ty cmtx a b 0 YStep dtransform round exch round exch % tx ty cmtx a b dy.x dy.y YStep div exch YStep div exch % tx ty cmtx a b c d 7 -3 roll astore % { a b c d tx ty } } if grestore } bind def /PATusp { false PATredef PATDict begin CColor PATsc end } bind def % left45 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 32 32 true [ 32 0 0 -32 0 32 ] {<808080804040404020202020101010100808080804040404 020202020101010180808080404040402020202010101010 080808080404040402020202010101018080808040404040 202020201010101008080808040404040202020201010101 808080804040404020202020101010100808080804040404 0202020201010101>} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P4 exch def % horizontal sawtooth lines 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 32 16 true [ 32 0 0 -16 0 16 ] {<000000000000000000000000000000000000000000000000 000000000100010002800280044004400820082010101010 20082008400440048002800200010001>} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P21 exch def % vertical sawtooth lines 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 16 32 true [ 16 0 0 -32 0 32 ] {<400020001000080004000200010000800100020004000800 100020004000800040002000100008000400020001000080 01000200040008001000200040008000>} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P22 exch def /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def F2psBegin10setmiterlimit0.060000.06000scF2psBegin 10 setmiterlimit 0.06000 0.06000 sc % % Fig objects follow % % Polyline n 7200 1200 m 1200 7200 l 7200 4200 l cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P21 [16 0 0 -8 80.00 80.00] PATmp PATsp ef gr PATusp % Polyline n 4200 1200 m 4200 4200 l 7200 1200 l 4200 1200 l cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P4 [16 0 0 -16 280.00 80.00] PATmp PATsp ef gr PATusp % Polyline n 1200 1200 m 1200 7200 l 4200 4200 l 4200 1200 l cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P22 [8 0 0 -16 80.00 80.00] PATmp PATsp ef gr PATusp /Times-Roman ff 540.00 scf sf 3375 3675 m gs 1 -1 sc (A) col0 sh gr /Times-Roman ff 540.00 scf sf 3675 3375 m gs 1 -1 sc (N) col0 sh gr % Arc 15.000 slw gs clippath 2953 5408 m 3042 5421 l 3075 5195 l 3005 5367 l 2986 5182 l cp eoclip n 5850.0 5700.0 2865.7 -174.0 -96.0 arc gs col0 s gr gr % arrowhead n 2986 5182 m 3005 5367 l 3075 5195 l col0 s % Polyline 30.000 slw gs clippath 1290 1185 m 1110 1185 l 1110 1626 l 1200 1266 l 1290 1626 l cp eoclip n 1200 7200 m 1200 1200 l gs col0 s gr gr % arrowhead n 1290 1626 m 1200 1266 l 1110 1626 l col0 s % Polyline gs clippath 7215 7290 m 7215 7110 l 6774 7110 l 7134 7200 l 6774 7290 l cp eoclip n 1200 7200 m 7200 7200 l gs col0 s gr gr % arrowhead n 6774 7290 m 7134 7200 l 6774 7110 l col0 s % Polyline n 4200 1200 m 4200 4200 l gs col0 s gr % Polyline n 1200 7200 m 7200 1200 l gs col0 s gr % Polyline n 4200 7125 m 4200 7275 l gs col0 s gr % Polyline n 1125 4200 m 1275 4200 l gs col0 s gr % Polyline 60.000 slw [15 90] 90 sd n 1230 7230 m 1230 1830 l gs col0 s gr [] 0 sd % Polyline [15 90] 90 sd n 1200 7200 m 7200 4200 l gs col0 s gr [] 0 sd % Polyline 30.000 slw [120] 0 sd n 3000 5400 m 6600 3600 l gs col0 s gr [] 0 sd % Polyline n 6675 3525 m 6525 3675 l gs col0 s gr % Polyline n 6525 3525 m 6675 3675 l gs col0 s gr % Polyline n 2925 5325 m 3075 5475 l gs col0 s gr % Polyline n 5475 2775 m 5625 2925 l gs col0 s gr /Times-Roman ff 420.00 scf sf 7020 7770 m gs 1 -1 sc (2) col0 sh gr /Times-Roman ff 420.00 scf sf 945 1665 m gs 1 -1 sc (3) col0 sh gr /Times-Roman ff 540.00 scf sf 6375 3450 m gs 1 -1 sc (G) col0 sh gr /Times-Roman ff 540.00 scf sf 5250 2700 m gs 1 -1 sc (X) col0 sh gr /Times-Roman ff 540.00 scf sf 2475 5325 m gs 1 -1 sc (W) col0 sh gr /Times-Roman ff 540.00 scf sf 4125 7725 m gs 1 -1 sc (2) col0 sh gr /Times-Roman ff 540.00 scf sf 1125 7725 m gs 1 -1 sc (1) col0 sh gr /Times-Roman ff 540.00 scf sf 6600 7650 m gs 1 -1 sc (m) col0 sh gr /Times-Roman ff 540.00 scf sf 900 7275 m gs 1 -1 sc (1) col0 sh gr /Times-Roman ff 540.00 scf sf 825 4425 m gs 1 -1 sc (3) col0 sh gr /Times-Roman ff 540.00 scf sf 525 1575 m gs 1 -1 sc (m) col0 sh gr F2psBegin10setmiterlimit0.060000.06000scF2psEnd rs end %%EndDocument endTexFig 2818 2631 a Fu(\(iii\))523 2793 y Fr(Fig.)15 b(5.)24 b Fu(A)g(graphical)h(represen)n(tation)g(of)g(the)f(simple)g(solution.) i(Let)e Fq(G)h Fu(b)r(e)f(the)g(input)f(distribu-)523 2884 y(tion.)g(\(i\))f(If)h Fq(G)e Ft(2)h(U)1078 2892 y Fj(0)1123 2884 y Ft([)10 b(M)1276 2892 y Fj(0)1311 2884 y Fu(,)23 b Fq(G)f Fu(is)h(w)n(ell-represen)n(ted)g(b)n(y)e(a)i(t) n(w)n(o-phase)f(Co)n(xian)2825 2852 y Fj(+)2899 2884 y Fu(distribution)g Fq(X)6 b Fu(.)523 2975 y(\(ii\))21 b(If)g Fq(G)g Ft(2)h(U)927 2944 y Fj(+)985 2975 y Ft([)7 b(M)1135 2944 y Fj(+)1186 2975 y Fu(,)21 b Fq(G)f Fu(is)i(w)n (ell-represen)n(ted)f(b)n(y)e Fq(A)2099 2944 y Fi(N)2157 2975 y Fu(\()p Fq(X)6 b Fu(\),)21 b(where)g Fq(X)27 b Fu(is)21 b(a)g(t)n(w)n(o-phase)g(Co)n(xian)3354 2944 y Fj(+)523 3067 y Fu(distribution.)28 b(\(iii\))g(If)g Fq(G)c Ft(2)h(L)p Fu(,)j Fq(G)f Fu(is)h(w)n(ell-represen)n(ted)g(b)n(y) f Fq(Z)5 b Fu(,)28 b(where)g Fq(Z)33 b Fu(is)28 b Fq(W)35 b Fu(=)24 b Fq(A)3042 3035 y Fi(N)3099 3067 y Fu(\()p Fq(X)6 b Fu(\))28 b(with)523 3158 y(probabilit)n(y)c Fq(p)g Fu(and)g(0)h(with)g(probabilit)n(y)f(1)15 b Ft(\000)f Fq(p)24 b Fu(and)g Fq(X)31 b Fu(is)25 b(a)f(t)n(w)n(o-phase)h(Co)n (xian)2915 3126 y Fj(+)2990 3158 y Fu(distribution.)523 3556 y Fo(A)n(nalyzing)31 b(the)j(Numb)-5 b(er)33 b(of)h(Phases)h(R)-5 b(e)g(quir)g(e)g(d)40 b Fw(The)27 b(pro)r(of)f(of)g(Theorem)g(2)g (relies)523 3656 y(on)h(the)h(follo)n(wing)f(theorem:)523 3812 y Fl(Theorem)j(3.)41 b Fm([18])23 b(L)l(et)e Fk(S)1392 3782 y Fv(\()p Fh(n)p Fv(\))1511 3812 y Fm(denote)h(the)f(set)g(of)h (distributions)g(that)f(ar)l(e)h(wel)t(l-r)l(epr)l(esente)l(d)523 3921 y(by)27 b(an)g Fn(n)p Fm(-phase)h(acyclic)g(PH)f(distribution.)h (L)l(et)e Fk(S)2122 3933 y Fd(V)2173 3884 y Fv(\()p Fh(n)p Fv(\))2297 3921 y Fm(and)h Fk(E)2506 3891 y Fv(\()p Fh(n)p Fv(\))2630 3921 y Fm(b)l(e)g(the)g(sets)f(de\014ne)l(d)h(by:)1129 4117 y Ft(S)1176 4125 y Fb(V)1223 4081 y Fj(\()p Fi(n)p Fj(\))1337 4117 y Fu(=)1420 4027 y Fe(n)1475 4117 y Fq(F)1535 4024 y Fe(\014)1535 4074 y(\014)1535 4123 y(\014)1563 4117 y Fq(m)1631 4081 y Fi(F)1631 4130 y Fj(2)1703 4117 y Fq(>)1794 4069 y(n)17 b Fu(+)g(1)p 1794 4100 179 4 v 1860 4168 a Fq(n)2010 4117 y Fc(and)28 b Fq(m)2227 4081 y Fi(F)2227 4130 y Fj(3)2298 4117 y Ft(\025)2389 4069 y Fq(n)18 b Fu(+)f(3)p 2389 4100 V 2389 4168 a Fq(n)h Fu(+)f(2)2578 4117 y Fq(m)2646 4081 y Fi(F)2646 4130 y Fj(2)2696 4027 y Fe(o)2765 4117 y Fu(;)1176 4304 y Ft(E)1224 4268 y Fj(\()p Fi(n)p Fj(\))1337 4304 y Fu(=)1420 4214 y Fe(n)1475 4304 y Fq(F)1535 4211 y Fe(\014)1535 4261 y(\014)1535 4311 y(\014)1563 4304 y Fq(m)1631 4268 y Fi(F)1631 4318 y Fj(2)1703 4304 y Fu(=)1794 4256 y Fq(n)g Fu(+)g(1)p 1794 4287 V 1860 4355 a Fq(n)2010 4304 y Fc(and)28 b Fq(m)2227 4268 y Fi(F)2227 4318 y Fj(3)2298 4304 y Fu(=)2389 4256 y Fq(n)18 b Fu(+)f(2)p 2389 4287 V 2455 4355 a Fq(n)2578 4214 y Fe(o)553 4524 y Fm(for)30 b(inte)l(gers)g Fn(n)23 b Fk(\025)g Fw(2)p Fm(.)29 b(Then)i Fk(S)1524 4494 y Fv(\()p Fh(n)p Fv(\))1644 4524 y Fk(\032)23 b(S)1782 4536 y Fd(V)1833 4488 y Fv(\()p Fh(n)p Fv(\))1949 4524 y Fk([)18 b(E)2073 4494 y Fv(\()p Fh(n)p Fv(\))2200 4524 y Fm(for)31 b(inte)l(gers)f Fn(n)23 b Fk(\025)f Fw(2)p Fm(.)523 4707 y(Pr)l(o)l(of)43 b(\(The)l(or)l(em)22 b(2\).)h Fw(W)-7 b(e)19 b(will)h(sho)n(w)e(that)h(\(i\))h(if)f(a)g (distribution)g Fn(G)g Fw(is)g(in)g Fk(S)2879 4719 y Fd(V)2930 4670 y Fv(\()p Fh(l)p Fv(\))3009 4707 y Fk(\\)q Fw(\()p Fk(U)27 b([)19 b(M)p Fw(\),)523 4816 y(then)31 b(at)g(most)f Fn(l)22 b Fw(+)e(1)30 b(phases)g(are)g(used,)g(and)h (\(ii\))g(if)g(a)g(distribution)f Fn(G)h Fw(is)g(in)g Fk(S)3100 4828 y Fd(V)3151 4779 y Fv(\()p Fh(l)p Fv(\))3249 4816 y Fk(\\)21 b(L)p Fw(,)523 4924 y(then)37 b(at)g(most)g Fn(l)26 b Fw(+)e(2)36 b(phases)g(are)g(used.)h(Since)g Fk(S)2188 4894 y Fv(\()p Fh(l)p Fv(\))2304 4924 y Fk(\032)h(S)2457 4936 y Fd(V)2508 4888 y Fv(\()p Fh(l)p Fv(\))2610 4924 y Fk([)25 b(E)2741 4894 y Fv(\()p Fh(l)p Fv(\))2856 4924 y Fw(b)n(y)36 b(Theorem)g(3,)p eop %%Page: 13 13 13 12 bop 523 448 a Fw(this)36 b(completes)e(the)i(pro)r(of.)f(Notice)g (that)g(the)h(simple)f(solution)g(is)g(not)g(de\014ned)h(when)523 548 y Fn(G)23 b Fk(2)h(E)741 518 y Fv(\()p Fh(l)p Fv(\))818 548 y Fw(.)648 648 y(\(i\))38 b(Supp)r(ose)f Fn(G)j Fk(2)f(U)34 b([)25 b(M)p Fw(.)37 b(If)h Fn(G)i Fk(2)f(S)1975 660 y Fd(V)2026 611 y Fv(\()p Fh(l)p Fv(\))2104 648 y Fw(,)e(then)h(b)n(y)f (\(12\))g(the)h(EC)f(distribution)523 747 y(pro)n(vided)25 b(b)n(y)h(the)h(simple)g(solution)e(has)h(at)g(most)g Fn(l)18 b Fw(+)d(1)26 b(phases.)g(\(ii\))h(Supp)r(ose)f Fn(G)d Fk(2)h(L)p Fw(.)j(If)523 872 y Fn(G)h Fk(2)g(S)749 884 y Fd(V)800 836 y Fv(\()p Fh(l)p Fv(\))877 872 y Fw(,)j(then)g Fn(m)1196 842 y Fh(W)1196 893 y Fv(2)1299 872 y Fw(=)1497 830 y Fh(m)1556 805 y Fg(G)1556 846 y Ff(2)p 1401 853 300 4 v 1401 905 a Fv(2)p Fh(m)1493 885 y Fg(G)1493 925 y Ff(2)1541 905 y Fd(\000)p Fh(m)1652 885 y Fg(G)1652 925 y Ff(3)1738 872 y Fn(>)1840 839 y Fh(l)p Fv(+2)p 1840 853 106 4 v 1840 901 a Fh(l)p Fv(+1)1955 872 y Fw(.)g(By)f (\(12\),)g(the)h(EC)f(distribution)g(pro)n(vided)523 991 y(b)n(y)d(the)h(simple)g(solution)f(has)g(at)h(most)f Fn(l)20 b Fw(+)e(2)27 b(phases.)1086 b Fk(u)-55 b(t)523 1248 y Fp(4)112 b(V)-9 b(arian)m(ts)37 b(of)g(Closed-F)-9 b(orm)37 b(Solutions)523 1439 y Fw(In)d(this)g(section,)g(w)n(e)g (presen)n(t)f(t)n(w)n(o)g(re\014nemen)n(ts)h(of)g(the)g(simple)g (solution)g(\(Section)g(3\),)523 1539 y(whic)n(h)28 b(w)n(e)f(refer)g (to)g(as)g(the)h(impro)n(v)n(ed)e(solution)h(and)h(the)g(n)n (umerically)f(stable)g(solution.)523 1780 y Fl(4.1)95 b(An)32 b(Impro)m(v)m(ed)f(Closed-F)-8 b(orm)29 b(Solution)523 1938 y Fw(W)-7 b(e)35 b(\014rst)g(describ)r(e)f(the)i(prop)r(erties)e (that)h(the)g(impro)n(v)n(ed)f(solution)g(satis\014es.)g(W)-7 b(e)35 b(then)523 2037 y(describ)r(e)30 b(the)h(high)g(lev)n(el)f (ideas)g(b)r(ehind)i(the)f(construction)e(of)i(the)g(impro)n(v)n(ed)e (solution.)523 2137 y(Figure)h(6)g(is)h(an)f(implemen)n(tation)h(of)g (the)g(impro)n(v)n(ed)e(solution.)h(See)h([17])f(for)g(details)g(on)523 2236 y(ho)n(w)d(the)h(high)f(lev)n(el)h(ideas)f(describ)r(ed)g(ab)r(o)n (v)n(e)f(are)h(realized)f(in)i(the)g(impro)n(v)n(ed)f(solution.)523 2428 y Fo(Pr)-5 b(op)g(erties)45 b(of)f(the)h(Impr)-5 b(ove)g(d)45 b(Solution)53 b Fw(This)37 b(solution)f(is)h(de\014ned)g (for)g(all)g(the)523 2527 y(input)i(distributions)e Fn(G)j Fk(2)g(P)7 b(H)1586 2539 y Fv(3)1661 2527 y Fw(and)37 b(uses)h(a)f(smaller)g(n)n(um)n(b)r(er)g(of)h(phases)f(than)h(the)523 2627 y(simple)j(solution.)f(Sp)r(eci\014cally)-7 b(,)41 b(the)h(n)n(um)n(b)r(er)e(of)h(phases)f(required)g(in)h(the)g(impro)n (v)n(ed)523 2726 y(solution)27 b(is)h(c)n(haracterized)d(b)n(y)i(the)h (follo)n(wing)f(theorem:)523 2876 y Fl(Theorem)j(4.)41 b Fm(Under)25 b(the)g(impr)l(ove)l(d)h(solution,)f(the)g(numb)l(er)f (of)i(phases)g(ne)l(e)l(de)l(d)f(to)f(wel)t(l-)523 2976 y(r)l(epr)l(esent)29 b(any)i(distribution)f Fn(G)g Fm(by)g(an)g(EC)h (distribution)f(is)g(at)g(most)f Fn(O)r(P)12 b(T)g Fw(\()p Fn(G)p Fw(\))19 b(+)f(1)p Fm(.)523 3125 y Fw(F)-7 b(or)27 b(a)g(pro)r(of)g(of)h(the)g(theorem,)f(see)g([17].)523 3316 y Fo(High)33 b(L)-5 b(evel)33 b(Ide)-5 b(as)42 b Fw(Consider)25 b(an)h(arbitrary)e(distribution)j Fn(G)c Fk(2)g(P)7 b(H)2819 3328 y Fv(3)2856 3316 y Fw(.)26 b(Our)g(approac)n (h)523 3416 y(consists)40 b(of)h(t)n(w)n(o)f(steps,)h(the)g(\014rst)g (of)g(whic)n(h)f(in)n(v)n(olv)n(es)g(constructing)g(a)g(baseline)g(EC) 523 3515 y(distribution,)35 b(and)f(the)h(second)f(of)h(whic)n(h)f(in)n (v)n(olv)n(es)f(reducing)h(the)h(n)n(um)n(b)r(er)f(of)h(phases)523 3615 y(in)28 b(this)h(baseline)f(solution.)f(If)i Fn(G)24 b Fk(2)g(P)7 b(H)1823 3579 y Fd(\000)1823 3637 y Fv(3)1879 3615 y Fw(,)28 b(then)h(the)f(baseline)g(solution)g(used)g(is)g(simply) 523 3715 y(giv)n(en)20 b(b)n(y)h(the)h(simple)f(solution)g(\(Section)g (3\).)g(If)h Fn(G)32 b(=)-51 b Fk(2)24 b(P)7 b(H)2352 3679 y Fd(\000)2352 3736 y Fv(3)2408 3715 y Fw(,)21 b(then)h(to)f (obtain)g(the)h(baseline)523 3840 y(EC)k(distributing)g(w)n(e)g (\014rst)f(\014nd)i(a)f(distribution)g Fn(W)35 b Fk(2)23 b(P)7 b(H)2418 3804 y Fd(\000)2418 3862 y Fv(3)2500 3840 y Fw(suc)n(h)26 b(that)2874 3798 y Fh(m)2933 3773 y Fg(W)2933 3815 y Ff(3)p 2874 3821 124 4 v 2874 3873 a Fh(m)2933 3853 y Fg(W)2933 3893 y Ff(2)3030 3840 y Fw(=)3128 3798 y Fh(m)3187 3773 y Fg(G)3187 3815 y Ff(3)p 3128 3821 108 4 v 3128 3873 a Fh(m)3187 3853 y Fg(G)3187 3893 y Ff(2)3271 3840 y Fw(and)523 3971 y Fn(m)596 3941 y Fh(W)596 3992 y Fv(2)706 3971 y Fn(<)35 b(m)879 3941 y Fh(G)879 3992 y Fv(2)970 3971 y Fw(and)f(then)h(set)g Fn(p)g Fw(suc)n(h)f(that)h Fn(G)g Fw(is)g(w)n(ell-represen)n(ted)e(b)n(y)h(distribution)h Fn(Z)6 b Fw(,)523 4071 y(where)38 b Fn(Z)6 b Fw(\()p Fk(\001)p Fw(\))41 b(=)f Fn(W)12 b Fw(\()p Fk(\001)p Fw(\))p Fn(p)26 b Fw(+)g(1)f Fk(\000)g Fn(p)p Fw(.)38 b(\(See)h(Section)f(3)g(for)g(an)g(explanation)f(of)h Fn(Z)6 b Fw(\).)77 b(The)523 4170 y(parameters)26 b(of)i(the)h(EC)e (distribution)h(that)g(w)n(ell-represen)n(ts)e Fn(W)40 b Fw(are)27 b(then)i(obtained)e(b)n(y)523 4270 y(the)h(simple)g (solution)f(\(Section)h(3\).)648 4361 y(Next,)21 b(w)n(e)g(describ)r(e) g(an)h(idea)f(to)g(reduce)g(the)h(n)n(um)n(b)r(er)f(of)g(phases)g(used) g(in)h(the)g(baseline)523 4453 y(EC)38 b(distribution.)g(The)g(simple)g (solution)g(\(Section)g(3\))g(is)g(based)g(on)f(the)i(fact)f(that)g(a) 523 4544 y(distribution)23 b Fn(X)30 b Fw(is)23 b(w)n(ell-represen)n (ted)e(b)n(y)i(a)g(t)n(w)n(o-phase)f(Co)n(xian)g(distribution)h(when)g Fn(X)30 b Fk(2)523 4635 y(U)575 4647 y Fv(0)631 4635 y Fk([)20 b(M)806 4647 y Fv(0)843 4635 y Fw(.)28 b(In)h(fact,)g(a)f (wider)g(range)f(of)h(distributions)h(are)e(w)n(ell-represen)n(ted)g(b) n(y)h(the)h(set)523 4727 y(of)f(t)n(w)n(o-phase)d(Co)n(xian)i (distributions.)g(In)h(particular,)e(if)1203 4910 y Fq(X)i Ft(2)1366 4820 y Fe(n)1422 4910 y Fq(F)1482 4816 y Fe(\014)1482 4866 y(\014)1482 4916 y(\014)1519 4861 y Fu(3)p 1519 4893 39 4 v 1519 4960 a(2)1589 4910 y Ft(\024)21 b Fq(m)1738 4874 y Fi(X)1738 4923 y Fj(2)1817 4910 y Ft(\024)g Fu(2)26 b(and)f Fq(m)2179 4874 y Fi(X)2179 4923 y Fj(3)2258 4910 y Fu(=)c(2)p Fq(m)2445 4874 y Fi(X)2445 4923 y Fj(2)2520 4910 y Ft(\000)c Fu(1)2635 4820 y Fe(o)2703 4910 y Fq(;)p eop %%Page: 14 14 14 13 bop 562 448 a Fw(then)39 b Fn(X)45 b Fw(is)39 b(w)n(ell-represen) n(ted)e(b)n(y)h(a)g(t)n(w)n(o-phase)f(Co)n(xian)h(distribution.)h(In)g (fact,)g(the)523 548 y(ab)r(o)n(v)n(e)30 b(solution)h(can)g(b)r(e)h (impro)n(v)n(ed)f(up)r(on)g(y)n(et)h(further.)f(Ho)n(w)n(ev)n(er,)f (for)h(readabilit)n(y)-7 b(,)30 b(w)n(e)523 648 y(p)r(ostp)r(one)e (this)g(to)f([17].)1295 617 y Fv(5)p 571 859 2787 4 v 569 952 4 94 v 582 925 a Fu(\()p Fq(n)p Fu(,)g Fq(p)p Fu(,)e Fq(\025)836 933 y Fi(Y)889 925 y Fu(,)h Fq(\025)981 933 y Fi(X)t Fj(1)1069 925 y Fu(,)g Fq(\025)1161 933 y Fi(X)t Fj(2)1249 925 y Fu(,)g Fq(p)1335 933 y Fi(X)1393 925 y Fu(\))f(=)g Fs(Improved)p Fu(\()p Fq(\026)1921 893 y Fi(G)1921 938 y Fj(1)1975 925 y Fu(,)h Fq(\026)2068 893 y Fi(G)2068 938 y Fj(2)2120 925 y Fu(,)g Fq(\026)2213 893 y Fi(G)2213 938 y Fj(3)2265 925 y Fu(\))p 3355 952 V 569 1049 4 97 v 583 1018 a(Input:)p 583 1046 207 4 v 24 w(the)f(\014rst)h(three)f(momen)n(ts)f(of)i(a)g(distribution)g Fq(G)p Fu(:)g Fq(\026)2352 986 y Fi(G)2352 1031 y Fj(1)2404 1018 y Fu(,)g Fq(\026)2497 986 y Fi(G)2497 1031 y Fj(2)2549 1018 y Fu(,)g(and)f Fq(\026)2791 986 y Fi(G)2791 1031 y Fj(3)2843 1018 y Fu(.)p 3355 1049 4 97 v 569 1144 4 96 v 583 1113 a(Output:)p 583 1141 269 4 v 24 w(parameters)g(of)i(the)e (EC)h(distribution,)h(\()p Fq(n)p Fu(,)f Fq(p)p Fu(,)f Fq(\025)2321 1121 y Fi(Y)2374 1113 y Fu(,)h Fq(\025)2466 1121 y Fi(X)t Fj(1)2554 1113 y Fu(,)g Fq(\025)2646 1121 y Fi(X)t Fj(2)2734 1113 y Fu(,)g Fq(p)2820 1121 y Fi(X)2878 1113 y Fu(\))p 3355 1144 4 96 v 569 1291 4 147 v 583 1236 a(1.)g Fq(m)736 1204 y Fi(G)736 1250 y Fj(2)808 1236 y Fu(=)940 1193 y Fi(\026)977 1172 y Fg(G)977 1212 y Ff(2)p 899 1219 167 4 v 899 1268 a Fj(\()p Fi(\026)960 1251 y Fg(G)960 1291 y Ff(1)1009 1268 y Fj(\))1033 1254 y Ff(2)1075 1236 y Fu(;)78 b Fq(m)1242 1204 y Fi(G)1242 1250 y Fj(3)1314 1236 y Fu(=)1448 1193 y Fi(\026)1485 1172 y Fg(G)1485 1212 y Ff(3)p 1405 1219 172 4 v 1405 1268 a Fi(\026)1442 1251 y Fg(G)1442 1291 y Ff(1)1491 1268 y Fi(\026)1528 1251 y Fg(G)1528 1291 y Ff(2)1586 1236 y Fu(.)p 3355 1291 4 147 v 569 1649 4 359 v 583 1489 a(2.)26 b Fq(p)21 b Fu(=)809 1296 y Fe(8)809 1370 y(>)809 1395 y(<)809 1545 y(>)809 1570 y(:)904 1340 y Fj(\()p Fi(m)982 1319 y Fg(G)982 1359 y Ff(2)1031 1340 y Fj(\))1055 1319 y Ff(2)1087 1340 y Fj(+2)p Fi(m)1218 1319 y Fg(G)1218 1359 y Ff(2)1267 1340 y Fb(\000)p Fj(1)p 904 1366 442 4 v 1018 1415 a(2\()p Fi(m)1126 1397 y Fg(G)1126 1437 y Ff(2)1175 1415 y Fj(\))1199 1401 y Ff(2)1379 1383 y Fu(if)26 b Fq(m)1517 1351 y Fi(G)1517 1396 y Fj(3)1590 1383 y Fq(>)21 b Fu(2)p Fq(m)1777 1351 y Fi(G)1777 1396 y Fj(2)1846 1383 y Ft(\000)16 b Fu(1)p Fq(;)39 b Fu(and)2255 1352 y Fj(1)p 2180 1366 182 4 v 2180 1415 a Fi(m)2234 1397 y Fg(G)2234 1437 y Ff(2)2283 1415 y Fb(\000)p Fj(1)2396 1383 y Fu(is)27 b(an)e(in)n(teger)q Fq(;)1031 1473 y Fj(1)p 904 1487 285 4 v 904 1535 a(2)p Fi(m)988 1518 y Fg(G)988 1558 y Ff(2)1037 1535 y Fb(\000)p Fi(m)1139 1518 y Fg(G)1139 1558 y Ff(3)1379 1503 y Fu(if)h Fq(m)1517 1472 y Fi(G)1517 1517 y Fj(3)1590 1503 y Fq(<)21 b Fu(2)p Fq(m)1777 1472 y Fi(G)1777 1517 y Fj(2)1846 1503 y Ft(\000)16 b Fu(1)p Fq(;)894 1622 y Fu(1)447 b(otherwise)q Fq(:)p 3355 1649 4 359 v 569 1776 4 128 v 583 1741 a Fu(3.)26 b Fq(\026)714 1710 y Fi(W)714 1755 y Fj(1)805 1741 y Fu(=)896 1699 y Fi(\026)933 1678 y Fg(G)933 1718 y Ff(1)p 896 1724 86 4 v 923 1767 a Fi(p)992 1741 y Fu(;)77 b Fq(m)1158 1710 y Fi(W)1158 1755 y Fj(2)1248 1741 y Fu(=)21 b Fq(pm)1436 1710 y Fi(G)1436 1755 y Fj(2)1487 1741 y Fu(;)77 b Fq(m)1653 1710 y Fi(W)1653 1755 y Fj(3)1744 1741 y Fu(=)21 b Fq(pm)1932 1710 y Fi(G)1932 1755 y Fj(3)1983 1741 y Fu(.)p 3355 1776 4 128 v 569 2075 4 299 v 583 1945 a(4.)26 b Fq(n)21 b Fu(=)816 1776 y Fe(8)816 1851 y(<)816 2001 y(:)902 1780 y(j)995 1827 y Fi(m)1049 1806 y Fg(W)1049 1846 y Ff(2)p 956 1853 198 4 v 956 1902 a Fi(m)1010 1884 y Fg(W)1010 1924 y Ff(2)1075 1902 y Fb(\000)p Fj(1)1163 1780 y Fe(k)1362 1870 y Fu(if)27 b Fq(m)1501 1838 y Fi(W)1501 1883 y Fj(3)1591 1870 y Fu(=)21 b(2)p Fq(m)1778 1838 y Fi(W)1778 1883 y Fj(2)1865 1870 y Ft(\000)c Fu(1)p Fq(;)39 b Fu(and)25 b Fq(m)2257 1838 y Fi(W)2257 1883 y Fj(2)2348 1870 y Fq(<)c Fu(2)902 1929 y Fe(j)995 1976 y Fi(m)1049 1956 y Fg(W)1049 1996 y Ff(2)p 956 2002 V 956 2051 a Fi(m)1010 2034 y Fg(W)1010 2074 y Ff(2)1075 2051 y Fb(\000)p Fj(1)1180 2019 y Fu(+)c(1)1295 1929 y Fe(k)1362 2019 y Fu(otherwise)r Fq(:)p 3355 2075 4 299 v 569 2222 4 147 v 583 2167 a Fu(5.)26 b Fq(m)736 2136 y Fi(X)736 2181 y Fj(2)815 2167 y Fu(=)906 2125 y Fj(\()p Fi(n)p Fb(\000)p Fj(3\))p Fi(m)1124 2104 y Fg(W)1124 2144 y Ff(2)1189 2125 y Fb(\000)p Fj(\()p Fi(n)p Fb(\000)p Fj(2\))p 906 2150 497 4 v 906 2199 a(\()p Fi(n)p Fb(\000)p Fj(2\))p Fi(m)1124 2182 y Fg(W)1124 2222 y Ff(2)1189 2199 y Fb(\000)p Fj(\()p Fi(n)p Fb(\000)p Fj(1\))1412 2167 y Fu(;)77 b Fq(\026)1556 2136 y Fi(X)1556 2181 y Fj(1)1636 2167 y Fu(=)1918 2125 y Fi(\026)1955 2104 y Fg(W)1955 2144 y Ff(1)p 1727 2150 486 4 v 1727 2199 a Fj(\()p Fi(n)p Fb(\000)p Fj(2\))p Fi(m)1945 2182 y Fg(X)1945 2222 y Ff(2)1999 2199 y Fb(\000)p Fj(\()p Fi(n)p Fb(\000)p Fj(3\))2222 2167 y Fu(.)p 3355 2222 4 147 v 569 2322 4 100 v 583 2291 a(6.)26 b Fq(\013)21 b Fu(=)g(\()p Fq(n)d Ft(\000)e Fu(2\)\()p Fq(m)1155 2259 y Fi(X)1155 2304 y Fj(2)1230 2291 y Ft(\000)h Fu(1\))1388 2225 y Fe(\000)1426 2291 y Fq(n)p Fu(\()p Fq(n)h Ft(\000)e Fu(1\)\()p Fq(m)1808 2259 y Fi(X)1808 2304 y Fj(2)1866 2291 y Fu(\))1896 2259 y Fj(2)1948 2291 y Ft(\000)g Fq(n)p Fu(\(2)p Fq(n)i Ft(\000)f Fu(5\))p Fq(m)2415 2259 y Fi(X)2415 2304 y Fj(2)2490 2291 y Fu(+)g(\()p Fq(n)g Ft(\000)g Fu(1\)\()p Fq(n)g Ft(\000)g Fu(3\))3043 2225 y Fe(\001)3082 2291 y Fu(.)p 3355 2322 V 569 2433 4 112 v 583 2402 a(7.)26 b Fq(\014)f Fu(=)817 2337 y Fe(\000)855 2402 y Fu(\()p Fq(n)18 b Ft(\000)e Fu(1\))p Fq(m)1161 2370 y Fi(X)1161 2415 y Fj(2)1236 2402 y Ft(\000)h Fu(\()p Fq(n)g Ft(\000)g Fu(2\))1551 2337 y Fe(\001)c(\000)1640 2402 y Fu(\()p Fq(n)18 b Ft(\000)e Fu(2\))p Fq(m)1946 2370 y Fi(X)1946 2415 y Fj(2)2021 2402 y Ft(\000)h Fu(\()p Fq(n)g Ft(\000)g Fu(3\))2336 2337 y Fe(\001)2374 2354 y Fj(2)2409 2402 y Fu(.)p 3355 2433 V 569 2580 4 147 v 583 2525 a(8.)26 b Fq(m)736 2493 y Fi(X)736 2539 y Fj(3)815 2525 y Fu(=)906 2482 y Fi(\014)s(m)998 2461 y Fg(W)998 2501 y Ff(3)1062 2482 y Fb(\000)p Fi(\013)p 906 2508 245 4 v 974 2557 a(m)1028 2540 y Fg(X)1028 2580 y Ff(2)1160 2525 y Fu(.)p 3355 2580 4 147 v 569 2829 4 250 v 583 2723 a(9.)g Fq(u)21 b Fu(=)814 2583 y Fe(\()893 2650 y Fu(1)330 b(if)26 b(3)p Fq(m)1437 2618 y Fi(X)1437 2664 y Fj(2)1517 2650 y Fu(=)21 b(2)p Fq(m)1704 2618 y Fi(X)1704 2664 y Fj(3)957 2727 y(6)p Fb(\000)p Fj(2)p Fi(m)1119 2706 y Fg(X)1119 2746 y Ff(3)p 903 2752 325 4 v 903 2801 a Fj(3)p Fi(m)987 2784 y Fg(X)987 2824 y Ff(2)1041 2801 y Fb(\000)p Fj(2)p Fi(m)1173 2784 y Fg(X)1173 2824 y Ff(3)1261 2769 y Fu(otherwise)1783 2723 y(;)77 b Fq(v)25 b Fu(=)2024 2583 y Fe(\()2102 2650 y Fu(0)486 b(if)27 b(3)p Fq(m)2803 2618 y Fi(X)2803 2664 y Fj(2)2882 2650 y Fu(=)21 b(2)p Fq(m)3069 2618 y Fi(X)3069 2664 y Fj(3)2229 2727 y(12)p Fb(\000)p Fj(6)p Fi(m)2421 2706 y Fg(X)2421 2746 y Ff(2)p 2112 2752 481 4 v 2112 2801 a Fi(m)2166 2784 y Fg(X)2166 2824 y Ff(2)2220 2801 y Fj(\(3)p Fi(m)2328 2784 y Fg(X)2328 2824 y Ff(2)2383 2801 y Fb(\000)p Fj(2)p Fi(m)2515 2784 y Fg(X)2515 2824 y Ff(3)2569 2801 y Fj(\))2626 2769 y Fu(otherwise)3148 2723 y(.)p 3355 2829 4 250 v 569 2985 4 157 v 583 2930 a(10.)26 b Fq(\025)751 2938 y Fi(X)t Fj(1)861 2930 y Fu(=)952 2890 y Fi(u)p Fj(+)1036 2835 y Ft(p)p 1099 2835 181 4 v 1099 2890 a Fi(u)1136 2876 y Ff(2)1168 2890 y Fb(\000)p Fj(4)p Fi(v)p 952 2913 328 4 v 1055 2962 a Fj(2)p Fi(\026)1122 2945 y Fg(X)1122 2985 y Ff(1)1289 2930 y Fu(;)g Fq(\025)1381 2938 y Fi(X)t Fj(2)1491 2930 y Fu(=)1582 2890 y Fi(u)p Fb(\000)1667 2835 y Ft(p)p 1730 2835 181 4 v 1730 2890 a Fi(u)1767 2876 y Ff(2)1800 2890 y Fb(\000)p Fj(4)p Fi(v)p 1582 2913 330 4 v 1686 2962 a Fj(2)p Fi(\026)1753 2945 y Fg(X)1753 2985 y Ff(1)1921 2930 y Fu(;)g Fq(p)2007 2938 y Fi(X)2086 2930 y Fu(=)2177 2887 y Fi(\025)2214 2898 y Fg(X)s Ff(2)2295 2887 y Fi(\026)2332 2867 y Fg(X)2332 2907 y Ff(1)2386 2887 y Fj(\()p Fi(\025)2447 2898 y Fg(X)s Ff(1)2528 2887 y Fi(\026)2565 2867 y Fg(X)2565 2907 y Ff(1)2619 2887 y Fb(\000)p Fj(1\))p 2177 2913 546 4 v 2344 2962 a Fi(\025)2381 2973 y Fg(X)s Ff(1)2463 2962 y Fi(\026)2500 2945 y Fg(X)2500 2985 y Ff(1)2732 2930 y Fu(;)g Fq(\025)2824 2938 y Fi(Y)2898 2930 y Fu(=)3136 2900 y Fj(1)p 2989 2914 326 4 v 2989 2962 a(\()p Fi(m)3067 2945 y Fg(X)3067 2985 y Ff(2)3121 2962 y Fb(\000)p Fj(1\))p Fi(\026)3260 2945 y Fg(X)3260 2985 y Ff(1)3324 2930 y Fu(.)p 3355 2985 4 157 v 571 2988 2787 4 v 873 3131 a Fr(Fig.)15 b(6.)25 b Fu(An)g(implemen)n(tation)f(of)j(the)e(impro)n(v)n (ed)f(closed-form)i(solution.)523 3641 y Fl(4.2)95 b(A)32 b(Numerically)e(Stable)h(Closed-F)-8 b(orm)29 b(Solution)523 3803 y Fw(The)g(impro)n(v)n(ed)f(solution)h(\(Section)g(4.1\))g(is)g (not)g(n)n(umerically)f(stable)h(when)g Fn(G)d Fk(2)g(U)37 b Fw(and)523 3903 y Fn(m)596 3873 y Fh(G)596 3923 y Fv(2)683 3903 y Fw(is)31 b(close)g(to)1088 3870 y Fh(l)p Fv(+1)p 1088 3884 106 4 v 1130 3931 a Fh(l)1234 3903 y Fw(for)g(in)n(tegers)f Fn(l)h Fk(\025)d Fw(1,)j(i.e.,)h(on)f(the)g(b)r(orders)f(b)r(et)n(w)n (een)i Fk(U)3026 3915 y Fh(i)3053 3903 y Fw('s.)f(In)h(this)523 4002 y(section,)21 b(w)n(e)h(presen)n(t)f(a)g(n)n(umerically)g(stable)g (solution.)g(W)-7 b(e)22 b(\014rst)g(describ)r(e)f(the)h(prop)r(erties) 523 4102 y(that)36 b(the)g(n)n(umerically)e(stable)h(solution)g (satis\014es.)g(W)-7 b(e)35 b(then)h(describ)r(e)f(the)h(high)f(lev)n (el)523 4202 y(ideas)28 b(b)r(ehind)i(the)f(construction)f(of)h(the)h (n)n(umerically)e(stable)g(solution.)h(Figure)f(6)g(is)h(an)523 4301 y(implemen)n(tation)22 b(of)f(the)h(n)n(umerically)f(stable)g (solution.)g(See)h([17])f(for)g(details)g(on)h(ho)n(w)f(the)523 4401 y(high)28 b(lev)n(el)f(ideas)g(describ)r(ed)g(ab)r(o)n(v)n(e)f (are)h(realized)g(in)g(the)h(n)n(umerically)f(stable)g(solution.)523 4597 y Fo(Pr)-5 b(op)g(erties)47 b(of)h(the)f(Numeric)-5 b(al)5 b(ly)46 b(Stable)g(Solution)56 b Fw(The)40 b(n)n(umerically)e (stable)523 4696 y(solution)26 b(uses)f(at)h(most)g(one)f(more)h(phase) f(than)h(the)h(impro)n(v)n(ed)d(solution)i(and)g(is)g(de\014ned)p 523 4748 473 4 v 546 4801 a Fj(5)606 4833 y Fu(While)h(this)f(further)h (impro)n(v)n(emen)n(t)c(reduces)k(the)f(n)n(um)n(b)r(er)e(of)j (necessary)g(phases)g(b)n(y)e(one)i(for)606 4924 y(man)n(y)d (distributions,)i(it)g(do)r(es)g(not)g(impro)n(v)n(e)e(the)i(w)n(orst)g (case)h(p)r(erformance.)p eop %%Page: 15 15 15 14 bop 1085 369 1759 4 v 1083 462 4 94 v 1097 434 a Fu(\()p Fq(n)p Fu(,)26 b Fq(p)p Fu(,)f Fq(\025)1350 442 y Fi(Y)1403 434 y Fu(,)h Fq(\025)1495 442 y Fi(X)t Fj(1)1583 434 y Fu(,)g Fq(\025)1675 442 y Fi(X)t Fj(2)1763 434 y Fu(,)g Fq(p)1849 442 y Fi(X)1907 434 y Fu(\))f(=)g Fs(Stable)p Fu(\()p Fq(\026)2357 403 y Fi(G)2357 448 y Fj(1)2410 434 y Fu(,)h Fq(\026)2503 403 y Fi(G)2503 448 y Fj(2)2555 434 y Fu(,)g Fq(\026)2648 403 y Fi(G)2648 448 y Fj(3)2700 434 y Fu(\))p 2841 462 V 1083 555 V 1097 528 a(If)f Fq(m)1241 496 y Fi(G)1241 541 y Fj(3)1314 528 y Ft(\024)c Fu(2)p Fq(m)1501 496 y Fi(G)1501 541 y Fj(2)1570 528 y Ft(\000)16 b Fu(1,)27 b(use)f Fs(Improved)p Fu(.)p 2841 555 V 1083 646 4 92 v 1097 619 a(Otherwise,)g(replace)h (steps)e(2-4)h(of)h Fs(Improved)g Fu(as)g(follo)n(ws:)p 2839 646 V 1083 854 4 208 v 1097 773 a(2.)f Fq(n)21 b Fu(=)1331 658 y Fe(\030)1389 714 y Fj(3)p Fi(m)1473 694 y Fg(G)1473 734 y Ff(2)1522 714 y Fb(\000)p Fj(2+)1647 653 y Fe(p)p 1730 653 442 4 v 61 x Fj(\()p Fi(m)1808 697 y Fg(G)1808 737 y Ff(2)1857 714 y Fj(\))1881 700 y Ff(2)1913 714 y Fb(\000)p Fj(2)p Fi(m)2045 697 y Fg(G)2045 737 y Ff(2)2095 714 y Fj(+2)p 1389 756 783 4 v 1650 805 a(2\()p Fi(m)1758 788 y Fg(G)1758 828 y Ff(2)1808 805 y Fb(\000)p Fj(1\))2182 658 y Fe(\031)2230 773 y Fu(.)p 2841 854 4 208 v 1083 977 4 124 v 1097 922 a(3.)26 b Fq(p)21 b Fu(=)1384 892 y Fj(1)p 1333 906 134 4 v 1333 954 a(2)p Fi(m)1417 937 y Fg(G)1417 977 y Ff(2)1489 857 y Fe(\000)1537 892 y Fi(n)p Fb(\000)p Fj(1)p 1537 906 117 4 v 1537 948 a Fi(n)p Fb(\000)p Fj(2)1681 922 y Fu(+)1806 892 y Fi(n)p 1767 906 V 1767 948 a(n)p Fb(\000)p Fj(1)1894 857 y Fe(\001)1932 922 y Fu(.)p 2841 977 4 124 v 1083 1104 4 128 v 1097 1069 a(4.)26 b Fq(\026)1228 1037 y Fi(W)1228 1083 y Fj(1)1319 1069 y Fu(=)1410 1026 y Fi(\026)1447 1005 y Fg(G)1447 1045 y Ff(1)p 1410 1052 86 4 v 1437 1094 a Fi(p)1506 1069 y Fu(;)77 b Fq(m)1672 1037 y Fi(W)1672 1083 y Fj(2)1762 1069 y Fu(=)21 b Fq(pm)1950 1037 y Fi(G)1950 1083 y Fj(2)2001 1069 y Fu(;)77 b Fq(m)2167 1037 y Fi(W)2167 1083 y Fj(3)2258 1069 y Fu(=)21 b Fq(pm)2446 1037 y Fi(G)2446 1083 y Fj(3)2497 1069 y Fu(.)p 2841 1104 4 128 v 1085 1107 1759 4 v 723 1251 a Fr(Fig.)14 b(7.)26 b Fu(An)f(implemen)n (tation)f(of)i(the)g(n)n(umerically)f(stable)h(closed-form)g(solution.) 523 1636 y Fw(for)20 b(all)h(the)g(input)g(distributions)g(in)g Fk(P)7 b(H)1807 1648 y Fv(3)1844 1636 y Fw(.)21 b(Sp)r(eci\014cally)-7 b(,)21 b(the)g(n)n(um)n(b)r(er)f(of)h(phases)f(required)523 1735 y(in)28 b(the)g(n)n(umerically)f(stable)g(solution)g(is)g(c)n (haracterized)f(b)n(y)h(the)h(follo)n(wing)f(theorem:)523 1924 y Fl(Theorem)j(5.)41 b Fm(Under)21 b(the)h(numeric)l(al)t(ly)g (stable)g(solution,)g(the)f(numb)l(er)g(of)h(phases)h(ne)l(e)l(de)l(d) 523 2023 y(to)e(wel)t(l-r)l(epr)l(esent)h(any)g(distribution)g Fn(G)f Fm(by)h(an)f(EC)h(distribution)g(is)g(at)f(most)g Fn(O)r(P)12 b(T)g Fw(\()p Fn(G)p Fw(\)+2)p Fm(.)523 2304 y Fw(A)28 b(pro)r(of)f(of)h(Theorem)e(5)h(is)h(giv)n(en)f(in)h([17].) 648 2411 y(The)h(EC)f(distribution,)h Fn(Z)6 b Fw(,)29 b(that)h(is)f(pro)n(vided)e(b)n(y)i(the)h(n)n(umerically)e(stable)h (solution)523 2510 y(is)f(n)n(umerically)e(stable)h(in)h(the)g(follo)n (wing)f(sense:)523 2699 y Fl(Prop)s(osition)j(1.)41 b Fm(L)l(et)25 b Fn(Z)31 b Fm(b)l(e)26 b(the)g(EC)g(distribution)g(pr)l (ovide)l(d)i(by)e(the)g(numeric)l(al)t(ly)g(stable)523 2798 y(solution,)j(wher)l(e)h(the)f(input)f(distribution)h Fn(G)g Fm(is)g(wel)t(l-r)l(epr)l(esente)l(d)h(by)f Fn(Z)6 b Fm(.)29 b(L)l(et)f(\()p Fn(n)p Fm(,)h Fn(p)p Fm(,)g Fn(\025)3322 2810 y Fh(Y)3379 2798 y Fm(,)523 2898 y Fn(\025)571 2910 y Fh(X)5 b Fv(1)668 2898 y Fm(,)26 b Fn(\025)767 2910 y Fh(X)5 b Fv(2)863 2898 y Fm(,)27 b Fn(p)957 2910 y Fh(X)1019 2898 y Fm(\))f(b)l(e)g(the)g(p)l(ar)l (ameters)g(of)g Fn(Z)6 b Fm(.)26 b(Supp)l(ose)g(that)g(e)l(ach)h(p)l (ar)l(ameter)f Fn(p)p Fm(,)g Fn(\025)3126 2910 y Fh(Y)3184 2898 y Fm(,)g Fn(\025)3283 2910 y Fh(X)5 b Fv(1)3379 2898 y Fm(,)523 2998 y Fn(\025)571 3010 y Fh(X)g Fv(2)668 2998 y Fm(,)34 b(and)g Fn(p)934 3010 y Fh(X)1031 2998 y Fm(has)g(an)g(err)l(or)g Fn(\001p)p Fm(,)g Fn(\001\025)1808 3010 y Fh(Y)1866 2998 y Fm(,)g Fn(\001\025)2042 3010 y Fh(X)5 b Fv(1)2139 2998 y Fm(,)34 b Fn(\001\025)2315 3010 y Fh(X)5 b Fv(2)2412 2998 y Fm(,)34 b(and)g Fn(\001p)2747 3010 y Fh(X)2810 2998 y Fm(,)g(r)l(esp)l(e)l(ctively,)i(in)523 3097 y(absolute)24 b(value.)h(L)l(et)f Fn(\001\026)1326 3067 y Fh(Z)1326 3118 y Fv(1)1403 3097 y Fw(=)e Fk(j)p Fn(\026)1563 3067 y Fh(Z)1563 3118 y Fv(1)1622 3097 y Fk(\000)6 b Fn(\026)1743 3067 y Fh(G)1743 3118 y Fv(1)1799 3097 y Fk(j)24 b Fm(b)l(e)g(the)g(err)l(or)g(of)h(the)f(\014rst)f (moment)g(of)i Fn(Z)30 b Fm(and)24 b(let)523 3197 y Fn(\001m)665 3167 y Fh(Z)665 3219 y(i)742 3197 y Fw(=)e Fk(j)p Fn(m)925 3167 y Fh(Z)925 3219 y(i)991 3197 y Fk(\000)12 b Fn(m)1141 3167 y Fh(G)1141 3219 y(i)1196 3197 y Fk(j)26 b Fm(b)l(e)h(the)g(err)l (or)g(of)h(the)e Fn(i)p Fm(-th)g(normalize)l(d)j(moment)d(of)h Fn(Z)33 b Fm(for)27 b Fn(i)c Fw(=)g(2)p Fn(;)14 b Fw(3)p Fm(.)523 3297 y(If)624 3259 y Fh(\001p)p 624 3277 90 4 v 652 3325 a(p)723 3297 y Fm(,)792 3262 y Fh(\001\025)886 3270 y Fg(Y)p 792 3277 145 4 v 820 3325 a Fh(\025)859 3333 y Fg(Y)947 3297 y Fm(,)1016 3262 y Fh(\001\025)1110 3270 y Fg(X)s Ff(1)p 1016 3277 177 4 v 1043 3325 a Fh(\025)1082 3333 y Fg(X)s Ff(1)1202 3297 y Fm(,)1271 3262 y Fh(\001\025)1365 3270 y Fg(X)s Ff(2)p 1271 3277 V 1299 3325 a Fh(\025)1338 3333 y Fg(X)s Ff(2)1457 3297 y Fm(,)34 b(and)1691 3259 y Fh(\001p)1780 3267 y Fg(X)p 1691 3277 144 4 v 1719 3325 a Fh(p)1753 3333 y Fg(X)1874 3297 y Fn(<)29 b(\017)g Fw(=)g(10)2209 3266 y Fd(\000)p Fv(5)2331 3297 y Fm(\(r)l(esp)l(e)l (ctively,)35 b Fn(\017)29 b Fw(=)g(10)3069 3266 y Fd(\000)p Fv(9)3157 3297 y Fm(\),)34 b(then)533 3398 y Fh(\001\026)628 3373 y Fg(Z)628 3415 y Ff(1)p 533 3421 142 4 v 561 3473 a Fh(\026)601 3453 y Fg(Z)601 3493 y Ff(1)708 3441 y Fn(<)23 b Fw(0)p Fn(:)p Fw(01)i Fm(and)1138 3398 y Fh(\001m)1252 3373 y Fg(Z)1252 3415 y(i)p 1138 3421 161 4 v 1166 3473 a Fh(m)1225 3453 y Fg(Z)1225 3494 y(i)1332 3441 y Fn(<)e Fw(0)p Fn(:)p Fw(01)i Fm(for)j Fn(i)22 b Fw(=)h(2)p Fn(;)14 b Fw(3)p Fm(,)26 b(pr)l(ovide)l(d)j(that)e(the)g(normalize)l(d)h (moments)523 3565 y(of)j Fn(G)f Fm(satis\014es)f(the)h(c)l(ondition)h (in)f(Figur)l(e)g(8)g(\(a\))g(\(r)l(esp)l(e)l(ctively,)i(\(b\)\).)523 3745 y Fw(In)f(Prop)r(osition)d(1,)i Fn(\017)h Fw(w)n(as)e(c)n(hosen)g (to)i(b)r(e)g(10)1974 3715 y Fd(\000)p Fv(5)2092 3745 y Fw(and)f(10)2340 3715 y Fd(\000)p Fv(9)2429 3745 y Fw(,)g(resp)r(ectiv)n(ely)-7 b(.)30 b(These)g(corre-)523 3845 y(sp)r(ond)20 b(to)h(the)g(precisions)e(of)h(the)h Fa(float)d Fw(\(six)j(decimal)f(digits\))h(and)f Fa(double)e Fw(\(ten)j(decimal)523 3945 y(digits\))31 b(data)f(t)n(yp)r(e)g(in)h (C,)g(resp)r(ectiv)n(ely)-7 b(.)60 b(In)31 b(Figure)e(8)i(\(b\),)g(it)g (is)f(imp)r(ossible)g(to)h(distin-)523 4044 y(guish)h(the)h(set)g(of)f (all)h(non-negativ)n(e)d(distributions)j(from)f(the)h(set)g(of)f (distributions)h(for)523 4144 y(whic)n(h)f(the)g(stabilit)n(y)f(guaran) n(tee)f(of)i(Prop)r(osition)e(1)h(holds.)h(Closed)f(form)g(form)n(ulas) g(for)523 4243 y(the)d(curv)n(es)e(in)i(Figure)f(8)g(and)h(a)f(pro)r (of)g(of)h(Prop)r(osition)e(1)h(are)f(giv)n(en)h(in)h([17].)523 4481 y Fo(High)33 b(L)-5 b(evel)34 b(Ide)-5 b(as)41 b Fw(Ac)n(hieving)26 b(the)h(n)n(umerical)f(stabilit)n(y)g(is)g(based)g (on)h(the)f(same)g(idea)523 4581 y(as)j(treating)g(input)i (distributions)e(whic)n(h)h(are)f(not)h(in)g Fk(P)7 b(H)2419 4544 y Fd(\000)2419 4602 y Fv(3)2475 4581 y Fw(.)30 b(Namely)-7 b(,)30 b(w)n(e)g(\014rst)f(\014nd)h(an)523 4706 y(EC)j(distribution)h Fn(W)45 b Fw(suc)n(h)33 b(that)1644 4664 y Fh(m)1703 4639 y Fg(W)1703 4680 y Ff(3)p 1644 4687 124 4 v 1644 4739 a Fh(m)1703 4719 y Fg(W)1703 4759 y Ff(2)1810 4706 y Fw(=)1917 4664 y Fh(m)1976 4639 y Fg(G)1976 4680 y Ff(3)p 1917 4687 108 4 v 1917 4739 a Fh(m)1976 4719 y Fg(G)1976 4759 y Ff(2)2068 4706 y Fw(and)g Fn(m)2308 4676 y Fh(W)2308 4727 y Fv(2)2417 4706 y Fn(<)f(m)2587 4676 y Fh(G)2587 4727 y Fv(2)2676 4706 y Fw(so)h(that)h(the)g(solution) 523 4825 y(is)d(n)n(umerically)g(stable)g(for)f Fn(W)12 b Fw(,)32 b(and)f(then)h(set)f Fn(p)g Fw(suc)n(h)g(that)h Fn(G)g Fw(is)f(w)n(ell-represen)n(ted)e(b)n(y)523 4924 y Fn(Z)6 b Fw(\()p Fk(\001)p Fw(\))23 b(=)g Fn(W)12 b Fw(\()p Fk(\001)p Fw(\))p Fn(p)19 b Fw(+)f(1)g Fk(\000)g Fn(p)p Fw(.)27 b(\(See)h(Section)g(3)f(for)g(an)g(explanation)g(of)h Fn(Z)6 b Fw(\).)p eop %%Page: 16 16 16 15 bop 509 365 a 11367087 9472573 1512980 11774935 36048404 40521564 startTexFig 509 365 a %%BeginDocument: stability_float2.eps %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The Mathworks, Inc. %%Title: stability_float2.eps %%CreationDate: 02/22/2003 14:19:24 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Pages: 1 %%BoundingBox: 23 179 548 616 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rm /rmoveto ldef /rl /rlineto ldef /s {show newpath} bdef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef /rc {rectclip} bdef /rf {rectfill} bdef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont} bdef /reencode {exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop} bdef /isroman {findfont /CharStrings get /Agrave known} bdef /FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS} bdef /csm {1 dpi2point div -1 dpi2point div scale neg translate dup landscapeMode eq {pop -90 rotate} {rotateMode eq {90 rotate} if} ifelse} bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L {lineto stroke} bdef /MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef /AP {{rlineto} repeat} bdef /PDlw -1 def /W {/PDlw currentlinewidth def setlinewidth} def /PP {closepath eofill} bdef /DP {closepath stroke} bdef /MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath} bdef /FR {MR stroke} bdef /PR {MR fill} bdef /L1i {{currentfile picstr readhexstring pop} image} bdef /tMatrix matrix def /MakeOval {newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix} bdef /FO {MakeOval stroke} bdef /PO {MakeOval fill} bdef /PD {currentlinewidth 2 div 0 360 arc fill PDlw -1 eq not {PDlw w /PDlw -1 def} if} def /FA {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef /PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef /FAn {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef /PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } bdef /PRR {MRR fill } bdef /MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath} bdef /FlrRR {MlrRR stroke } bdef /PlrRR {MlrRR fill } bdef /MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath} bdef /FtbRR {MtbRR stroke } bdef /PtbRR {MtbRR fill } bdef /stri 6 array def /dtri 6 array def /smat 6 array def /dmat 6 array def /tmat1 6 array def /tmat2 6 array def /dif 3 array def /asub {/ind2 exch def /ind1 exch def dup dup ind1 get exch ind2 get sub exch } bdef /tri_to_matrix { 2 0 asub 3 1 asub 4 0 asub 5 1 asub dup 0 get exch 1 get 7 -1 roll astore } bdef /compute_transform { dmat dtri tri_to_matrix tmat1 invertmatrix smat stri tri_to_matrix tmat2 concatmatrix } bdef /ds {stri astore pop} bdef /dt {dtri astore pop} bdef /db {2 copy /cols xdef /rows xdef mul dup string currentfile exch readhexstring pop /bmap xdef pop pop} bdef /it {gs np dtri aload pop moveto lineto lineto cp c cols rows 8 compute_transform {bmap} image gr}bdef /il {newpath moveto lineto stroke}bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 23 179 548 616 MathWorks begin bpage %%EndPageSetup %%BeginObject: obj1 bplot /dpi2point 12 def portraitMode 0216 7344 csm 65 -48 6306 5233 MR c np 91 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef c0 1 j 1 sg 0 0 6913 5186 PR 6 w 0 4226 5356 0 0 -4226 899 4615 4 MP PP -5356 0 0 4226 5356 0 0 -4226 899 4615 5 MP stroke 4 w DO SO 6 w 0 sg 899 4615 mt 6255 4615 L 899 389 mt 6255 389 L 899 4615 mt 899 389 L 6255 4615 mt 6255 389 L 899 4615 mt 6255 4615 L 899 4615 mt 899 389 L 899 4615 mt 899 4561 L 899 389 mt 899 442 L %%IncludeResource: font Helvetica /Helvetica /ISOLatin1Encoding 240 FMSR 833 4872 mt (1) s 1791 4615 mt 1791 4561 L 1791 389 mt 1791 442 L 1625 4872 mt (1.5) s 2684 4615 mt 2684 4561 L 2684 389 mt 2684 442 L 2618 4872 mt (2) s 3577 4615 mt 3577 4561 L 3577 389 mt 3577 442 L 3411 4872 mt (2.5) s 4469 4615 mt 4469 4561 L 4469 389 mt 4469 442 L 4403 4872 mt (3) s 5362 4615 mt 5362 4561 L 5362 389 mt 5362 442 L 5196 4872 mt (3.5) s 6255 4615 mt 6255 4561 L 6255 389 mt 6255 442 L 6189 4872 mt (4) s 899 4615 mt 952 4615 L 6255 4615 mt 6201 4615 L 731 4704 mt (1) s 899 4086 mt 952 4086 L 6255 4086 mt 6201 4086 L 531 4175 mt (1.5) s 899 3558 mt 952 3558 L 6255 3558 mt 6201 3558 L 731 3647 mt (2) s 899 3030 mt 952 3030 L 6255 3030 mt 6201 3030 L 531 3119 mt (2.5) s 899 2502 mt 952 2502 L 6255 2502 mt 6201 2502 L 731 2591 mt (3) s 899 1973 mt 952 1973 L 6255 1973 mt 6201 1973 L 531 2062 mt (3.5) s 899 1445 mt 952 1445 L 6255 1445 mt 6201 1445 L 731 1534 mt (4) s 899 917 mt 952 917 L 6255 917 mt 6201 917 L 531 1006 mt (4.5) s 899 389 mt 952 389 L 6255 389 mt 6201 389 L 731 478 mt (5) s 899 4615 mt 6255 4615 L 899 389 mt 6255 389 L 899 4615 mt 899 389 L 6255 4615 mt 6255 389 L gs 899 389 5357 4227 MR c np 24 w 48 -29 71 -42 69 -41 69 -41 68 -41 68 -41 66 -39 67 -40 65 -39 65 -39 64 -38 64 -38 63 -38 62 -37 62 -37 61 -37 60 -36 60 -36 60 -35 58 -35 59 -35 57 -34 57 -34 57 -34 55 -33 56 -34 55 -32 54 -33 53 -32 54 -32 52 -31 52 -31 52 -31 51 -31 51 -30 50 -30 49 -29 49 -30 49 -29 48 -29 48 -28 47 -28 47 -28 46 -28 46 -27 45 -27 45 -27 44 -27 44 -26 44 -26 43 -26 43 -25 42 -26 42 -25 41 -24 41 -25 41 -24 40 -24 40 -24 40 -24 39 -23 38 -23 39 -23 37 -23 38 -22 37 -22 37 -22 36 -22 36 -22 1804 -1077 916 4593 71 MP stroke 317 -4204 916 4593 2 MP stroke DA 48 -29 71 -41 69 -41 69 -41 68 -41 68 -39 66 -40 67 -39 65 -39 65 -38 64 -38 64 -38 63 -37 62 -37 62 -37 61 -36 60 -36 60 -35 60 -35 58 -35 59 -34 57 -34 57 -34 57 -33 55 -34 56 -32 55 -33 54 -32 53 -32 54 -31 52 -31 52 -31 52 -31 51 -30 51 -30 50 -29 49 -30 49 -29 49 -29 48 -28 48 -28 47 -28 47 -28 46 -27 46 -27 45 -27 45 -27 44 -26 44 -26 44 -26 43 -25 43 -26 42 -25 42 -24 41 -25 41 -24 41 -24 40 -24 40 -24 40 -23 39 -23 38 -23 39 -23 37 -22 38 -22 37 -22 37 -22 36 -22 36 -21 1804 -1067 916 4604 71 MP stroke 8 -4226 916 4615 2 MP stroke gr 24 w DA 3423 5058 mt (m) s %%IncludeResource: font Helvetica /Helvetica /ISOLatin1Encoding 192 FMSR 3622 5178 mt (2) s %%IncludeResource: font Helvetica /Helvetica /ISOLatin1Encoding 240 FMSR 313 2656 mt -90 rotate (m) s 90 rotate %%IncludeResource: font Helvetica /Helvetica /ISOLatin1Encoding 192 FMSR 433 2457 mt -90 rotate (3) s 90 rotate %%IncludeResource: font Helvetica /Helvetica /ISOLatin1Encoding 240 FMSR 2223 198 mt (Stability region \(err=10) s %%IncludeResource: font Helvetica /Helvetica /ISOLatin1Encoding 192 FMSR 4643 78 mt (-5) s %%IncludeResource: font Helvetica /Helvetica /ISOLatin1Encoding 240 FMSR 4861 198 mt (\)) s SO 6 w end eplot %%EndObject epage end showpage %%Trailer %%EOF %%EndDocument endTexFig 1021 1657 a Fu(\(a\))26 b Fq(\017)21 b Fu(=)g(10)1354 1625 y Fb(\000)p Fj(5)1972 365 y 11367087 9472573 1512980 11774935 36048404 40521564 startTexFig 1972 365 a %%BeginDocument: stability_double.eps %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The Mathworks, Inc. %%Title: stability_double.eps %%CreationDate: 02/22/2003 14:24:21 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Pages: 1 %%BoundingBox: 23 179 548 616 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rm /rmoveto ldef /rl /rlineto ldef /s {show newpath} bdef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef /rc {rectclip} bdef /rf {rectfill} bdef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont} bdef /reencode {exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop} bdef /isroman {findfont /CharStrings get /Agrave known} bdef /FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS} bdef /csm {1 dpi2point div -1 dpi2point div scale neg translate dup landscapeMode eq {pop -90 rotate} {rotateMode eq {90 rotate} if} ifelse} bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L {lineto stroke} bdef /MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef /AP {{rlineto} repeat} bdef /PDlw -1 def /W {/PDlw currentlinewidth def setlinewidth} def /PP {closepath eofill} bdef /DP {closepath stroke} bdef /MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath} bdef /FR {MR stroke} bdef /PR {MR fill} bdef /L1i {{currentfile picstr readhexstring pop} image} bdef /tMatrix matrix def /MakeOval {newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix} bdef /FO {MakeOval stroke} bdef /PO {MakeOval fill} bdef /PD {currentlinewidth 2 div 0 360 arc fill PDlw -1 eq not {PDlw w /PDlw -1 def} if} def /FA {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef /PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef /FAn {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef /PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } bdef /PRR {MRR fill } bdef /MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath} bdef /FlrRR {MlrRR stroke } bdef /PlrRR {MlrRR fill } bdef /MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath} bdef /FtbRR {MtbRR stroke } bdef /PtbRR {MtbRR fill } bdef /stri 6 array def /dtri 6 array def /smat 6 array def /dmat 6 array def /tmat1 6 array def /tmat2 6 array def /dif 3 array def /asub {/ind2 exch def /ind1 exch def dup dup ind1 get exch ind2 get sub exch } bdef /tri_to_matrix { 2 0 asub 3 1 asub 4 0 asub 5 1 asub dup 0 get exch 1 get 7 -1 roll astore } bdef /compute_transform { dmat dtri tri_to_matrix tmat1 invertmatrix smat stri tri_to_matrix tmat2 concatmatrix } bdef /ds {stri astore pop} bdef /dt {dtri astore pop} bdef /db {2 copy /cols xdef /rows xdef mul dup string currentfile exch readhexstring pop /bmap xdef pop pop} bdef /it {gs np dtri aload pop moveto lineto lineto cp c cols rows 8 compute_transform {bmap} image gr}bdef /il {newpath moveto lineto stroke}bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 23 179 548 616 MathWorks begin bpage %%EndPageSetup %%BeginObject: obj1 bplot /dpi2point 12 def portraitMode 0216 7344 csm 65 -48 6306 5233 MR c np 91 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef c0 1 j 1 sg 0 0 6913 5186 PR 6 w 0 4226 5356 0 0 -4226 899 4615 4 MP PP -5356 0 0 4226 5356 0 0 -4226 899 4615 5 MP stroke 4 w DO SO 6 w 0 sg 899 4615 mt 6255 4615 L 899 389 mt 6255 389 L 899 4615 mt 899 389 L 6255 4615 mt 6255 389 L 899 4615 mt 6255 4615 L 899 4615 mt 899 389 L 899 4615 mt 899 4561 L 899 389 mt 899 442 L %%IncludeResource: font Helvetica /Helvetica /ISOLatin1Encoding 240 FMSR 833 4872 mt (1) s 1791 4615 mt 1791 4561 L 1791 389 mt 1791 442 L 1625 4872 mt (1.5) s 2684 4615 mt 2684 4561 L 2684 389 mt 2684 442 L 2618 4872 mt (2) s 3577 4615 mt 3577 4561 L 3577 389 mt 3577 442 L 3411 4872 mt (2.5) s 4469 4615 mt 4469 4561 L 4469 389 mt 4469 442 L 4403 4872 mt (3) s 5362 4615 mt 5362 4561 L 5362 389 mt 5362 442 L 5196 4872 mt (3.5) s 6255 4615 mt 6255 4561 L 6255 389 mt 6255 442 L 6189 4872 mt (4) s 899 4615 mt 952 4615 L 6255 4615 mt 6201 4615 L 731 4704 mt (1) s 899 4086 mt 952 4086 L 6255 4086 mt 6201 4086 L 531 4175 mt (1.5) s 899 3558 mt 952 3558 L 6255 3558 mt 6201 3558 L 731 3647 mt (2) s 899 3030 mt 952 3030 L 6255 3030 mt 6201 3030 L 531 3119 mt (2.5) s 899 2502 mt 952 2502 L 6255 2502 mt 6201 2502 L 731 2591 mt (3) s 899 1973 mt 952 1973 L 6255 1973 mt 6201 1973 L 531 2062 mt (3.5) s 899 1445 mt 952 1445 L 6255 1445 mt 6201 1445 L 731 1534 mt (4) s 899 917 mt 952 917 L 6255 917 mt 6201 917 L 531 1006 mt (4.5) s 899 389 mt 952 389 L 6255 389 mt 6201 389 L 731 478 mt (5) s 899 4615 mt 6255 4615 L 899 389 mt 6255 389 L 899 4615 mt 899 389 L 6255 4615 mt 6255 389 L gs 899 389 5357 4227 MR c np 24 w 48 -29 71 -42 69 -41 69 -41 68 -41 68 -41 66 -39 67 -40 65 -39 65 -39 64 -38 64 -38 63 -38 62 -37 62 -37 61 -37 60 -36 60 -36 60 -35 58 -35 59 -35 57 -34 57 -34 57 -34 55 -33 56 -34 55 -32 54 -33 53 -32 54 -32 52 -31 52 -31 52 -31 51 -31 51 -30 50 -30 49 -29 49 -30 49 -29 48 -29 48 -28 47 -28 47 -28 46 -28 46 -27 45 -27 45 -27 44 -27 44 -26 44 -26 43 -26 43 -25 42 -26 42 -25 41 -24 41 -25 41 -24 40 -24 40 -24 40 -24 39 -23 38 -23 39 -23 37 -23 38 -22 37 -22 37 -22 36 -22 36 -22 1804 -1077 916 4593 71 MP stroke 0 -4103 916 4492 2 MP stroke DA 48 -29 71 -41 69 -41 69 -41 68 -41 68 -39 66 -40 67 -39 65 -39 65 -38 64 -38 64 -38 63 -37 62 -37 62 -37 61 -36 60 -36 60 -35 60 -35 58 -35 59 -34 57 -34 57 -34 57 -33 55 -34 56 -32 55 -33 54 -32 53 -32 54 -31 52 -31 52 -31 52 -31 51 -30 51 -30 50 -29 49 -30 49 -29 49 -29 48 -28 48 -28 47 -28 47 -28 46 -27 46 -27 45 -27 45 -27 44 -26 44 -26 44 -26 43 -25 43 -26 42 -25 42 -24 41 -25 41 -24 41 -24 40 -24 40 -24 40 -23 39 -23 38 -23 39 -23 37 -22 38 -22 37 -22 37 -22 36 -22 36 -21 1804 -1067 916 4604 71 MP stroke 8 -4226 916 4615 2 MP stroke gr 24 w DA 3423 5058 mt (m) s %%IncludeResource: font Helvetica /Helvetica /ISOLatin1Encoding 192 FMSR 3622 5178 mt (2) s %%IncludeResource: font Helvetica /Helvetica /ISOLatin1Encoding 240 FMSR 313 2656 mt -90 rotate (m) s 90 rotate %%IncludeResource: font Helvetica /Helvetica /ISOLatin1Encoding 192 FMSR 433 2457 mt -90 rotate (3) s 90 rotate %%IncludeResource: font Helvetica /Helvetica /ISOLatin1Encoding 240 FMSR 2223 198 mt (Stability region \(err=10) s %%IncludeResource: font Helvetica /Helvetica /ISOLatin1Encoding 192 FMSR 4643 78 mt (-9) s %%IncludeResource: font Helvetica /Helvetica /ISOLatin1Encoding 240 FMSR 4861 198 mt (\)) s SO 6 w end eplot %%EndObject epage end showpage %%Trailer %%EOF %%EndDocument endTexFig 2482 1657 a Fu(\(b\))k Fq(\017)c Fu(=)h(10)2820 1625 y Fb(\000)p Fj(9)523 1819 y Fr(Fig.)15 b(8.)j Fu(If)i(the)e(normalized) i(momen)n(ts)d(of)j Fq(G)f Fu(lie)h(b)r(et)n(w)n(een)f(the)g(t)n(w)n(o) g(solid)h(lines,)g(then)f(the)g(normal-)523 1910 y(ized)30 b(momen)n(ts)d(of)j(the)f(EC)h(distribution)f Fq(Z)5 b Fu(,)30 b(pro)n(vided)f(b)n(y)f(the)h(n)n(umerically)g(stable)h (solution,)523 2001 y(are)35 b(insensitiv)n(e)f(to)g(the)g(small)g(c)n (hange)g(\()p Fq(\017)h Fu(=)g(10)2021 1969 y Fb(\000)p Fj(5)2138 2001 y Fu(for)g(\(a\))f(and)g Fq(\017)h Fu(=)g(10)2791 1969 y Fb(\000)p Fj(9)2908 2001 y Fu(for)g(\(b\)\))e(in)h(the)523 2092 y(parameters)26 b(of)i Fq(Z)5 b Fu(.)27 b(The)f(dotted)g(lines)h (delineate)g(the)f(set)h(of)g(all)h(nonnegativ)n(e)e(distributions)h Fq(G)523 2184 y Fu(\()p Fq(m)621 2152 y Fi(G)621 2197 y Fj(3)694 2184 y Ft(\025)21 b Fq(m)843 2152 y Fi(G)843 2197 y Fj(2)915 2184 y Ft(\025)g Fu(1\).)523 2488 y Fp(5)112 b(Conclusion)523 2719 y Fw(In)37 b(this)h(pap)r(er,)e(w)n(e)h(prop)r (ose)f(a)h(closed-form)e(solution)i(for)g(the)g(parameters)e(of)i(a)g (PH)523 2819 y(distribution,)29 b Fn(P)12 b Fw(,)29 b(that)g(w)n (ell-represen)n(ts)d(a)j(giv)n(en)f(distribution)h Fn(G)p Fw(.)g(Our)f(solution)g(is)h(the)523 2919 y(\014rst)19 b(that)h(ac)n(hiev)n(es)d(all)i(of)g(the)h(follo)n(wing)e(goals:)g (\(i\))i(the)f(\014rst)g(three)g(momen)n(ts)g(of)g Fn(G)h Fw(and)f Fn(P)523 3018 y Fw(agree,)25 b(\(ii\))i(an)n(y)e(distribution) h Fn(G)h Fw(that)f(is)g(w)n(ell-represen)n(ted)f(b)n(y)g(a)h(PH)g (distribution)g(\(i.e.,)523 3118 y Fn(G)d Fk(2)h(P)7 b(H)825 3130 y Fv(3)863 3118 y Fw(\))26 b(can)f(b)r(e)i(w)n (ell-represen)n(ted)d(b)n(y)h Fn(P)12 b Fw(,)26 b(\(iii\))h(the)f(n)n (um)n(b)r(er)g(of)g(phases)f(used)h(in)g Fn(P)38 b Fw(is)523 3218 y(at)27 b(most)g Fn(O)r(P)12 b(T)g Fw(\()p Fn(G)p Fw(\))18 b(+)f Fn(c)p Fw(,)27 b(where)g Fn(c)g Fw(is)g(a)g(small)g (constan)n(t,)f(\(iv\))i(the)f(solution)g(is)g(expressed)523 3317 y(in)h(closed)f(form.)g(Also,)g(the)h(n)n(umerical)f(stabilit)n(y) h(of)f(the)h(solution)f(is)h(discussed.)648 3423 y(The)22 b(k)n(ey)f(idea)h(is)g(the)h(de\014nition)g(and)f(use)g(of)g(EC)g (distributions,)g(a)g(subset)g(of)g(PH)g(dis-)523 3523 y(tributions.)i(The)h(set)f(of)g(EC)g(distributions)g(is)g(de\014ned)h (so)e(that)i(it)f(includes)h(minimal)f(PH)523 3623 y(distributions,)29 b(in)h(the)g(sense)f(that)g(for)g(an)n(y)g(distribution,)g Fn(G)p Fw(,)h(that)g(is)f(w)n(ell-represen)n(ted)523 3722 y(b)n(y)f Fn(n)p Fw(-phase)g(acyclic)f(PH)h(distribution,)h(there) f(exists)g(an)g(EC)g(distribution,)h Fn(E)5 b Fw(,)28 b(with)h(at)523 3822 y(most)j Fn(n)22 b Fw(+)f(1)32 b(phases)g(suc)n(h) g(that)h Fn(G)g Fw(is)f(w)n(ell-represen)n(ted)f(b)n(y)h Fn(E)5 b Fw(.)32 b(This)h(prop)r(ert)n(y)e(of)i(the)523 3922 y(set)26 b(of)h(EC)f(distributions)g(is)g(the)h(k)n(ey)f(to)g(ac)n (hieving)f(the)i(ab)r(o)n(v)n(e)e(goals)g(\(i\),)i(\(ii\),)g(and)f (\(iii\).)523 4021 y(Also,)k(the)h(EC)f(distribution)h(is)f(de\014ned)h (so)e(that)i(it)g(has)f(a)g(small)g(n)n(um)n(b)r(er)g(\(six\))h(of)f (free)523 4121 y(parameters.)35 b(This)i(prop)r(ert)n(y)e(of)i(the)g (EC)f(distribution)g(is)h(the)g(k)n(ey)f(to)g(ac)n(hieving)g(the)523 4220 y(ab)r(o)n(v)n(e)g(goal)g(\(iv\).)i(The)g(same)f(ideas)g(are)f (applied)h(to)h(further)f(reduce)g(the)h(degrees)e(of)523 4320 y(freedom)27 b(of)g(the)g(EC)g(distribution.)g(That)g(is,)g(w)n(e) f(constrain)g(one)h(of)g(the)g(six)g(parameters)523 4420 y(of)e(the)g(EC)g(distribution)g(without)g(excluding)g(minimal)g(PH)g (distributions)g(from)g(the)g(set)523 4519 y(of)j(EC)f(distributions.) 648 4625 y(W)-7 b(e)19 b(pro)n(vide)g(a)g(complete)g(c)n (haracterization)e(of)i(the)h(EC)f(distribution)h(with)g(resp)r(ect)f (to)523 4725 y(the)j(normalized)f(momen)n(ts;)g(the)i(c)n (haracterization)c(is)i(enabled)h(b)n(y)g(the)g(simple)g(de\014nition) 523 4825 y(of)c(the)h(EC)f(distribution.)h(The)g(analysis)e(is)h(an)g (elegan)n(t)g(induction)h(based)f(on)g(the)h(recursiv)n(e)523 4924 y(de\014nition)26 b(of)h(the)f(EC)g(distribution;)g(the)g (inductiv)n(e)h(analysis)d(is)i(enabled)g(b)n(y)g(a)g(solution)p eop %%Page: 17 17 17 16 bop 523 448 a Fw(to)21 b(a)f(non)n(trivial)f(recursiv)n(e)g(form) n(ula.)h(Based)g(on)g(the)h(c)n(haracterization,)d(w)n(e)j(pro)n(vide)e (three)523 548 y(v)-5 b(arian)n(ts)26 b(of)i(closed-form)e(solutions)g (for)h(the)h(parameters)e(of)h(the)h(EC)f(distribution)h(that)523 648 y(w)n(ell-represen)n(ts)d(an)n(y)h(input)i(distribution,)f Fn(G)p Fw(,)h(that)f(can)g(b)r(e)g(w)n(ell-represen)n(ted)f(b)n(y)g(a)h (PH)523 747 y(distribution)h(\()p Fn(G)23 b Fk(2)h(P)7 b(H)1310 759 y Fv(3)1348 747 y Fw(\).)648 850 y(One)23 b Fm(take-home)28 b(lesson)d Fw(from)e(this)i(pap)r(er)e(is)h(that)h (the)f(momen)n(t-matc)n(hing)g(problem)523 949 y(is)29 b(b)r(etter)h(solv)n(ed)f(with)g(resp)r(ect)h(to)f(the)h(ab)r(o)n(v)n (e)e(four)h(goals)f(b)n(y)h(sewing)g(together)f(t)n(w)n(o)h(or)523 1049 y(more)34 b(t)n(yp)r(es)g(of)g(distributions,)g(so)g(that)h(one)f (can)g(gain)f(the)i(b)r(est)g(prop)r(erties)e(of)i(b)r(oth.)523 1149 y(The)k(EC)g(distribution)g(sews)f(the)h(t)n(w)n(o-phase)f(Co)n (xian)f(distribution)j(and)e(the)i(Erlang)523 1248 y(distribution.)31 b(The)g(p)r(oin)n(t)g(is)g(that)g(these)g(t)n(w)n(o)f(distributions)g (pro)n(vide)g(sev)n(eral)f(di\013eren)n(t)523 1348 y(and)e(complemen)n (tary)g(desirable)f(prop)r(erties.)648 1451 y(F)-7 b(uture)33 b(w)n(ork)e(includes)j(assessing)d(the)i(minimalit)n(y)g(of)g(our)f (solution)h(with)g(resp)r(ect)523 1550 y(to)25 b(general)e(\(cyclic\))i (PH)g(distributions.)f(If)h(our)f(solution)h(is)f(not)h(close)f(to)g (minimal,)h(then)523 1650 y(\014nding)f(a)g(minimal)g(cyclic)f(PH)h (distribution)g(that)g(w)n(ell-represen)n(ts)e(an)n(y)h(giv)n(en)g (distribu-)523 1750 y(tion)30 b Fn(G)g Fw(is)g(also)e(imp)r(ortan)n(t.) i(While)g(acyclic)f(PH)h(distributions)g(are)f(w)n(ell)g(c)n (haracterized)523 1849 y(in)c([18],)f(the)i(minim)n(um)f(n)n(um)n(b)r (er)g(of)g(phases)f(required)g(for)g(a)h(general)e(\(cyclic\))j(PH)e (distri-)523 1949 y(bution)k(to)f(w)n(ell-represen)n(t)f(a)h(giv)n(en)g (distribution)h(is)f(not)h(kno)n(wn.)523 2213 y Fl(Ac)m(kno)m (wledgemen)m(t)48 b Fw(W)-7 b(e)33 b(w)n(ould)g(lik)n(e)f(to)h(thank)g (Miklos)f(T)-7 b(elek)33 b(for)g(his)g(help)g(in)g(im-)523 2313 y(pro)n(ving)26 b(the)i(presen)n(tation)f(and)g(qualit)n(y)g(of)h (this)f(pap)r(er.)523 2593 y Fp(References)561 2800 y Fu(1.)43 b(D.)34 b(Aldous)g(and)g(L.)h(Shepp.)59 b(The)34 b(least)i(v)l(ariable)e(phase)h(t)n(yp)r(e)e(distribution)i(is)g (Erlang.)663 2891 y Fc(Communic)l(ations)28 b(in)f(Statistics)i(-)f (Stotchastic)i(Mo)l(dels)p Fu(,)d(3:467)g({)f(473,)h(1987.)561 2985 y(2.)43 b(T.)26 b(Altiok.)34 b(On)25 b(the)g(phase-t)n(yp)r(e)f (appro)n(ximations)i(of)g(general)g(distributions.)35 b Fc(IIE)26 b(T)-6 b(r)l(ans-)663 3077 y(actions)p Fu(,)27 b(17:110)h({)e(116,)h(1985.)561 3171 y(3.)43 b(A.)29 b(F)-6 b(eldmann)27 b(and)i(W.)g(Whitt.)44 b(Fitting)30 b(mixtures)e(of)i(exp)r(onen)n(tials)f(to)g(long-tail)i(distri-)663 3262 y(butions)26 b(to)g(analyze)h(net)n(w)n(ork)g(p)r(erformance)f(mo) r(dels.)37 b Fc(Performanc)l(e)29 b(Evaluation)p Fu(,)f(32:245)663 3354 y({)e(279,)h(1998.)561 3448 y(4.)43 b(H.)20 b(F)-6 b(rank)n(e,)20 b(J.)h(Jann,)f(J.)h(Moreira,)h(P)-6 b(.)21 b(P)n(attnaik,)f(and)g(M.)h(Jette.)26 b(An)20 b(ev)l(aluation)g(of)h (parallel)663 3539 y(job)k(sc)n(heduling)h(for)g(ASCI)f (blue-paci\014c.)33 b(In)25 b Fc(Pr)l(o)l(c)l(e)l(e)l(dings)k(of)e(Sup) l(er)l(c)l(omputing)j('99)p Fu(,)25 b(pages)663 3631 y(679)h({)g(691,)i(No)n(v)n(em)n(b)r(er)23 b(1999.)561 3725 y(5.)43 b(M.)37 b(Harc)n(hol-Balter,)i(C.)e(Li,)h(T.)f(Osogami,)h (A.)f(Sc)n(heller-W)-6 b(olf,)38 b(and)e(M.)i(S.)f(Squillan)n(te.)663 3816 y(Analysis)21 b(of)g(task)f(assignmen)n(t)h(with)f(cycle)h (stealing)h(under)e(cen)n(tral)h(queue.)k(In)20 b Fc(Pr)l(o)l(c)l(e)l (e)l(dings)663 3908 y(of)27 b(ICDCS)g('03)p Fu(,)f(pages)g(628{637,)j (Ma)n(y)d(2003.)561 4002 y(6.)43 b(A.)27 b(Horv\023)-38 b(ath)26 b(and)h(M.)h(T)-6 b(elek.)39 b(Appro)n(ximating)26 b(hea)n(vy)g(tailed)i(b)r(eha)n(vior)f(with)g(phase)h(t)n(yp)r(e)663 4093 y(distributions.)23 b(In)18 b Fc(A)l(dvanc)l(es)k(in)f(Matrix-A)n (nalytic)h(Metho)l(ds)h(for)e(Sto)l(chastic)i(Mo)l(dels)p Fu(,)c(pages)663 4185 y(191)26 b({)g(214.)i(Notable)e(Publications,)h (July)e(2000.)561 4279 y(7.)43 b(A.)24 b(Horv\023)-38 b(ath)25 b(and)f(M.)i(T)-6 b(elek.)33 b(Ph\014t:)25 b(A)g(general)h (phase-t)n(yp)r(e)d(\014tting)i(to)r(ol.)34 b(In)25 b Fc(Pr)l(o)l(c)l(e)l(e)l(dings)663 4370 y(of)i(Performanc)l(e)i(TOOLS)e (2002)p Fu(,)g(pages)f(82)h({)f(91,)g(April)g(2002.)561 4465 y(8.)43 b(M.)25 b(A.)g(Johnson.)35 b(Selecting)25 b(parameters)g(of)h(phase)g(distributions:)f(Com)n(bining)g(nonlinear) 663 4556 y(programming,)e(heuristics,)i(and)f(Erlang)h(distributions.) 32 b Fc(ORSA)26 b(Journal)h(on)f(Computing)p Fu(,)663 4647 y(5:69)h({)f(83,)h(1993.)561 4742 y(9.)43 b(M.)26 b(A.)g(Johnson)g(and)g(M.)h(F.)f(T)-6 b(aa\013e.)36 b(An)25 b(in)n(v)n(estigation)i(of)g(phase-distribution)e(momen)n(t-)663 4833 y(matc)n(hing)h(algorithms)i(for)g(use)g(in)f(queueing)g(mo)r (dels.)39 b Fc(Queueing)29 b(Systems)p Fu(,)h(8:129)f({)e(147,)663 4924 y(1991.)p eop %%Page: 18 18 18 17 bop 523 448 a Fu(10.)43 b(M.)25 b(A.)f(Johnson)h(and)g(M.)g(R.)f (T)-6 b(aa\013e.)34 b(Matc)n(hing)25 b(momen)n(ts)d(to)j(phase)g (distributions:)g(Mix-)663 540 y(tures)k(of)i(Erlang)g(distributions)f (of)g(common)e(order.)47 b Fc(Communic)l(ations)32 b(in)f(Statistics)i (|)663 631 y(Sto)l(chastic)c(Mo)l(dels)p Fu(,)e(5:711)g({)f(743,)h (1989.)523 722 y(11.)43 b(M.)25 b(A.)f(Johnson)h(and)f(M.)g(R.)h(T)-6 b(aa\013e.)33 b(Matc)n(hing)24 b(momen)n(ts)f(to)h(phase)h (distributions:)g(Den-)663 814 y(sit)n(y)i(function)h(shap)r(es.)42 b Fc(Communic)l(ations)30 b(in)f(Statistics)i(|)f(Sto)l(chastic)h(Mo)l (dels)p Fu(,)e(6:283)h({)663 905 y(306,)d(1990.)523 996 y(12.)43 b(M.)18 b(A.)f(Johnson)h(and)e(M.)i(R.)g(T)-6 b(aa\013e.)21 b(Matc)n(hing)d(momen)n(ts)d(to)j(phase)f(distributions:) h(Nonlin-)663 1088 y(ear)24 b(programming)f(approac)n(hes.)32 b Fc(Communic)l(ations)26 b(in)g(Statistics)h(|)f(Sto)l(chastic)i(Mo)l (dels)p Fu(,)663 1179 y(6:259)f({)f(281,)h(1990.)523 1270 y(13.)43 b(S.)31 b(Karlin)g(and)f(W.)h(Studden.)49 b Fc(Tchebyche\013)34 b(Systems:)f(With)f(Applic)l(ations)i(in)d(A)n (nalysis)663 1362 y(and)d(Statistics)p Fu(.)36 b(John)26 b(Wiley)g(and)f(Sons,)h(1966.)523 1453 y(14.)43 b(R.)32 b(E.)h(A.)g(Kha)n(y)n(ari,)f(R.)h(Sadre,)f(and)h(B.)g(Ha)n(v)n(erk)n (ort.)54 b(Fitting)33 b(w)n(orld-wide)h(w)n(eb)e(request)663 1544 y(traces)26 b(with)g(the)f(EM-algorithm.)35 b Fc(Performanc)l(e)29 b(Evalutation)p Fu(,)f(52:175)f({)f(191,)i(2003.)523 1636 y(15.)43 b(R.)29 b(Marie.)45 b(Calculating)31 b(equilibrium)d (probabilities)j(for)e Fq(\025)p Fu(\()p Fq(n)p Fu(\))p Fq(=c)2639 1645 y Fi(k)2678 1636 y Fq(=)p Fu(1)p Fq(=n)i Fu(queues.)43 b(In)29 b Fc(Pr)l(o-)663 1727 y(c)l(e)l(e)l(dings)g(of)e Fu(P)n(erformance)f(1980,)i(pages)e(117)h({)f(125,)h(1980.)523 1818 y(16.)43 b(M.)33 b(F.)f(Neuts.)53 b Fc(Matrix-Ge)l(ometric)37 b(Solutions)d(in)f(Sto)l(chastic)j(Mo)l(dels:)d(A)n(n)h(A)n(lgorithmic) 663 1910 y(Appr)l(o)l(ach)p Fu(.)i(The)26 b(Johns)g(Hopkins)f(Univ)n (ersit)n(y)g(Press,)i(1981.)523 2001 y(17.)43 b(T.)31 b(Osogami)f(and)g(M.)h(Harc)n(hol-Balter.)49 b(A)29 b(closed-form)i (solution)g(for)g(mapping)e(general)663 2092 y(distributions)38 b(to)h(minimal)e(PH)h(distributions.)73 b(T)-6 b(ec)n(hnical)39 b(Rep)r(ort)f(CMU-CS-03-114,)663 2183 y(Sc)n(ho)r(ol)26 b(of)g(Computer)f(Science,)h(Carnegie)i(Mellon)e(Univ)n(ersit)n(y)-6 b(,)25 b(2003.)523 2275 y(18.)43 b(T.)26 b(Osogami)g(and)f(M.)h(Harc)n (hol-Balter.)36 b(Necessary)26 b(and)f(su\016cien)n(t)h(conditions)g (for)g(repre-)663 2366 y(sen)n(ting)21 b(general)g(distributions)g(b)n (y)f(Co)n(xians.)27 b(In)20 b Fc(Pr)l(o)l(c)l(e)l(e)l(dings)25 b(of)e(TOOLS)g('03)p Fu(,)e(Septem)n(b)r(er)663 2457 y(2003.)523 2549 y(19.)43 b(T.)31 b(Osogami,)f(M.)h(Harc)n(hol-Balter,) h(and)e(A.)f(Sc)n(heller-W)-6 b(olf.)49 b(Analysis)30 b(of)h(cycle)f(stealing)663 2640 y(with)c(switc)n(hing)g(cost.)35 b(In)25 b Fc(Pr)l(o)l(c)l(e)l(e)l(dings)30 b(of)d(Sigmetrics)i('03)p Fu(,)c(pages)i(184{195,)i(June)c(2003.)523 2731 y(20.)43 b(A.)24 b(Risk)l(a,)h(V.)g(Diev,)f(and)h(E.)g(Smirni.)32 b(E\016cien)n(t)24 b(\014tting)h(of)g(long-tailed)h(data)f(sets)g(in)n (to)g(PH)663 2823 y(distributions.)34 b Fc(Performanc)l(e)29 b(Evaluation)p Fu(,)e(2003)g(\(to)f(app)r(ear\).)523 2914 y(21.)43 b(C.)33 b(Sauer)f(and)g(K.)h(Chandy)-6 b(.)54 b(Appro)n(ximate)31 b(analysis)i(of)g(cen)n(tral)g(serv)n(er)g (mo)r(dels.)55 b Fc(IBM)663 3005 y(Journal)28 b(of)f(R)l(ese)l(ar)l(ch) j(and)e(Development)p Fu(,)f(19:301)h({)e(313,)h(1975.)523 3097 y(22.)43 b(L.)33 b(Sc)n(hmic)n(kler.)57 b(Meda:)34 b(Mixed)g(Erlang)h(distributions)e(as)i(phase-t)n(yp)r(e)d(represen)n (tations)663 3188 y(of)j(empirical)h(distribution)f(functions.)62 b Fc(Communic)l(ations)37 b(in)e(Statistics)j(|)f(Sto)l(chastic)663 3279 y(Mo)l(dels)p Fu(,)26 b(8:131)i({)e(156,)h(1992.)523 3371 y(23.)43 b(M.)36 b(Squillan)n(te.)64 b(Matrix-analytic)36 b(metho)r(ds)f(in)h(sto)r(c)n(hastic)h(parallel-serv)n(er)g(sc)n (heduling)663 3462 y(mo)r(dels.)47 b(In)30 b Fc(A)l(dvanc)l(es)j(in)e (Matrix-A)n(nalytic)i(Metho)l(ds)g(for)f(Sto)l(chastic)i(Mo)l(dels)p Fu(.)d(Notable)663 3553 y(Publications,)c(July)f(1998.)523 3645 y(24.)43 b(D.)26 b(Starobinski)h(and)f(M.)h(Sidi.)37 b(Mo)r(deling)27 b(and)g(analysis)g(of)g(p)r(o)n(w)n(er-tail)h (distributions)f(via)663 3736 y(classical)h(teletra\016c)e(metho)r(ds.) 34 b Fc(Queueing)28 b(Systems)p Fu(,)g(36:243)g({)e(267,)h(2000.)523 3827 y(25.)43 b(M.)31 b(T)-6 b(elek)31 b(and)f(A.)h(Heindl.)49 b(Matc)n(hing)31 b(momen)n(ts)e(for)j(acyclic)f(discrete)h(and)e(con)n (tin)n(uous)663 3919 y(phase-t)n(yp)r(e)20 b(distributions)h(of)h (second)g(order.)27 b Fc(International)e(Journal)f(of)f(Simulation)p Fu(,)e(3:47)663 4010 y({)26 b(57,)g(2003.)523 4101 y(26.)43 b(W.)25 b(Whitt.)33 b(Appro)n(ximating)24 b(a)i(p)r(oin)n(t)f(pro)r (cess)h(b)n(y)e(a)i(renew)n(al)g(pro)r(cess:)g(Tw)n(o)h(basic)f(meth-) 663 4193 y(o)r(ds.)35 b Fc(Op)l(er)l(ations)29 b(R)l(ese)l(ar)l(ch)p Fu(,)f(30:125)g({)e(147,)h(1982.)523 4284 y(27.)43 b(Y.)17 b(Zhang,)i(H.)e(F)-6 b(rank)n(e,)18 b(J.)h(Moreira,)h(and)d(A.)h(Siv)l (asubramaniam.)i(An)d(in)n(tegrated)i(approac)n(h)663 4375 y(to)37 b(parallel)j(sc)n(heduling)d(using)h(gang-sc)n(heduling,)h (bac)n(k\014lling,)f(and)f(migration.)70 b Fc(IEEE)663 4467 y(T)-6 b(r)l(ansactions)29 b(on)f(Par)l(al)t(lel)g(and)g (Distribute)l(d)h(Systems)p Fu(,)e(14:236)h({)e(247,)h(2003.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF