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%!PS-Adobe-2.0 %%Creator: dvips 5.47 Copyright 1986-91 Radical Eye Software %%Title: sp.pro %%EndComments % /Times-Roman findfont 10 scalefont setfont % /Line1 (In Proceedings of Performance Tools '98) def /Line1a (In ) def /Line1b (Proceedings of Performance Tools '98) def % Line1 stringwidth pop 612 exch sub 2 div 772 moveto % /Times-Roman findfont 10 scalefont setfont Line1a show % /Times-Italic findfont 10 scalefont setfont Line1b show % % line 2 % /Line2 (Lecture Notes in Computer Science, Vol. 1468, pp. 231-242, September 1998) def /Line2a (Lecture Notes in Computer Science, ) def /Line2b (Vol. 1468, pp. 231-242, September 1998) def % Line2 stringwidth pop 612 exch sub 2 div 757 moveto % % /Times-Italic findfont 10 scalefont setfont Line2a show % /Times-Roman findfont 10 scalefont setfont Line2b show % %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: outline.dvi %%Pages: 12 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments 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861 y Fu(?)r(;)s Ft(1)1429 891 y Fv(,)g(Mark)g(E.)g(Cro)n(v)n(ella)2108 861 y Fu(??)r(;)t Ft(2)2239 891 y Fv(,)h(and)f(Cristina)g(D.)i(Murta)3114 861 y Fu(?)14 b(?)h(?)5 b(;)s Ft(2)1264 1033 y Fs(1)1337 1065 y Fr(Lab)r(oratory)27 b(for)f(Computer)f(Science,)h(MIT)1454 1157 y Fq(harchol@theory.lcs.mit.edu)1041 1216 y Fs(2)1114 1248 y Fr(Departmen)n(t)e(of)j(Computer)e(Science,)h(Boston)g(Univ)n (ersit)n(y)1514 1339 y Fp(f)p Fq(crovella,murta)p Fp(g)p Fq(@bu.edu)759 1718 y Fo(Abstract.)43 b Fr(W)-6 b(e)17 b(consider)i(a)g(distributed)e(serv)n(er)h(system)g(mo)r(del)f(and)h (ask)h(whic)n(h)759 1809 y(p)r(olicy)34 b(should)g(b)r(e)g(used)f(for)i (assigning)h(tasks)e(to)g(hosts.)g(In)f(our)h(mo)r(del)f(eac)n(h)759 1901 y(host)27 b(pro)r(cesses)h(tasks)f(in)g(First-Come-First-Serv)n(e) f(order)h(and)g(the)f(task's)i(ser-)759 1992 y(vice)37 b(demand)f(is)h(kno)n(wn)g(in)g(adv)l(ance.)g(W)-6 b(e)37 b(consider)g(four)h(task)f(assignmen)n(t)759 2083 y(p)r(olicies)25 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y(w)n(e)e(\014nd)e(that)g(when)h(the)g(task)g(sizes)h(are)g(not)f (highly)f(v)l(ariable,)i(the)f(Dynamic)759 2905 y(p)r(olicy)22 b(is)g(preferable.)h(Ho)n(w)n(ev)n(er)e(when)g(task)h(sizes)g(sho)n(w)g (the)f(degree)h(of)g(v)l(ariabil-)759 2997 y(it)n(y)h(more)f(c)n (haracteristic)i(of)g(empirically)f(measured)f(computer)g(w)n (orkloads,)i(the)759 3088 y(Size-Based)32 b(p)r(olicy)f(is)g(the)f(b)r (est)h(c)n(hoice.)g(W)-6 b(e)31 b(use)f(the)h(resulting)g(observ)l (ations)759 3179 y(to)g(argue)g(in)g(fa)n(v)n(or)g(of)h(a)f(sp)r (eci\014c)g(size-based)g(p)r(olicy)-6 b(,)31 b(SIT)-6 b(A-E,)30 b(that)g(can)h(out-)759 3271 y(p)r(erform)g(the)f(Dynamic)f (p)r(olicy)i(b)n(y)f(almost)g(2)h(orders)g(of)h(magnitude)d(and)i(can) 759 3362 y(outp)r(erform)22 b(other)g(task)g(assignmen)n(t)f(p)r (olicies)j(b)n(y)d(man)n(y)f(orders)j(of)f(magnitude,)759 3453 y(under)j(a)h(realistic)i(task)d(size)i(distribution.)523 3755 y Fm(1)112 b(In)m(tro)s(duction)523 3978 y Fv(T)-7 b(o)38 b(build)h(high-capacit)n(y)e(serv)n(er)f(systems,)i(dev)n(elop)r (ers)f(are)g(increasingly)g(turning)h(to)523 4078 y(distributed)28 b(designs)g(b)r(ecause)f(of)h(their)g(scalabilit)n(y)f(and)g (cost-e\013ectiv)n(eness.)g(Examples)523 4177 y(of)h(this)h(trend)f (include)g(distributed)h(W)-7 b(eb)28 b(serv)n(ers,)f(distributed)h (database)f(serv)n(ers,)f(and)523 4277 y(high)19 b(p)r(erformance)e (computing)i(clusters.)f(In)h(suc)n(h)f(a)g(system,)h(requests)e(for)h (service)g(arriv)n(e)523 4377 y(and)32 b(m)n(ust)f(b)r(e)h(assigned)f (to)h(one)f(of)g(the)i(host)e(mac)n(hines)g(for)g(pro)r(cessing.)f(The) i(rule)g(for)523 4476 y(assigning)26 b(tasks)h(to)g(host)h(mac)n(hines) f(is)g(kno)n(wn)g(as)g(the)h Fl(task)i(assignment)f(p)l(olicy)p Fv(.)p 523 4565 473 4 v 538 4619 a Fk(?)600 4650 y Fr(Supp)r(orted)24 b(b)n(y)h(the)g(NSF)g(P)n(ostdo)r(ctoral)j(F)-6 b(ello)n(wship)27 b(in)e(the)h(Mathematical)g(Sciences.)506 4710 y Fk(??)600 4742 y Fr(Supp)r(orted)e(in)i(part)f(b)n(y)g(NSF)g(Gran)n(ts)h (CCR-9501822)j(and)c(CCR-9706685.)446 4801 y Fk(?)14 b(?)g(?)600 4833 y Fr(Supp)r(orted)42 b(b)n(y)f(a)j(gran)n(t)f(from)f (CAPES,)h(Brazil.)i(P)n(ermanen)n(t)d(address:)i(Depto.)e(de)h(In-)600 4924 y(form\023)-38 b(atica,)27 b(Univ)n(ersidade)e(F)-6 b(ederal)26 b(do)g(P)n(aran\023)-38 b(a,)27 b(Curitiba,)f(PR)g(81531,)i (Brazil.)p eop %%Page: 2 2 2 1 bop 648 448 a Fv(In)27 b(this)h(pap)r(er)g(w)n(e)f(concen)n(trate)f (on)i(the)g(particular)e(mo)r(del)i(of)f(a)h(distributed)g(serv)n(er) 523 548 y(system)f(in)h(whic)n(h)f(eac)n(h)f(incoming)h(task)g(is)g (immediately)g(assigned)f(to)i(a)e(host)h(mac)n(hine,)523 648 y(and)43 b(eac)n(h)e(host)i(mac)n(hine)f(pro)r(cesses)f(its)i (assigned)f(tasks)f(in)i(\014rst-come-\014rst-serv)n(ed)523 747 y(\(F)n(CFS\))32 b(order.)d(W)-7 b(e)31 b(also)f(assume)g(that)h (the)g(task's)f(service)g(demand)h(is)f(kno)n(wn)h(in)g(ad-)523 847 y(v)-5 b(ance.)29 b(Our)g(motiv)-5 b(ation)30 b(for)f(considering)f (this)i(mo)r(del)g(is)f(that)h(it)g(is)f(an)h(abstraction)e(of)523 946 y(some)f(existing)g(distributed)h(serv)n(ers,)e(describ)r(ed)h(in)h (Section)g(3.)648 1046 y(W)-7 b(e)30 b(consider)f(four)g(task)g (assignmen)n(t)g(p)r(olicies)h(commonly)f(prop)r(osed)g(for)g(suc)n(h)g (dis-)523 1146 y(tributed)e(serv)n(er)d(systems:)i(Round-Robin,)g(in)h (whic)n(h)f(tasks)f(are)h(assigned)f(to)h(hosts)g(in)g(a)523 1245 y(cyclical)e(fashion;)h(Random,)f(in)i(whic)n(h)e(eac)n(h)h(task)f (is)h(assigned)e(to)i(eac)n(h)f(host)h(with)h(equal)523 1345 y(probabilit)n(y;)32 b(Size-Based,)g(in)h(whic)n(h)g(all)g(tasks)f (within)h(a)g(certain)f(size)h(range)e(are)h(sen)n(t)523 1445 y(to)f(an)f(individual)h(host;)f(and)h(Dynamic)f(\(also)g(kno)n (wn)g(as)g(Least-W)-7 b(ork-Remaining\))29 b(in)523 1544 y(whic)n(h)34 b(an)g(incoming)f(task)g(is)h(assigned)f(to)h(the)g(host) g(with)g(the)g(least)g(amoun)n(t)f(of)h(out-)523 1644 y(standing)h(w)n(ork)e(left)j(to)f(do)g(\(based)g(on)g(the)g(sum)g(of)g (the)h(sizes)e(of)h(those)g(tasks)f(in)i(the)523 1743 y(queue\).)648 1843 y(Our)h(goal)g(is)h(to)h(study)f(the)h(in\015uence) f(of)h(task)e(size)h(v)-5 b(ariabilit)n(y)38 b(on)g(the)h(decision)523 1943 y(of)d(whic)n(h)h(task)e(assignmen)n(t)h(p)r(olicy)g(is)g(b)r (est.)h(W)-7 b(e)36 b(are)g(motiv)-5 b(ated)36 b(in)h(this)f(resp)r (ect)g(b)n(y)523 2042 y(the)30 b(increasing)f(evidence)g(for)h(high)f (v)-5 b(ariabilit)n(y)29 b(in)h(task)g(size)f(distributions,)h (witnessed)523 2142 y(in)h(man)n(y)g(measuremen)n(ts)e(of)i(computer)g (w)n(orkloads.)d(In)k(particular,)d(measuremen)n(ts)h(of)523 2242 y(man)n(y)j(computer)g(w)n(orkloads)d(ha)n(v)n(e)i(b)r(een)i(sho)n (wn)f(to)g(\014t)g(a)g(hea)n(vy-tailed)f(distributions)523 2341 y(with)c(v)n(ery)f(high)g(v)-5 b(ariance,)27 b(as)f(describ)r(ed)i (in)g(Section)f(2.2.)648 2441 y(In)h(comparing)g(task)g(assignmen)n(t)f (p)r(olicies,)i(w)n(e)f(mak)n(e)g(use)g(of)h(sim)n(ulations)f(and)g (also)523 2540 y(analysis)21 b(or)g(analytic)g(appro)n(ximations.)f(W) -7 b(e)22 b(sho)n(w)f(that)h(the)g(v)-5 b(ariabilit)n(y)21 b(of)h(the)g(task)f(size)523 2640 y(distribution)k(mak)n(es)e(a)i (crucial)f(di\013erence)g(in)h(c)n(ho)r(osing)e(a)i(task)f(assignmen)n (t)f(p)r(olicy)-7 b(,)25 b(and)523 2740 y(w)n(e)k(use)g(the)h (resulting)f(observ)-5 b(ations)28 b(to)h(argue)f(for)h(a)g(sp)r (eci\014c)h(task)f(assignmen)n(t)f(p)r(olicy)523 2839 y(that)g(w)n(orks)e(w)n(ell)h(under)h(conditions)f(of)g(high)h(task)f (size)g(v)-5 b(ariance.)523 3092 y Fm(2)112 b(Bac)m(kground)38 b(and)h(Previous)d(W)-9 b(ork)523 3277 y Fj(2.1)95 b(F)-8 b(undamen)m(tal)31 b(Results)f(in)h(T)-8 b(ask)33 b(Assignmen)m(t)523 3430 y Fv(The)27 b(problem)f(of)h(task)f(assignmen)n(t)g(in)h(a)f(mo)r (del)h(lik)n(e)f(ours)g(has)g(b)r(een)h(extensiv)n(ely)f(stud-)523 3530 y(ied,)38 b(but)h(man)n(y)e(basic)g(questions)g(remain)h(op)r(en.) g(In)g(the)g(case)f(where)g(task)h(sizes)f(are)523 3629 y(unkno)n(wn,)d(the)h(follo)n(wing)f(results)g(exist:)g(Under)h(an)f (exp)r(onen)n(tial)g(task)g(size)g(distribu-)523 3729 y(tion,)g(the)g(optimalit)n(y)g(of)g(Shortest-Line)f(task)g(assignmen)n (t)g(p)r(olicy)g(\(send)i(the)f(task)f(to)523 3828 y(the)38 b(host)f(with)h(the)g(shortest)f(queue\))g(w)n(as)g(pro)n(v)n(en)f(b)n (y)h(Winston)h([14])e(and)i(extended)523 3928 y(b)n(y)27 b(W)-7 b(eb)r(er)27 b([12])f(to)h(include)g(task)g(size)g (distributions)f(with)i(nondecreasing)d(failure)i(rate.)523 4028 y(The)g(actual)f(p)r(erformance)f(of)h(the)h(Shortest-Line)f(p)r (olicy)g(is)h(not)f(kno)n(wn)g(exactly)-7 b(,)26 b(but)h(is)523 4127 y(appro)n(ximated)32 b(b)n(y)h(Nelson)g(and)g(Phillips)g([9].)g (In)h(fact)f(as)g(the)g(v)-5 b(ariabilit)n(y)33 b(of)g(the)h(task)523 4227 y(size)22 b(distribution)g(gro)n(ws,)e(the)j(Shortest-Line)e(p)r (olicy)h(is)g(no)g(longer)e(optimal,)i(Whitt)i([13].)648 4327 y(In)e(the)i(case)d(where)i(the)g(individual)g(task)f(sizes)g(are) g(kno)n(wn,)g(as)g(in)h(our)f(mo)r(del,)h(equiv-)523 4426 y(alen)n(t)30 b(optimalit)n(y)h(and)f(p)r(erformance)g(results)g (ha)n(v)n(e)f(not)i(b)r(een)g(dev)n(elop)r(ed)f(for)h(the)g(task)523 4526 y(assignmen)n(t)23 b(problem,)h(to)h(the)f(b)r(est)h(of)g(our)e (kno)n(wledge.)g(F)-7 b(or)24 b(the)h(scenario)e(in)h(whic)n(h)h(the) 523 4625 y(ages)32 b(of)i(the)f(tasks)g(curren)n(tly)g(serving)f(are)g (kno)n(wn,)h(W)-7 b(eb)r(er)34 b([12])e(has)h(sho)n(wn)g(that)h(the)523 4725 y(Shortest-Exp)r(ected-Dela)n(y)24 b(rule)i(is)g(optimal)g(for)f (task)g(size)h(distributions)g(with)g(increas-)523 4825 y(ing)d(failure)g(rate,)g(and)h(Whitt)g([13])f(has)g(sho)n(wn)g(that)g (there)h(exist)f(task)g(size)g(distributions)523 4924 y(for)k(whic)n(h)h(the)g(Shortest-Exp)r(ected-Dela)n(y)d(rule)j(is)f (not)h(optimal.)p eop %%Page: 3 3 3 2 bop 1302 365 a 10656645 10656645 13024788 18945146 27299430 33219788 startTexFig 1302 365 a %%BeginDocument: UNIXlifetimes-loglog.ps 50 dict begin % This is a standard prolog for Postscript generated by Tk's canvas % widget. % @(#) prolog.ps 1.2 94/12/09 10:53:18 % The definitions below just define all of the variables used in % any of the procedures here. This is needed for obscure reasons % explained on p. 716 of the Postscript manual (Section H.2.7, % "Initializing Variables," in the section on Encapsulated Postscript). /baseline 0 def /stipimage 0 def /height 0 def /justify 0 def /lineLength 0 def /spacing 0 def /stipple 0 def /strings 0 def /xoffset 0 def /yoffset 0 def /tmpstip null def % Define the array ISOLatin1Encoding (which specifies how characters are % encoded for ISO-8859-1 fonts), if it isn't already present (Postscript % level 2 is supposed to define it, but level 1 doesn't). systemdict /ISOLatin1Encoding known not { /ISOLatin1Encoding [ /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /minus /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /dotlessi /grave /acute /circumflex /tilde /macron /breve /dotaccent /dieresis /space /ring /cedilla /space /hungarumlaut /ogonek /caron /space /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen /registered /macron /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedillar /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis ] def } if % Override the setfont procedure with a new procedure that re-encodes % the font to use the ISO Latin-1 style. The body of this procedure % comes from Section 5.6.1 of the Postscript book. /realsetfont /setfont load def /setfont { dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end % I'm not sure why it's necessary to use "definefont" on this new % font, but it seems to be important; just use the name "Temporary" % for the font. /Temporary exch definefont realsetfont } bind def % StrokeClip % % This procedure converts the current path into a clip area under % the assumption of stroking. It's a bit tricky because some Postscript % interpreters get errors during strokepath for dashed lines. If % this happens then turn off dashes and try again. /StrokeClip { {strokepath} stopped { (This Postscript printer gets limitcheck overflows when) = (stippling dashed lines; lines will be printed solid instead.) = [] 0 setdash strokepath} if clip } bind def % desiredSize EvenPixels closestSize % % The procedure below is used for stippling. Given the optimal size % of a dot in a stipple pattern in the current user coordinate system, % compute the closest size that is an exact multiple of the device's % pixel size. This allows stipple patterns to be displayed without % aliasing effects. /EvenPixels { % Compute exact number of device pixels per stipple dot. dup 0 matrix currentmatrix dtransform dup mul exch dup mul add sqrt % Round to an integer, make sure the number is at least 1, and compute % user coord distance corresponding to this. dup round dup 1 lt {pop 1} if exch div mul } bind def % width height string StippleFill -- % % Given a path already set up and a clipping region generated from % it, this procedure will fill the clipping region with a stipple % pattern. "String" contains a proper image description of the % stipple pattern and "width" and "height" give its dimensions. Each % stipple dot is assumed to be about one unit across in the current % user coordinate system. This procedure trashes the graphics state. /StippleFill { % The following code is needed to work around a NeWSprint bug. /tmpstip 1 index def % Change the scaling so that one user unit in user coordinates % corresponds to the size of one stipple dot. 1 EvenPixels dup scale % Compute the bounding box occupied by the path (which is now % the clipping region), and round the lower coordinates down % to the nearest starting point for the stipple pattern. Be % careful about negative numbers, since the rounding works % differently on them. pathbbox 4 2 roll 5 index div dup 0 lt {1 sub} if cvi 5 index mul 4 1 roll 6 index div dup 0 lt {1 sub} if cvi 6 index mul 3 2 roll % Stack now: width height string y1 y2 x1 x2 % Below is a doubly-nested for loop to iterate across this area % in units of the stipple pattern size, going up columns then % across rows, blasting out a stipple-pattern-sized rectangle at % each position 6 index exch { 2 index 5 index 3 index { % Stack now: width height string y1 y2 x y gsave 1 index exch translate 5 index 5 index true matrix tmpstip imagemask grestore } for pop } for pop pop pop pop pop } bind def % -- AdjustColor -- % Given a color value already set for output by the caller, adjusts % that value to a grayscale or mono value if requested by the CL % variable. /AdjustColor { CL 2 lt { currentgray CL 0 eq { .5 lt {0} {1} ifelse } if setgray } if } bind def % x y strings lineLength spacing xoffset yoffset justify stipple DrawText -- % This procedure does all of the real work of drawing text. The % color and font must already have been set by the caller, and the % following arguments must be on the stack: % % x, y - Coordinates at which to draw text. % strings - An array of strings, one for each line of the text item, % in order from top to bottom. % lineLength - Minimum line length: needed to justify text properly. % spacing - Spacing between lines. % xoffset - Horizontal offset for text bbox relative to x and y: 0 for % nw/w/sw anchor, -0.5 for n/center/s, and -1.0 for ne/e/se. % yoffset - Vertical offset for text bbox relative to x and y: 0 for % nw/n/ne anchor, +0.5 for w/center/e, and +1.0 for sw/s/se. % justify - 0 for left justification, 0.5 for center, 1 for right justify. % stipple - Boolean value indicating whether or not text is to be % drawn in stippled fashion. If text is stippled, % procedure StippleText must have been defined to call % StippleFill in the right way. % % Also, when this procedure is invoked, the color and font must already % have been set for the text. /DrawText { /stipple exch def /justify exch def /yoffset exch def /xoffset exch def /spacing exch def /lineLength exch def /strings exch def % First scan through all of the text to find the widest line (if it's % longer than the "lineLength" argument). strings { stringwidth pop dup lineLength gt {/lineLength exch def} {pop} ifelse newpath } forall % Compute the baseline offset and the actual font height. 0 0 moveto (TXygqPZ) false charpath pathbbox dup /baseline exch def exch pop exch sub /height exch def pop newpath % Translate coordinates first so that the origin is at the upper-left % corner of the text's bounding box. Remember that x and y for % positioning are still on the stack. translate lineLength xoffset mul strings length 1 sub spacing mul height add yoffset mul translate % Now use the baseline and justification information to translate so % that the origin is at the baseline and positioning point for the % first line of text. justify lineLength mul baseline neg translate % Iterate over each of the lines to output it. For each line, % compute its width again so it can be properly justified, then % display it. strings { dup stringwidth pop justify neg mul 0 moveto stipple { % The text is stippled, so turn it into a path and print % by calling StippledText, which in turn calls StippleFill. % Unfortunately, many Postscript interpreters will get % overflow errors if we try to do the whole string at % once, so do it a character at a time. gsave /char (X) def { char 0 3 -1 roll put currentpoint gsave char true charpath clip StippleText grestore char stringwidth translate moveto } forall grestore } {show} ifelse 0 spacing neg translate } forall } bind def /CL 2 def save 306.0 396.0 translate 0.426 0.426 scale -253 -253 translate 0 507 moveto 507 507 lineto 507 0 lineto 0 0 lineto closepath clip newpath gsave 80 447 moveto 400 0 rlineto 0 -380 rlineto -400 0 rlineto closepath 1 setlinewidth 0 setlinejoin 2 setlinecap 0.000 0.000 0.000 setrgbcolor AdjustColor stroke grestore gsave /Helvetica findfont 18 scalefont setfont 0.000 0.000 0.000 setrgbcolor AdjustColor 280 487 [ (Distribution of process lifetimes \050log plot\051) ] 0 20 -0.5 0.5 0 false DrawText grestore gsave /Helvetica findfont 18 scalefont setfont 0.000 0.000 0.000 setrgbcolor AdjustColor 280 464 [ (\050fraction of processes with duration > T\051) ] 0 20 -0.5 0.5 0 false DrawText grestore gsave /Helvetica findfont 18 scalefont setfont 0.000 0.000 0.000 setrgbcolor AdjustColor 495 12 [ (Duration \050T secs.\051) ] 0 20 -1 1 0 false DrawText grestore gsave /Helvetica findfont 18 scalefont setfont 0.000 0.000 0.000 setrgbcolor AdjustColor 65 447 [ (1) ] 0 20 -1 0.5 0 false DrawText grestore gsave /Helvetica findfont 18 scalefont setfont 0.000 0.000 0.000 setrgbcolor AdjustColor 65 390 [ (1/2) ] 0 20 -1 0.5 0 false DrawText grestore gsave /Helvetica findfont 18 scalefont setfont 0.000 0.000 0.000 setrgbcolor AdjustColor 65 332.174 [ (1/4) ] 0 20 -1 0.5 0 false DrawText grestore gsave /Helvetica findfont 18 scalefont setfont 0.000 0.000 0.000 setrgbcolor AdjustColor 65 275.174 [ (1/8) ] 0 20 -1 0.5 0 false DrawText grestore gsave /Helvetica findfont 18 scalefont setfont 0.000 0.000 0.000 setrgbcolor AdjustColor 65 218.174 [ (1/16) ] 0 20 -1 0.5 0 false DrawText grestore gsave /Helvetica findfont 18 scalefont setfont 0.000 0.000 0.000 setrgbcolor AdjustColor 65 160.348 [ (1/32) ] 0 20 -1 0.5 0 false DrawText grestore gsave /Helvetica findfont 18 scalefont setfont 0.000 0.000 0.000 setrgbcolor AdjustColor 65 103.348 [ (1/64) ] 0 20 -1 0.5 0 false DrawText grestore gsave /Helvetica findfont 18 scalefont setfont 0.000 0.000 0.000 setrgbcolor AdjustColor 80 46.9999999999999 [ (1) ] 0 20 -0.5 0.5 0 false DrawText grestore gsave /Helvetica findfont 18 scalefont setfont 0.000 0.000 0.000 setrgbcolor AdjustColor 140 46.9999999999999 [ (2) ] 0 20 -0.5 0.5 0 false DrawText grestore gsave /Helvetica findfont 18 scalefont setfont 0.000 0.000 0.000 setrgbcolor AdjustColor 200.87 46.9999999999999 [ (4) ] 0 20 -0.5 0.5 0 false DrawText grestore gsave /Helvetica findfont 18 scalefont setfont 0.000 0.000 0.000 setrgbcolor AdjustColor 260.87 46.9999999999999 [ (8) ] 0 20 -0.5 0.5 0 false DrawText grestore gsave /Helvetica findfont 18 scalefont setfont 0.000 0.000 0.000 setrgbcolor AdjustColor 320.87 46.9999999999999 [ (16) ] 0 20 -0.5 0.5 0 false DrawText grestore gsave /Helvetica findfont 18 scalefont setfont 0.000 0.000 0.000 setrgbcolor AdjustColor 381.739 46.9999999999999 [ (32) ] 0 20 -0.5 0.5 0 false DrawText grestore gsave /Helvetica findfont 18 scalefont setfont 0.000 0.000 0.000 setrgbcolor AdjustColor 441.739 46.9999999999999 [ (64) ] 0 20 -0.5 0.5 0 false DrawText grestore gsave matrix currentmatrix 80.5 445.6599 translate 0.499999999999993 0.5 scale 1 0 moveto 0 0 1 0 360 arc setmatrix 1 setlinewidth 0 setlinejoin 2 setlinecap 0.000 0.000 0.000 setrgbcolor AdjustColor stroke grestore gsave matrix currentmatrix 82.2217 444.6273 translate 0.500000000000007 0.5 scale 1 0 moveto 0 0 1 0 360 arc 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orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack 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 landscapeMode eq {90 rotate} if } 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 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389 mt 1969 443 L 1869 4785 mt (0.4) s 2505 4616 mt 2505 4562 L 2505 389 mt 2505 443 L 2405 4785 mt (0.6) s 3041 4616 mt 3041 4562 L 3041 389 mt 3041 443 L 2941 4785 mt (0.8) s 3577 4616 mt 3577 4562 L 3577 389 mt 3577 443 L 3537 4785 mt (1) s 4112 4616 mt 4112 4562 L 4112 389 mt 4112 443 L 4012 4785 mt (1.2) s 4648 4616 mt 4648 4562 L 4648 389 mt 4648 443 L 4548 4785 mt (1.4) s 5184 4616 mt 5184 4562 L 5184 389 mt 5184 443 L 5084 4785 mt (1.6) s 5719 4616 mt 5719 4562 L 5719 389 mt 5719 443 L 5619 4785 mt (1.8) s 6255 4616 mt 6255 4562 L 6255 389 mt 6255 443 L 6215 4785 mt (2) s 898 4616 mt 925 4616 L 6255 4616 mt 6228 4616 L 898 4616 mt 952 4616 L 6255 4616 mt 6201 4616 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 4669 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 4580 mt (7) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 4434 mt 925 4434 L 6255 4434 mt 6228 4434 L 898 4328 mt 925 4328 L 6255 4328 mt 6228 4328 L 898 4252 mt 925 4252 L 6255 4252 mt 6228 4252 L 898 4194 mt 925 4194 L 6255 4194 mt 6228 4194 L 898 4146 mt 925 4146 L 6255 4146 mt 6228 4146 L 898 4106 mt 925 4106 L 6255 4106 mt 6228 4106 L 898 4071 mt 925 4071 L 6255 4071 mt 6228 4071 L 898 4040 mt 925 4040 L 6255 4040 mt 6228 4040 L 898 4012 mt 925 4012 L 6255 4012 mt 6228 4012 L 898 4012 mt 952 4012 L 6255 4012 mt 6201 4012 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 4065 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 3976 mt (8) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 3830 mt 925 3830 L 6255 3830 mt 6228 3830 L 898 3724 mt 925 3724 L 6255 3724 mt 6228 3724 L 898 3649 mt 925 3649 L 6255 3649 mt 6228 3649 L 898 3590 mt 925 3590 L 6255 3590 mt 6228 3590 L 898 3542 mt 925 3542 L 6255 3542 mt 6228 3542 L 898 3502 mt 925 3502 L 6255 3502 mt 6228 3502 L 898 3467 mt 925 3467 L 6255 3467 mt 6228 3467 L 898 3436 mt 925 3436 L 6255 3436 mt 6228 3436 L 898 3408 mt 925 3408 L 6255 3408 mt 6228 3408 L 898 3408 mt 952 3408 L 6255 3408 mt 6201 3408 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 3461 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 3372 mt (9) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 3227 mt 925 3227 L 6255 3227 mt 6228 3227 L 898 3120 mt 925 3120 L 6255 3120 mt 6228 3120 L 898 3045 mt 925 3045 L 6255 3045 mt 6228 3045 L 898 2986 mt 925 2986 L 6255 2986 mt 6228 2986 L 898 2938 mt 925 2938 L 6255 2938 mt 6228 2938 L 898 2898 mt 925 2898 L 6255 2898 mt 6228 2898 L 898 2863 mt 925 2863 L 6255 2863 mt 6228 2863 L 898 2832 mt 925 2832 L 6255 2832 mt 6228 2832 L 898 2804 mt 925 2804 L 6255 2804 mt 6228 2804 L 898 2804 mt 952 2804 L 6255 2804 mt 6201 2804 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 2857 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 2768 mt (10) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 2623 mt 925 2623 L 6255 2623 mt 6228 2623 L 898 2516 mt 925 2516 L 6255 2516 mt 6228 2516 L 898 2441 mt 925 2441 L 6255 2441 mt 6228 2441 L 898 2382 mt 925 2382 L 6255 2382 mt 6228 2382 L 898 2335 mt 925 2335 L 6255 2335 mt 6228 2335 L 898 2294 mt 925 2294 L 6255 2294 mt 6228 2294 L 898 2259 mt 925 2259 L 6255 2259 mt 6228 2259 L 898 2228 mt 925 2228 L 6255 2228 mt 6228 2228 L 898 2201 mt 925 2201 L 6255 2201 mt 6228 2201 L 898 2201 mt 952 2201 L 6255 2201 mt 6201 2201 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 2254 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 2165 mt (11) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 2019 mt 925 2019 L 6255 2019 mt 6228 2019 L 898 1912 mt 925 1912 L 6255 1912 mt 6228 1912 L 898 1837 mt 925 1837 L 6255 1837 mt 6228 1837 L 898 1778 mt 925 1778 L 6255 1778 mt 6228 1778 L 898 1731 mt 925 1731 L 6255 1731 mt 6228 1731 L 898 1690 mt 925 1690 L 6255 1690 mt 6228 1690 L 898 1655 mt 925 1655 L 6255 1655 mt 6228 1655 L 898 1624 mt 925 1624 L 6255 1624 mt 6228 1624 L 898 1597 mt 925 1597 L 6255 1597 mt 6228 1597 L 898 1597 mt 952 1597 L 6255 1597 mt 6201 1597 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 1650 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 1561 mt (12) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 1415 mt 925 1415 L 6255 1415 mt 6228 1415 L 898 1309 mt 925 1309 L 6255 1309 mt 6228 1309 L 898 1233 mt 925 1233 L 6255 1233 mt 6228 1233 L 898 1175 mt 925 1175 L 6255 1175 mt 6228 1175 L 898 1127 mt 925 1127 L 6255 1127 mt 6228 1127 L 898 1086 mt 925 1086 L 6255 1086 mt 6228 1086 L 898 1051 mt 925 1051 L 6255 1051 mt 6228 1051 L 898 1020 mt 925 1020 L 6255 1020 mt 6228 1020 L 898 993 mt 925 993 L 6255 993 mt 6228 993 L 898 993 mt 952 993 L 6255 993 mt 6201 993 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 1046 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 957 mt (13) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 811 mt 925 811 L 6255 811 mt 6228 811 L 898 705 mt 925 705 L 6255 705 mt 6228 705 L 898 629 mt 925 629 L 6255 629 mt 6228 629 L 898 571 mt 925 571 L 6255 571 mt 6228 571 L 898 523 mt 925 523 L 6255 523 mt 6228 523 L 898 483 mt 925 483 L 6255 483 mt 6228 483 L 898 448 mt 925 448 L 6255 448 mt 6228 448 L 898 417 mt 925 417 L 6255 417 mt 6228 417 L 898 389 mt 925 389 L 6255 389 mt 6228 389 L 898 389 mt 952 389 L 6255 389 mt 6201 389 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 442 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 353 mt (14) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 4616 mt 6255 4616 L 898 389 mt 6255 389 L 898 4616 mt 898 389 L 6255 4616 mt 6255 389 L 898 389 mt 898 389 L 6255 389 mt 6255 389 L gs 898 389 5358 4228 MR c np 268 192 268 237 268 269 267 289 268 298 268 300 268 296 268 283 268 258 267 220 268 170 268 121 268 81 268 56 268 40 268 31 267 25 268 20 268 16 1166 901 20 MP stroke gr 36 36 1166 901 FO 36 36 1434 917 FO 36 36 1702 937 FO 36 36 1969 962 FO 36 36 2237 993 FO 36 36 2505 1033 FO 36 36 2773 1089 FO 36 36 3041 1170 FO 36 36 3309 1291 FO 36 36 3577 1461 FO 36 36 3844 1681 FO 36 36 4112 1939 FO 36 36 4380 2222 FO 36 36 4648 2518 FO 36 36 4916 2818 FO 36 36 5184 3116 FO 36 36 5451 3405 FO 36 36 5719 3674 FO 36 36 5987 3911 FO 36 36 6255 4103 FO gs 898 389 5358 4228 MR c np gr 2055 274 mt (Second Moment of Bounded Pareto Distribution) s 3401 4945 mt (alpha) s end eplot epage end showpage %%EndDocument endTexFig 523 1744 a Fo(Fig.)15 b(2.)40 b Fr(P)n(arameters)i(of)g(the)e(Bounded) h(P)n(areto)h(Distribution)f(\(left\);)h(Second)f(Momen)n(t)f(of)523 1836 y Fi(B)t Fr(\()p Fi(k)r(;)13 b Fr(10)767 1804 y Fs(10)833 1836 y Fi(;)g(\013)p Fr(\))26 b(as)g(a)g(function)g(of)g Fi(\013)p Fr(,)g(when)g Fo(E)13 b Fp(f)p Fi(X)6 b Fp(g)22 b Fr(=)f(3000)27 b(\(righ)n(t\).)p 591 1960 2747 4 v 589 2051 4 92 v 603 2024 a(Num)n(b)r(er)c(of)k(hosts)p 1812 2051 V 668 w Fi(h)21 b Fr(=)g(8.)p 3336 2051 V 591 2054 2747 4 v 589 2146 4 92 v 603 2118 a(System)i(load)p 1812 2146 V 817 w Fi(\032)d Fr(=)i Fi(:)p Fr(8.)p 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y(Figure)h(2)h(\(righ)n(t\))g(sho)n(ws)f(the)h(second)g(momen)n(t) g Fj(E)2253 3669 y Fd(\010)2302 3737 y Fg(X)2378 3707 y Ft(2)2414 3669 y Fd(\011)2503 3737 y Fv(of)g(this)g(distribution)g (as)f(a)523 3836 y(function)19 b(of)g Fg(\013)g Fv(for)f Fg(p)23 b Fv(=)g(10)1352 3806 y Ft(10)1421 3836 y Fv(,)c(where)f Fg(k)k Fv(is)d(c)n(hosen)e(to)i(k)n(eep)f Fj(E)c Fh(f)o Fg(X)7 b Fh(g)18 b Fv(constan)n(t)g(at)h(3000,)d(\(0)23 b Fg(<)523 3936 y(k)j Fh(\024)d Fv(1500\).)j(The)i(\014gure)f(sho)n(ws) f(that)i(the)h(second)e(momen)n(t)g(explo)r(des)h(exp)r(onen)n(tially)f (as)523 4036 y Fg(\013)h Fv(declines.)f(F)-7 b(urthermore,)26 b(the)i(Bounded)f(P)n(areto)e(distribution)i(also)g(still)g(exhibits)h (the)523 4135 y(hea)n(vy-tailed)e(prop)r(ert)n(y)h(and)g(\(to)h(some)f (exten)n(t\))i(the)f(decreasing)e(failure)h(rate)g(prop)r(ert)n(y)523 4235 y(of)h(the)g(un)n(b)r(ounded)g(P)n(areto)d(distribution.)648 4336 y(Giv)n(en)k(the)g(ab)r(o)n(v)n(e)f(mo)r(del)i(of)f(a)g (distributed)h(serv)n(er)e(system,)h(w)n(e)g(ask)f(ho)n(w)h(to)g (select)523 4435 y(the)f(b)r(est)g(task)f(assignmen)n(t)g(p)r(olicy)-7 b(.)27 b(The)h(follo)n(wing)e(four)i(are)e(common)h(c)n(hoices:)523 4624 y Fj(Random)40 b Fv(:)27 b(an)g(incoming)g(task)f(is)h(sen)n(t)g (to)g(host)g Fg(i)f Fv(with)i(probabilit)n(y)e(1)p Fg(=h)p Fv(.)g(This)h(p)r(olicy)664 4724 y(equalizes)g(the)h(exp)r(ected)g(n)n (um)n(b)r(er)f(of)h(tasks)e(at)i(eac)n(h)f(host.)523 4825 y Fj(Round-Robin)40 b Fv(:)f(tasks)f(are)f(assigned)h(to)g(hosts)g (in)h(cyclical)f(fashion)h(with)g(the)g Fg(i)p Fv(th)664 4924 y(task)26 b(b)r(eing)g(assigned)f(to)i(host)f Fg(i)c Fv(mo)r(d)i Fg(h)p Fv(.)i(This)g(p)r(olicy)g(also)g(equalizes)f(the)i (exp)r(ected)p eop %%Page: 6 6 6 5 bop 664 448 a Fv(n)n(um)n(b)r(er)23 b(of)g(tasks)f(at)h(eac)n(h)g (host,)g(and)g(t)n(ypically)f(has)h(less)f(v)-5 b(ariabilit)n(y)23 b(in)g(in)n(terarriv)-5 b(al)664 548 y(times)28 b(than)g(Random.)523 644 y Fj(Size-Based)41 b Fv(:)23 b(Eac)n(h)f(host)i(serv)n(es)d(tasks)i (whose)f(service)h(demand)g(falls)g(in)h(a)f(designated)664 743 y(range.)g(This)h(p)r(olicy)g(attempts)h(to)f(k)n(eep)g(small)g (tasks)f(from)h(getting)g(\\stuc)n(k")f(b)r(ehind)664 843 y(large)j(tasks.)523 939 y Fj(Dynamic)41 b Fv(:)21 b(Eac)n(h)e(incoming)h(task)g(is)h(assigned)e(to)i(the)g(host)f(with)h (the)g(smallest)f(amoun)n(t)664 1039 y(of)28 b(outstanding)f(w)n(ork,)f (whic)n(h)h(is)h(the)g(sum)f(of)h(the)g(sizes)f(of)g(the)h(tasks)f(in)h (the)f(host's)664 1138 y(queue)37 b(plus)g(the)g(w)n(ork)f(remaining)g (on)g(that)h(task)g(curren)n(tly)e(b)r(eing)i(serv)n(ed.)f(This)664 1238 y(p)r(olicy)k(is)h(optimal)f(from)g(the)h(standp)r(oin)n(t)f(of)h (an)f(individual)g(task,)g(and)g(from)g(a)664 1338 y(system)28 b(standp)r(oin)n(t)f(attempts)h(to)g(ac)n(hiev)n(e)e(instan)n(taneous)g (load)h(balance.)648 1502 y(In)35 b(this)h(pap)r(er)f(w)n(e)g(compare)f (these)i(p)r(olicies)f(as)g(a)g(function)h(of)g(the)g(v)-5 b(ariabilit)n(y)34 b(of)523 1601 y(task)28 b(sizes.)g(The)g(e\013ectiv) n(eness)g(of)h(these)f(task)g(assignmen)n(t)g(sc)n(hemes)f(will)i(b)r (e)g(measured)523 1701 y(in)f(terms)g(of)f(mean)h(w)n(aiting)f(time)h (and)g(mean)g(slo)n(wdo)n(wn,)e(where)h(a)h(task's)f(slo)n(wdo)n(wn)f (is)523 1800 y(its)g(w)n(aiting)e(time)i(divided)g(b)n(y)f(its)h (service)f(demand.)g(All)h(means)f(are)g(p)r(er-task)f(a)n(v)n(erages.) 523 2035 y Fj(3.1)95 b(A)32 b(New)f(Size-Based)g(T)-8 b(ask)33 b(Assignmen)m(t)c(P)m(olicy:)j(SIT)-8 b(A-E)523 2186 y Fv(Before)36 b(delving)h(in)n(to)g(sim)n(ulation)f(and)h (analytic)f(results,)h(w)n(e)g(need)g(to)g(sp)r(ecify)g(a)g(few)523 2285 y(more)27 b(parameters)f(of)h(the)h(size-based)f(p)r(olicy)-7 b(.)648 2385 y(In)23 b(size-based)f(task)h(assignmen)n(t,)g(a)g(size)g (range)f(is)h(asso)r(ciated)f(with)i(eac)n(h)f(host)g(and)g(a)523 2485 y(task)j(is)g(sen)n(t)g(to)h(the)g(appropriate)d(host)j(based)e (on)i(its)f(size.)g(In)h(practice)f(the)h(size)f(ranges)523 2584 y(asso)r(ciated)d(with)h(the)h(hosts)e(are)g(often)i(c)n(hosen)e (somewhat)g(arbitrarily)-7 b(.)22 b(There)i(migh)n(t)g(b)r(e)523 2684 y(a)31 b(15-min)n(ute)g(queue)h(for)f(tasks)g(of)g(size)h(b)r(et)n (w)n(een)f(0)g(and)h(15)f(min)n(utes,)h(a)f(3-hour)f(queue)523 2784 y(for)36 b(tasks)g(of)h(size)g(b)r(et)n(w)n(een)g(15)f(min)n(utes) h(and)f(3)h(hours,)f(a)g(6-hour)g(queue,)h(a)f(12-hour)523 2883 y(queue)29 b(and)g(an)g(18-hour)f(queue,)h(for)g(example.)g (\(This)h(example)e(is)i(used)f(in)h(practice)e(at)523 2983 y(the)g(Cornell)f(Theory)f(Cen)n(ter)h(IBM)h(SP2)f(job)g(sc)n (heduler)g([6].\))648 3082 y(In)i(this)g(pap)r(er)g(w)n(e)g(c)n(ho)r (ose)e(a)i(more)f(formal)h(algorithm)f(for)g(size-based)g(task)h (assign-)523 3182 y(men)n(t,)c(whic)n(h)f(w)n(e)f(refer)h(to)g(as)f (SIT)-7 b(A-E)24 b(|)g(Size)h(In)n(terv)-5 b(al)23 b(T)-7 b(ask)23 b(Assignmen)n(t)h(with)h(Equal)523 3282 y(Load.)32 b(The)g(idea)g(is)g(simple:)h(de\014ne)f(the)h(size)f(range)f(asso)r (ciated)g(with)i(eac)n(h)e(host)h(suc)n(h)523 3381 y(that)27 b(the)f(total)g(w)n(ork)f(\(load\))h(directed)g(to)g(eac)n(h)g(host)g (is)g(the)g(same.)g(The)g(motiv)-5 b(ation)27 b(for)523 3481 y(doing)g(this)h(is)f(that)h(balancing)f(the)h(load)f(minimizes)h (mean)f(w)n(aiting)g(time.)648 3581 y(The)j(mec)n(hanism)h(for)f(ac)n (hieving)f(balanced)i(exp)r(ected)g(load)f(at)g(the)i(hosts)e(is)h(to)f (use)523 3680 y(the)25 b Fl(task)i(size)h(distribution)d Fv(to)g(de\014ne)g(the)g(cuto\013)g(p)r(oin)n(ts)f(\(de\014ning)h(the)g (ranges\))f(so)g(that)523 3780 y(the)29 b(exp)r(ected)g(w)n(ork)e (directed)i(to)g(eac)n(h)e(host)i(is)f(the)h(same.)g(The)f(task)h(size) f(distribution)523 3879 y(is)e(easy)g(to)g(obtain)h(b)n(y)f(main)n (taining)g(a)g(histogram)f(\(in)i(the)g(dispatc)n(her)e(unit\))j(of)e (all)h(task)523 3979 y(sizes)g(witnessed)h(o)n(v)n(er)d(a)j(p)r(erio)r (d)f(of)h(time.)648 4079 y(More)33 b(precisely)-7 b(,)33 b(let)h Fg(F)12 b Fv(\()p Fg(x)p Fv(\))35 b(=)f(Pr)o Fh(f)p Fg(X)39 b Fh(\024)34 b Fg(x)p Fh(g)g Fv(denote)g(the)g(cum)n (ulativ)n(e)g(distribution)523 4178 y(function)e(of)f(task)f(sizes)h (with)g(\014nite)h(mean)f Fg(M)9 b Fv(.)31 b(Let)g Fg(k)j Fv(denote)d(the)g(smallest)g(task)f(size,)523 4278 y Fg(p)f Fv(\(p)r(ossibly)f(equal)h(to)g(in\014nit)n(y\))g(denote)g(the)g (largest)e(task)i(size,)f(and)h Fg(h)g Fv(b)r(e)g(the)g(n)n(um)n(b)r (er)523 4378 y(of)d(hosts.)g(Then)h(w)n(e)e(determine)i(\\cuto\013)f(p) r(oin)n(ts")g Fg(x)2184 4390 y Fu(i)2212 4378 y Fv(,)h Fg(i)c Fv(=)g(0)14 b Fg(:)g(:)g(:)f(h)26 b Fv(where)g Fg(k)g Fv(=)c Fg(x)3084 4390 y Ft(0)3145 4378 y Fg(<)h(x)3280 4390 y Ft(1)3340 4378 y Fg(<)523 4477 y(x)570 4489 y Ft(2)631 4477 y Fg(<)f(:)14 b(:)g(:)23 b(<)g(x)973 4489 y Fu(h)p Ff(\000)p Ft(1)1124 4477 y Fg(<)g(x)1259 4489 y Fu(h)1326 4477 y Fv(=)f Fg(p)p Fv(,)28 b(suc)n(h)f(that)538 4590 y Fd(Z)621 4610 y Fu(x)659 4618 y Fc(1)584 4778 y Fu(x)622 4786 y Fc(0)654 4778 y Ft(=)p Fu(k)759 4703 y Fg(x)19 b Fh(\001)g Fg(dF)12 b Fv(\()p Fg(x)p Fv(\))24 b(=)1198 4590 y Fd(Z)1281 4610 y Fu(x)1319 4618 y Fc(2)1244 4778 y Fu(x)1282 4786 y Fc(1)1369 4703 y Fg(x)19 b Fh(\001)f Fg(dF)12 b Fv(\()p Fg(x)p Fv(\))24 b(=)f Fh(\001)14 b(\001)g(\001)23 b Fv(=)2015 4590 y Fd(Z)2098 4610 y Fu(x)2136 4619 y Fb(h)2174 4610 y Ft(=)p Fu(p)2061 4778 y(x)2099 4787 y Fb(h)p Fa(\000)p Fc(1)2277 4703 y Fg(x)c Fh(\001)f Fg(dF)12 b Fv(\()p Fg(x)p Fv(\))24 b(=)2725 4646 y Fg(M)p 2725 4684 90 4 v 2746 4760 a(h)2848 4703 y Fv(=)2945 4574 y Fd(R)3001 4594 y Fu(p)2985 4670 y(k)3053 4641 y Fg(x)19 b Fh(\001)f Fg(dF)12 b Fv(\()p Fg(x)p Fv(\))p 2945 4684 436 4 v 3139 4760 a Fg(h)523 4924 y Fv(and)27 b(assign)g(to)g(the)h Fg(i)p Fv(th)g(host)f(all)h(tasks)e(ranging)g(in) i(size)g(from)f Fg(x)2613 4936 y Fu(i)p Ff(\000)p Ft(1)2754 4924 y Fv(to)g Fg(x)2902 4936 y Fu(i)2930 4924 y Fv(.)p eop %%Page: 7 7 7 6 bop 648 448 a Fv(SIT)-7 b(A-E)34 b(as)h(de\014ned)g(can)g(b)r(e)h (applied)f(to)g Fl(any)g Fv(task)g(size)g(distribution)g(with)g (\014nite)523 548 y(mean.)27 b(In)h(the)g(remainder)f(of)g(the)h(pap)r (er)g(w)n(e)f(will)h(alw)n(a)n(ys)d(assume)i(the)h(task)f(size)g (distri-)523 648 y(bution)h(is)f(the)h(Bounded)g(P)n(areto)d (distribution,)j Fg(B)t Fv(\()p Fg(k)s(;)14 b(p;)g(\013)p Fv(\).)523 906 y Fm(4)112 b(Sim)m(ulation)35 b(Results)523 1099 y Fv(In)c(this)h(section)f(w)n(e)g(compare)e(the)j(Random,)f (Round-Robin,)g(SIT)-7 b(A-E,)31 b(and)g(Dynamic)523 1198 y(p)r(olicies)c(via)g(sim)n(ulation.)h(Sim)n(ulation)f(parameters) f(are)g(as)h(sho)n(wn)g(in)h(T)-7 b(able)27 b(1.)648 1298 y(Sim)n(ulating)c(a)h(serv)n(er)e(system)i(with)g(hea)n (vy-tailed,)f(highly)g(v)-5 b(ariable)23 b(service)g(times)h(is)523 1398 y(di\016cult)d(b)r(ecause)e(the)h(system)g(approac)n(hes)d(steady) i(state)h(v)n(ery)f(slo)n(wly)f(and)i(usually)f(from)523 1497 y(b)r(elo)n(w)h([2].)h(This)f(o)r(ccurs)g(b)r(ecause)g(the)h (running)g(a)n(v)n(erage)d(of)i(task)g(sizes)g(is)h(t)n(ypically)f(at)h (the)523 1597 y(outset)j(w)n(ell)f(b)r(elo)n(w)g(the)h(true)g(mean;)g (the)g(true)f(mean)h(isn't)g(ac)n(hiev)n(ed)e(un)n(til)i(enough)f (large)523 1696 y(tasks)18 b(arriv)n(e.)f(The)h(consequence)g(for)g(a)h (system)f(lik)n(e)g(our)g(o)n(wn)g(is)h(that)g(sim)n(ulation)f(outputs) 523 1796 y(app)r(ear)j(more)h(optimistic)g(than)h(they)f(w)n(ould)g(in) h(steady-state.)e(T)-7 b(o)22 b(mak)n(e)f(our)h(sim)n(ulation)523 1896 y(measuremen)n(ts)29 b(less)g(sensitiv)n(e)g(to)h(the)g(startup)f (transien)n(t,)g(w)n(e)h(run)f(our)g(sim)n(ulation)h(for)523 1995 y(4)12 b Fh(\002)g Fv(10)738 1965 y Ft(5)798 1995 y Fv(arriv)-5 b(als)23 b(and)h(then)h(capture)f(data)g(from)h(the)f (next)h(single)f(arriv)-5 b(al)23 b(to)h(the)h(system)523 2095 y(only)-7 b(.)24 b(Eac)n(h)f(data)g(p)r(oin)n(t)h(sho)n(wn)f(in)i (our)e(plots)h(is)g(the)g(a)n(v)n(erage)d(of)j(400)f(indep)r(enden)n(t) h(runs,)523 2195 y(eac)n(h)j(of)g(whic)n(h)h(started)f(from)g(an)h (empt)n(y)f(system.)648 2294 y(W)-7 b(e)33 b(consider)f Fg(\013)h Fv(v)-5 b(alues)33 b(in)g(the)h(range)d(1.1)i(\(high)g(v)-5 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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 % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack 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 landscapeMode eq {90 rotate} if } 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 /PP { closepath fill } 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 { 2 copy moveto lineto stroke } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 648 92 5756 4885 MR c np 76 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 /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 1 j c1 0 0 6913 5188 PR 6 w DO 4 w SO 6 w c0 898 4616 mt 6255 4616 L 898 389 mt 6255 389 L 898 4616 mt 898 389 L 6255 4616 mt 6255 389 L 898 4616 mt 898 4616 L 6255 4616 mt 6255 4616 L 898 4616 mt 6255 4616 L 898 4616 mt 898 389 L 898 4616 mt 898 4616 L 898 4616 mt 898 4562 L 898 389 mt 898 443 L 798 4785 mt (1.1) s 1568 4616 mt 1568 4562 L 1568 389 mt 1568 443 L 1468 4785 mt (1.2) s 2237 4616 mt 2237 4562 L 2237 389 mt 2237 443 L 2137 4785 mt (1.3) s 2907 4616 mt 2907 4562 L 2907 389 mt 2907 443 L 2807 4785 mt (1.4) s 3576 4616 mt 3576 4562 L 3576 389 mt 3576 443 L 3476 4785 mt (1.5) s 4246 4616 mt 4246 4562 L 4246 389 mt 4246 443 L 4146 4785 mt (1.6) s 4916 4616 mt 4916 4562 L 4916 389 mt 4916 443 L 4816 4785 mt (1.7) s 5585 4616 mt 5585 4562 L 5585 389 mt 5585 443 L 5485 4785 mt (1.8) s 6255 4616 mt 6255 4562 L 6255 389 mt 6255 443 L 6155 4785 mt (1.9) s 898 4616 mt 925 4616 L 6255 4616 mt 6228 4616 L 898 4616 mt 952 4616 L 6255 4616 mt 6201 4616 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 650 4669 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 810 4580 mt (3) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 4362 mt 925 4362 L 6255 4362 mt 6228 4362 L 898 4213 mt 925 4213 L 6255 4213 mt 6228 4213 L 898 4107 mt 925 4107 L 6255 4107 mt 6228 4107 L 898 4025 mt 925 4025 L 6255 4025 mt 6228 4025 L 898 3958 mt 925 3958 L 6255 3958 mt 6228 3958 L 898 3902 mt 925 3902 L 6255 3902 mt 6228 3902 L 898 3853 mt 925 3853 L 6255 3853 mt 6228 3853 L 898 3809 mt 925 3809 L 6255 3809 mt 6228 3809 L 898 3771 mt 925 3771 L 6255 3771 mt 6228 3771 L 898 3771 mt 952 3771 L 6255 3771 mt 6201 3771 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 650 3824 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 810 3735 mt (4) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 3516 mt 925 3516 L 6255 3516 mt 6228 3516 L 898 3367 mt 925 3367 L 6255 3367 mt 6228 3367 L 898 3262 mt 925 3262 L 6255 3262 mt 6228 3262 L 898 3180 mt 925 3180 L 6255 3180 mt 6228 3180 L 898 3113 mt 925 3113 L 6255 3113 mt 6228 3113 L 898 3056 mt 925 3056 L 6255 3056 mt 6228 3056 L 898 3007 mt 925 3007 L 6255 3007 mt 6228 3007 L 898 2964 mt 925 2964 L 6255 2964 mt 6228 2964 L 898 2925 mt 925 2925 L 6255 2925 mt 6228 2925 L 898 2925 mt 952 2925 L 6255 2925 mt 6201 2925 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 650 2978 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 810 2889 mt (5) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 2671 mt 925 2671 L 6255 2671 mt 6228 2671 L 898 2522 mt 925 2522 L 6255 2522 mt 6228 2522 L 898 2416 mt 925 2416 L 6255 2416 mt 6228 2416 L 898 2334 mt 925 2334 L 6255 2334 mt 6228 2334 L 898 2267 mt 925 2267 L 6255 2267 mt 6228 2267 L 898 2211 mt 925 2211 L 6255 2211 mt 6228 2211 L 898 2162 mt 925 2162 L 6255 2162 mt 6228 2162 L 898 2118 mt 925 2118 L 6255 2118 mt 6228 2118 L 898 2080 mt 925 2080 L 6255 2080 mt 6228 2080 L 898 2080 mt 952 2080 L 6255 2080 mt 6201 2080 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 650 2133 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 810 2044 mt (6) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 1825 mt 925 1825 L 6255 1825 mt 6228 1825 L 898 1676 mt 925 1676 L 6255 1676 mt 6228 1676 L 898 1571 mt 925 1571 L 6255 1571 mt 6228 1571 L 898 1489 mt 925 1489 L 6255 1489 mt 6228 1489 L 898 1422 mt 925 1422 L 6255 1422 mt 6228 1422 L 898 1365 mt 925 1365 L 6255 1365 mt 6228 1365 L 898 1316 mt 925 1316 L 6255 1316 mt 6228 1316 L 898 1273 mt 925 1273 L 6255 1273 mt 6228 1273 L 898 1234 mt 925 1234 L 6255 1234 mt 6228 1234 L 898 1234 mt 952 1234 L 6255 1234 mt 6201 1234 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 650 1287 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 810 1198 mt (7) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 980 mt 925 980 L 6255 980 mt 6228 980 L 898 831 mt 925 831 L 6255 831 mt 6228 831 L 898 725 mt 925 725 L 6255 725 mt 6228 725 L 898 643 mt 925 643 L 6255 643 mt 6228 643 L 898 577 mt 925 577 L 6255 577 mt 6228 577 L 898 520 mt 925 520 L 6255 520 mt 6228 520 L 898 471 mt 925 471 L 6255 471 mt 6228 471 L 898 428 mt 925 428 L 6255 428 mt 6228 428 L 898 389 mt 925 389 L 6255 389 mt 6228 389 L 898 389 mt 952 389 L 6255 389 mt 6201 389 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 650 442 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 810 353 mt (8) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 4616 mt 6255 4616 L 898 389 mt 6255 389 L 898 4616 mt 898 389 L 6255 4616 mt 6255 389 L 898 389 mt 898 389 L 6255 389 mt 6255 389 L gs 898 389 5358 4228 MR c np gr 36 36 898 771 FO 36 36 1568 1003 FO 36 36 2237 1600 FO 36 36 2907 1656 FO 36 36 3576 2164 FO 36 36 4246 2902 FO 36 36 4916 2895 FO 36 36 5585 3259 FO 36 36 6255 3155 FO gs 898 389 5358 4228 MR c np 670 -104 669 364 670 -7 670 738 669 508 670 56 669 597 670 232 898 771 9 MP stroke gr 36 36 898 984 FO 36 36 1568 1440 FO 36 36 2237 1701 FO 36 36 2907 1817 FO 36 36 3576 2359 FO 36 36 4246 2814 FO 36 36 4916 3082 FO 36 36 5585 3093 FO 36 36 6255 3234 FO gs 898 389 5358 4228 MR c np DD 670 141 669 11 670 268 670 455 669 542 670 116 669 261 670 456 898 984 9 MP stroke SO gr 36 36 898 4155 FO 36 36 1568 4134 FO 36 36 2237 3947 FO 36 36 2907 4030 FO 36 36 3576 3594 FO 36 36 4246 3570 FO 36 36 4916 3990 FO 36 36 5585 3906 FO 36 36 6255 4003 FO gs 898 389 5358 4228 MR c np DA 670 97 669 -84 670 420 670 -24 669 -436 670 83 669 -187 670 -21 898 4155 9 MP stroke SO gr 36 36 898 2501 FO 36 36 1568 2849 FO 36 36 2237 3264 FO 36 36 2907 3846 FO 36 36 3576 4124 FO 36 36 4246 4292 FO 36 36 4916 4304 FO 36 36 5585 4436 FO 36 36 6255 4604 FO gs 898 389 5358 4228 MR c np DO 670 168 669 132 670 12 670 168 669 278 670 582 669 415 670 348 898 2501 9 MP stroke gr DO 2658 274 mt (Simulated mean waiting time) s 3401 4945 mt (alpha) s 4447 1033 mt (- Random) s 4447 1369 mt (-. Round-Robin) s 4447 1624 mt (... Dynamic) s 4447 1878 mt (-- SITA-E) s SO end eplot epage end showpage %%EndDocument endTexFig 1994 2709 a 11188078 10069263 4587520 11730944 36110336 40370176 startTexFig 1994 2709 a %%BeginDocument: simmeanslow.eps % MathWorks dictionary /MathWorks 150 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 /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % 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 % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack 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 landscapeMode eq {90 rotate} if } 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 /PP { closepath fill } 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 { 2 copy moveto lineto stroke } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 592 92 5812 4885 MR c np 76 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 /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 1 j c1 0 0 6913 5188 PR 6 w DO 4 w SO 6 w c0 898 4616 mt 6255 4616 L 898 389 mt 6255 389 L 898 4616 mt 898 389 L 6255 4616 mt 6255 389 L 898 4616 mt 898 4616 L 6255 4616 mt 6255 4616 L 898 4616 mt 6255 4616 L 898 4616 mt 898 389 L 898 4616 mt 898 4616 L 898 4616 mt 898 4562 L 898 389 mt 898 443 L 798 4785 mt (1.1) s 1568 4616 mt 1568 4562 L 1568 389 mt 1568 443 L 1468 4785 mt (1.2) s 2237 4616 mt 2237 4562 L 2237 389 mt 2237 443 L 2137 4785 mt (1.3) s 2907 4616 mt 2907 4562 L 2907 389 mt 2907 443 L 2807 4785 mt (1.4) s 3576 4616 mt 3576 4562 L 3576 389 mt 3576 443 L 3476 4785 mt (1.5) s 4246 4616 mt 4246 4562 L 4246 389 mt 4246 443 L 4146 4785 mt (1.6) s 4916 4616 mt 4916 4562 L 4916 389 mt 4916 443 L 4816 4785 mt (1.7) s 5585 4616 mt 5585 4562 L 5585 389 mt 5585 443 L 5485 4785 mt (1.8) s 6255 4616 mt 6255 4562 L 6255 389 mt 6255 443 L 6155 4785 mt (1.9) s 898 4616 mt 925 4616 L 6255 4616 mt 6228 4616 L 898 4616 mt 952 4616 L 6255 4616 mt 6201 4616 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 594 4669 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 754 4580 mt (-1) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 4404 mt 925 4404 L 6255 4404 mt 6228 4404 L 898 4280 mt 925 4280 L 6255 4280 mt 6228 4280 L 898 4192 mt 925 4192 L 6255 4192 mt 6228 4192 L 898 4124 mt 925 4124 L 6255 4124 mt 6228 4124 L 898 4068 mt 925 4068 L 6255 4068 mt 6228 4068 L 898 4021 mt 925 4021 L 6255 4021 mt 6228 4021 L 898 3980 mt 925 3980 L 6255 3980 mt 6228 3980 L 898 3944 mt 925 3944 L 6255 3944 mt 6228 3944 L 898 3912 mt 925 3912 L 6255 3912 mt 6228 3912 L 898 3912 mt 952 3912 L 6255 3912 mt 6201 3912 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 594 3965 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 754 3876 mt (0) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 3699 mt 925 3699 L 6255 3699 mt 6228 3699 L 898 3575 mt 925 3575 L 6255 3575 mt 6228 3575 L 898 3487 mt 925 3487 L 6255 3487 mt 6228 3487 L 898 3419 mt 925 3419 L 6255 3419 mt 6228 3419 L 898 3363 mt 925 3363 L 6255 3363 mt 6228 3363 L 898 3316 mt 925 3316 L 6255 3316 mt 6228 3316 L 898 3275 mt 925 3275 L 6255 3275 mt 6228 3275 L 898 3239 mt 925 3239 L 6255 3239 mt 6228 3239 L 898 3207 mt 925 3207 L 6255 3207 mt 6228 3207 L 898 3207 mt 952 3207 L 6255 3207 mt 6201 3207 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 594 3260 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 754 3171 mt (1) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 2995 mt 925 2995 L 6255 2995 mt 6228 2995 L 898 2871 mt 925 2871 L 6255 2871 mt 6228 2871 L 898 2783 mt 925 2783 L 6255 2783 mt 6228 2783 L 898 2715 mt 925 2715 L 6255 2715 mt 6228 2715 L 898 2659 mt 925 2659 L 6255 2659 mt 6228 2659 L 898 2612 mt 925 2612 L 6255 2612 mt 6228 2612 L 898 2571 mt 925 2571 L 6255 2571 mt 6228 2571 L 898 2535 mt 925 2535 L 6255 2535 mt 6228 2535 L 898 2503 mt 925 2503 L 6255 2503 mt 6228 2503 L 898 2503 mt 952 2503 L 6255 2503 mt 6201 2503 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 594 2556 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 754 2467 mt (2) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 2290 mt 925 2290 L 6255 2290 mt 6228 2290 L 898 2166 mt 925 2166 L 6255 2166 mt 6228 2166 L 898 2078 mt 925 2078 L 6255 2078 mt 6228 2078 L 898 2010 mt 925 2010 L 6255 2010 mt 6228 2010 L 898 1954 mt 925 1954 L 6255 1954 mt 6228 1954 L 898 1907 mt 925 1907 L 6255 1907 mt 6228 1907 L 898 1866 mt 925 1866 L 6255 1866 mt 6228 1866 L 898 1830 mt 925 1830 L 6255 1830 mt 6228 1830 L 898 1798 mt 925 1798 L 6255 1798 mt 6228 1798 L 898 1798 mt 952 1798 L 6255 1798 mt 6201 1798 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 594 1851 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 754 1762 mt (3) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 1586 mt 925 1586 L 6255 1586 mt 6228 1586 L 898 1462 mt 925 1462 L 6255 1462 mt 6228 1462 L 898 1374 mt 925 1374 L 6255 1374 mt 6228 1374 L 898 1306 mt 925 1306 L 6255 1306 mt 6228 1306 L 898 1250 mt 925 1250 L 6255 1250 mt 6228 1250 L 898 1203 mt 925 1203 L 6255 1203 mt 6228 1203 L 898 1162 mt 925 1162 L 6255 1162 mt 6228 1162 L 898 1126 mt 925 1126 L 6255 1126 mt 6228 1126 L 898 1094 mt 925 1094 L 6255 1094 mt 6228 1094 L 898 1094 mt 952 1094 L 6255 1094 mt 6201 1094 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 594 1147 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 754 1058 mt (4) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 881 mt 925 881 L 6255 881 mt 6228 881 L 898 757 mt 925 757 L 6255 757 mt 6228 757 L 898 669 mt 925 669 L 6255 669 mt 6228 669 L 898 601 mt 925 601 L 6255 601 mt 6228 601 L 898 545 mt 925 545 L 6255 545 mt 6228 545 L 898 498 mt 925 498 L 6255 498 mt 6228 498 L 898 457 mt 925 457 L 6255 457 mt 6228 457 L 898 421 mt 925 421 L 6255 421 mt 6228 421 L 898 389 mt 925 389 L 6255 389 mt 6228 389 L 898 389 mt 952 389 L 6255 389 mt 6201 389 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 594 442 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 754 353 mt (5) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 4616 mt 6255 4616 L 898 389 mt 6255 389 L 898 4616 mt 898 389 L 6255 4616 mt 6255 389 L 898 389 mt 898 389 L 6255 389 mt 6255 389 L gs 898 389 5358 4228 MR c np gr 36 36 898 584 FO 36 36 1568 1107 FO 36 36 2237 1428 FO 36 36 2907 1484 FO 36 36 3576 2032 FO 36 36 4246 2630 FO 36 36 4916 2658 FO 36 36 5585 2977 FO 36 36 6255 2887 FO gs 898 389 5358 4228 MR c np 670 -90 669 319 670 28 670 598 669 548 670 56 669 321 670 523 898 584 9 MP stroke gr 36 36 898 771 FO 36 36 1568 1274 FO 36 36 2237 1619 FO 36 36 2907 1652 FO 36 36 3576 2199 FO 36 36 4246 2603 FO 36 36 4916 2866 FO 36 36 5585 2812 FO 36 36 6255 3028 FO gs 898 389 5358 4228 MR c np DD 670 216 669 -54 670 263 670 404 669 547 670 33 669 345 670 503 898 771 9 MP stroke SO gr 36 36 898 3523 FO 36 36 1568 3667 FO 36 36 2237 3718 FO 36 36 2907 3695 FO 36 36 3576 3689 FO 36 36 4246 3680 FO 36 36 4916 3686 FO 36 36 5585 3730 FO 36 36 6255 3718 FO gs 898 389 5358 4228 MR c np DA 670 -12 669 44 670 6 670 -9 669 -6 670 -23 669 51 670 144 898 3523 9 MP stroke SO gr 36 36 898 2002 FO 36 36 1568 2434 FO 36 36 2237 2856 FO 36 36 2907 3398 FO 36 36 3576 3642 FO 36 36 4246 3841 FO 36 36 4916 3846 FO 36 36 5585 4012 FO 36 36 6255 4136 FO gs 898 389 5358 4228 MR c np DO 670 124 669 166 670 5 670 199 669 244 670 542 669 422 670 432 898 2002 9 MP stroke gr DO 2723 274 mt (Simulated mean slowdown) s 3401 4945 mt (alpha) s 4447 1303 mt (- Random) s 4447 1571 mt (-. Round-Robin) s 4447 1851 mt (... Dynamic) s 4447 2063 mt (-- SITA-E) s SO end eplot epage end showpage %%EndDocument endTexFig 1193 4076 a Fr(\(a\))1341 b(\(b\))523 4251 y Fo(Fig.)15 b(3.)34 b Fr(Mean)h(W)-6 b(aiting)35 b(Time)f(\(a\))g(and)h(Mean)g(Slo) n(wdo)n(wn)g(\(b\))e(under)h(Sim)n(ulation)g(of)h(F)-6 b(our)523 4342 y(T)g(ask)26 b(Assignmen)n(t)f(P)n(olicies)i(as)g(a)f(F) -6 b(unction)25 b(of)h Fi(\013)p Fr(.)648 4625 y Fv(Figure)e(3)i(sho)n (ws)e(the)i(p)r(erformance)e(of)i(the)g(system)f(for)g(all)g(four)g(p)r (olicies,)h(as)f(a)g(func-)523 4725 y(tion)36 b(of)g Fg(\013)g Fv(\(note)g(the)g(logarithmic)f(scale)g(on)h(the)g Fg(y)i Fv(axis\).)e(Figure)f(3\(a\))h(sho)n(ws)e(mean)523 4825 y(w)n(aiting)24 b(time)h(and)g(3\(b\))f(sho)n(ws)g(mean)g(slo)n (wdo)n(wn.)g(Belo)n(w)f(w)n(e)h(simply)h(summarize)f(these)523 4924 y(results;)j(in)h(the)g(next)g(section,)f(w)n(e)g(will)h(use)g (analysis)e(to)h(explain)h(these)f(results.)p eop %%Page: 8 8 8 7 bop 648 448 a Fv(First)30 b(of)g(all,)g(observ)n(e)e(that)j(the)f (p)r(erformance)f(of)h(the)h(system)f(under)g(the)g(Random)523 548 y(and)j(Round)h(Robin)g(p)r(olicies)f(is)h(similar,)e(and)i(that)g (b)r(oth)g(cases)e(p)r(erform)h(m)n(uc)n(h)h(more)523 648 y(p)r(o)r(orly)e(than)g(the)h(other)f(t)n(w)n(o)g(\(SIT)-7 b(A-E)33 b(and)f(Dynamic\).)h(As)g Fg(\013)g Fv(declines,)f(b)r(oth)h (of)g(the)523 747 y(p)r(erformance)28 b(metrics)h(under)g(the)h(Random) f(and)g(Round-Robin)g(p)r(olicies)g(explo)r(de)g(ap-)523 847 y(pro)n(ximately)j(exp)r(onen)n(tially)-7 b(.)32 b(This)h(giv)n(es)f(an)h(indication)g(of)g(the)g(sev)n(ere)f(impacts)h (that)523 946 y(hea)n(vy-tailed)24 b(w)n(orkloads)f(can)i(ha)n(v)n(e)f (in)h(systems)g(with)h(naiv)n(e)e(task)h(assignmen)n(t)f(p)r(olicies.) 648 1057 y(The)g(Dynamic)h(p)r(olicy)g(sho)n(ws)e(the)j(b)r(ene\014ts)f (of)g(instan)n(taneous)e(load)h(balancing.)g(Dy-)523 1157 y(namic)33 b(is)g(on)g(the)h(order)e(of)h(100)f(times)i(b)r(etter) f(for)g(b)r(oth)h(metrics)f(when)g(compared)f(to)523 1257 y(Random)20 b(and)h(Round)g(Robin.)f(F)-7 b(or)20 b(large)g Fg(\013)p Fv(,)h(this)g(means)f(that)h(Dynamic)g(p)r(erforms) f(quite)523 1356 y(w)n(ell|with)33 b(mean)f(slo)n(wdo)n(wn)e(less)i (than)g(1.)g(Ho)n(w)n(ev)n(er)f(as)g(the)i(v)-5 b(ariabilit)n(y)31 b(in)i(task)e(size)523 1456 y(increases)d(\(as)i Fg(\013)d Fh(!)g Fv(1\),)j(Dynamic)f(is)h(unable)g(to)g(main)n(tain)f(go)r(o)r(d) g(p)r(erformance.)g(It)h(to)r(o)523 1555 y(su\013ers)24 b(from)g(roughly)f(exp)r(onen)n(tial)h(explosion)f(in)h(p)r(erformance) g(metrics)g(as)f Fg(\013)i Fv(declines.)648 1666 y(In)j(con)n(trast,)e (the)j(b)r(eha)n(vior)d(of)i(SIT)-7 b(A-E)28 b(is)g(quite)g(di\013eren) n(t)g(from)g(that)g(of)g(the)h(other)523 1766 y(three.)f(Ov)n(er)e(the) i(en)n(tire)f(range)f(of)i Fg(\013)g Fv(v)-5 b(alues)27 b(studied,)h(the)g(p)r(erformance)f(of)g(the)h(system)523 1866 y(under)k(SIT)-7 b(A-E)33 b(is)f(relativ)n(ely)f(unc)n(hanged,)h (with)h(mean)f(slo)n(wdo)n(wn)f(alw)n(a)n(ys)g(b)r(et)n(w)n(een)h(2)523 1965 y(and)25 b(3.)f(This)g(is)h(the)g(most)g(striking)e(asp)r(ect)i (of)g(our)e(data:)i(in)g(a)f(range)f(of)i Fg(\013)g Fv(in)g(whic)n(h)f (p)r(er-)523 2065 y(formance)19 b(metrics)h(for)g(Random,)f(Round)h (Robin,)h(and)f(Dynamic)g(all)g(explo)r(de,)g(SIT)-7 b(A-E's)523 2164 y(p)r(erformance)27 b(remains)f(remark)-5 b(ably)26 b(insensitiv)n(e)i(to)f(increase)f(in)i(task)f(size)g(v)-5 b(ariabilit)n(y)e(.)648 2275 y(As)25 b(a)g(result)g(w)n(e)g(\014nd)g (that)h(when)f(task)g(size)g(is)g(less)g(v)-5 b(ariable,)24 b(Dynamic)i(task)f(assign-)523 2375 y(men)n(t)19 b(exhibits)h(b)r (etter)f(p)r(erformance;)f(but)i(when)g(task)e(sizes)h(sho)n(w)f(the)i (v)-5 b(ariabilit)n(y)18 b(that)h(is)523 2474 y(more)24 b(c)n(haracteristic)e(of)j(empirical)e(measuremen)n(ts)h(\()p Fg(\013)g Fh(\031)e Fv(1)p Fg(:)p Fv(1\),)i(SIT)-7 b(A-E's)24 b(p)r(erformance)523 2574 y(can)j(b)r(e)h(on)g(the)g(order)e(of)h(100)g (times)g(b)r(etter)h(than)g(that)g(of)g(Dynamic.)648 2685 y(In)40 b([4])g(w)n(e)g(sim)n(ulate)g(a)f(range)g(of)h(loads)g(\() p Fg(\032)p Fv(\))h(and)f(sho)n(w)f(that)h(as)g(load)f(increases,)523 2785 y(SIT)-7 b(A-E)27 b(b)r(ecomes)h(preferable)e(to)i(Dynamic)f(o)n (v)n(er)f(a)h(larger)f(range)g(of)i Fg(\013)p Fv(.)648 2895 y(The)d(remark)-5 b(able)24 b(consistency)g(of)h(system)g(p)r (erformance)f(under)i(the)f(SIT)-7 b(A-E)25 b(p)r(olicy)523 2995 y(across)f(the)i(range)f(of)h Fg(\013)g Fv(from)g(1.1)f(to)h(1.9)f (is)h(di\016cult)h(to)f(understand)g(using)f(the)i(to)r(ols)e(of)523 3095 y(sim)n(ulation)30 b(alone.)f(F)-7 b(or)30 b(that)g(reason)f(the)h (next)h(section)e(dev)n(elops)g(analysis)g(of)h(SIT)-7 b(A-E)523 3194 y(and)27 b(the)g(other)g(p)r(olicies,)g(and)g(uses)g (that)g(analysis)f(to)h(explain)f(SIT)-7 b(A-E's)27 b(p)r(erformance.) 523 3516 y Fm(5)112 b(Analysis)37 b(of)g(T)-9 b(ask)38 b(Assignmen)m(t)e(P)m(olicies)523 3771 y Fv(T)-7 b(o)28 b(understand)f(the)i(di\013erences)f(b)r(et)n(w)n(een)g(the)g(p)r (erformance)f(of)h(the)g(four)g(task)g(assign-)523 3871 y(men)n(t)41 b(p)r(olicies,)g(w)n(e)f(pro)n(vide)g(a)h(full)g(analysis) f(of)h(the)g(Round-Robin,)g(Random,)f(and)523 3971 y(SIT)-7 b(A-E)27 b(p)r(olicies,)h(and)f(an)g(appro)n(ximation)f(of)i(the)g (Dynamic)f(p)r(olicy)-7 b(.)648 4081 y(In)30 b(the)h(analysis)e(b)r (elo)n(w)h(w)n(e)g(will)h(rep)r(eatedly)f(mak)n(e)f(use)i(of)f(the)h(P) n(ollaczek-Kinc)n(hin)523 4181 y(form)n(ula)c(b)r(elo)n(w)g(whic)n(h)g (analyzes)g(the)h(M/G/1)e(F)n(CFS)i(queue:)649 4386 y Fj(E)13 b Fh(f)p Fv(W)-7 b(aiting)27 b(Time)q Fh(g)22 b Fv(=)h Fg(\025)p Fj(E)1547 4319 y Fd(\010)1596 4386 y Fg(X)1672 4352 y Ft(2)1708 4319 y Fd(\011)1770 4386 y Fg(=)p Fv(2\(1)18 b Fh(\000)g Fg(\032)p Fv(\))175 b([P)n (ollaczek-Kinc)n(hin)25 b(form)n(ula])792 4520 y Fj(E)13 b Fh(f)p Fv(Slo)n(wdo)n(wn)o Fh(g)22 b Fv(=)h Fj(E)14 b Fh(f)o Fg(W)r(=X)7 b Fh(g)22 b Fv(=)h Fj(E)14 b Fh(f)o Fg(W)e Fh(g)18 b(\001)g Fj(E)2276 4453 y Fd(\010)2324 4520 y Fg(X)2400 4486 y Ff(\000)p Ft(1)2489 4453 y Fd(\011)523 4725 y Fv(where)40 b Fg(\025)g Fv(denotes)g(the)g(rate)g(of)g(the)g (arriv)-5 b(al)39 b(pro)r(cess,)g Fg(X)46 b Fv(denotes)40 b(the)g(service)f(time)523 4825 y(distribution,)20 b(and)f Fg(\032)h Fv(denotes)f(the)h(utilization)f(\()p Fg(\032)24 b Fv(=)e Fg(\025)p Fj(E)14 b Fh(f)p Fg(X)7 b Fh(g)o Fv(\).)20 b(The)f(slo)n(wdo)n(wn)f(form)n(ulas)523 4924 y(follo)n(w)27 b(from)g(the)h(fact)g(that)g Fg(W)39 b Fv(and)28 b Fg(X)34 b Fv(are)26 b(indep)r(enden)n(t)j(for)e(a)g(F)n(CFS)h(queue.)p eop %%Page: 9 9 9 8 bop 648 448 a Fv(Observ)n(e)16 b(that)j(ev)n(ery)e(metric)h(for)g (the)h(simple)f(F)n(CFS)h(queue)f(is)h(dep)r(enden)n(t)g(on)f Fj(E)3196 381 y Fd(\010)3245 448 y Fg(X)3321 418 y Ft(2)3357 381 y Fd(\011)3406 448 y Fv(,)523 548 y(the)36 b(second)f(momen)n(t)h (of)f(the)h(service)f(time.)h(Recall)f(that)h(if)g(the)g(w)n(orkload)e (is)h(hea)n(vy-)523 648 y(tailed,)28 b(the)g(second)f(momen)n(t)g(of)h (the)g(service)e(time)i(explo)r(des,)g(as)f(sho)n(wn)f(in)i(Figure)f (2.)523 797 y Fl(R)l(andom)h(T)-6 b(ask)28 b(Assignment.)d Fv(The)g(Random)g(p)r(olicy)g(simply)g(p)r(erforms)g(Bernoulli)f (split-)523 897 y(ting)f(on)f(the)h(input)h(stream,)e(with)h(the)g (result)g(that)g(eac)n(h)f(host)g(b)r(ecomes)h(an)f(indep)r(enden)n(t) 523 996 y Fg(M)t(=B)t Fv(\()p Fg(k)s(;)14 b(p;)g(\013)p Fv(\))p Fg(=)p Fv(1)22 b(queue.)g(The)g(load)g(at)g(the)h Fg(i)p Fv(th)g(host,)f(is)g(equal)g(to)g(the)h(system)f(load,)g(that) 523 1096 y(is,)g Fg(\032)667 1108 y Fu(i)717 1096 y Fv(=)h Fg(\032)p Fv(.)f(So)f(the)h(P)n(ollaczek-Kinc)n(hin)d(form)n(ula)i (applies)g(directly)-7 b(,)21 b(and)h(all)f(p)r(erformance)523 1196 y(metrics)g(are)g(prop)r(ortional)e(to)j(the)f(second)g(momen)n(t) h(of)f Fg(B)t Fv(\()p Fg(k)s(;)14 b(p;)g(\013)p Fv(\).)22 b(P)n(erformance)d(is)j(gen-)523 1295 y(erally)27 b(p)r(o)r(or)g(b)r (ecause)g(the)h(second)f(momen)n(t)g(of)h(the)g Fg(B)t Fv(\()p Fg(k)s(;)14 b(p;)g(\013)p Fv(\))28 b(is)f(high.)523 1445 y Fl(R)l(ound)i(R)l(obin.)f Fv(The)g(Round)f(Robin)h(p)r(olicy)f (splits)g(the)h(incoming)f(stream)g(so)g(eac)n(h)f(host)523 1544 y(sees)36 b(an)g Fg(E)884 1556 y Fu(h)927 1544 y Fg(=B)t Fv(\()p Fg(k)s(;)14 b(p;)g(\013)p Fv(\))p Fg(=)p Fv(1)36 b(queue,)g(with)h(utilization)f Fg(\032)2350 1556 y Fu(i)2416 1544 y Fv(=)h Fg(\032)p Fv(.)f(This)g(system)h(has)e (p)r(er-)523 1644 y(formance)30 b(close)f(to)i(the)g(Random)f(case)f (since)i(it)g(still)g(sees)e(high)i(v)-5 b(ariabilit)n(y)29 b(in)i(service)523 1743 y(times,)d(whic)n(h)f(dominates)h(p)r (erformance.)523 1893 y Fl(SIT)-6 b(A-E.)42 b Fv(The)f(SIT)-7 b(A-E)41 b(p)r(olicy)h(also)e(p)r(erforms)h(Bernoulli)g(splitting)g(on) h(the)g(arriv)-5 b(al)523 1993 y(stream)24 b(\(whic)n(h)h(follo)n(ws)f (from)g(our)g(assumption)g(that)h(task)f(sizes)h(are)e(indep)r(enden)n (t\).)j(By)523 2092 y(the)e(de\014nition)h(of)e(SIT)-7 b(A-E,)24 b Fg(\032)1484 2104 y Fu(i)1535 2092 y Fv(=)e Fg(\032)p Fv(.)i(Ho)n(w)n(ev)n(er)e(the)i(task)g(sizes)f(at)h(eac)n(h)f (queue)h(are)f(deter-)523 2192 y(mined)31 b(b)n(y)f(the)g(particular)f (v)-5 b(alues)30 b(of)g(the)h(in)n(terv)-5 b(al)29 b(cuto\013s,)i Fh(f)p Fg(x)2600 2204 y Fu(i)2627 2192 y Fh(g)p Fg(;)14 b(i)27 b Fv(=)g(0)p Fg(;)14 b(:::;)g(h)p Fv(.)29 b(In)i(fact,)523 2291 y(host)d Fg(i)g Fv(sees)f(a)h Fg(M)t(=B)t Fv(\()p Fg(x)1271 2303 y Fu(i)p Ff(\000)p Ft(1)1384 2291 y Fg(;)14 b(x)1468 2303 y Fu(i)1496 2291 y Fg(;)g(\013)p Fv(\))p Fg(=)p Fv(1)28 b(queue.)g(The)g(reason)e(for)i(this)g(is)g(that)h (partitioning)523 2391 y(the)24 b(Bounded)f(P)n(areto)f(distribution)i (in)n(to)f(con)n(tiguous)g(regions)f(and)h(renormalizing)f(eac)n(h)523 2491 y(of)30 b(the)h(resulting)e(regions)g(to)h(unit)h(probabilit)n(y)e (yields)h(a)g(new)g(set)g(of)g(Bounded)g(P)n(areto)523 2590 y(distributions.)k(In)g([4])f(w)n(e)h(sho)n(w)f(ho)n(w)g(to)g (calculate)h(the)g(set)f(of)h Fg(x)2681 2602 y Fu(i)2709 2590 y Fv(s)g(for)f(the)h Fg(B)t Fv(\()p Fg(k)s(;)14 b(p;)g(\013)p Fv(\))523 2690 y(distribution,)29 b(and)f(w)n(e)h(presen) n(t)f(the)h(resulting)f(form)n(ulas)f(that)i(pro)n(vide)e(full)j (analysis)d(of)523 2790 y(the)h(system)f(under)h(the)g(SIT)-7 b(A-E)27 b(p)r(olicy)h(for)f(all)g(the)h(p)r(erformance)e(metrics.)523 2939 y Fl(Dynamic.)37 b Fv(The)g(Dynamic)f(p)r(olicy)g(is)g(not)h (analytically)e(tractable,)g(whic)n(h)h(is)g(wh)n(y)g(w)n(e)523 3039 y(p)r(erformed)j(the)h(sim)n(ulation)f(study)-7 b(.)40 b(Ho)n(w)n(ev)n(er,)d(in)j([4])f(w)n(e)g(pro)n(v)n(e)f(that)i(a) f(distributed)523 3138 y(system)e(of)g(the)h(t)n(yp)r(e)g(in)f(this)h (pap)r(er)f(with)g Fg(h)h Fv(hosts)e(whic)n(h)i(p)r(erforms)e(Dynamic)i (task)523 3238 y(assignmen)n(t)30 b(is)h(actually)g(equiv)-5 b(alen)n(t)31 b(to)g(an)g(M/G/h)g(queue.)g(F)-7 b(ortunately)g(,)31 b(there)g(exist)523 3337 y(kno)n(wn)c(appro)n(ximations)e(for)i(the)h (p)r(erformance)f(metrics)g(of)h(the)g(M/G/h)f(queue)g([15]:)1080 3488 y Fj(E)1157 3420 y Fd(\010)1205 3488 y Fg(Q)1271 3503 y Fu(M)s(=G=h)1500 3420 y Fd(\011)1571 3488 y Fv(=)c Fj(E)1735 3420 y Fd(\010)1784 3488 y Fg(Q)1850 3503 y Fu(M)s(=)l(M)s(=h)2089 3420 y Fd(\011)2156 3488 y Fh(\001)18 b Fj(E)2274 3420 y Fd(\010)2322 3488 y Fg(X)2398 3453 y Ft(2)2435 3420 y Fd(\011)2497 3488 y Fg(=)p Fj(E)13 b Fh(f)p Fg(X)7 b Fh(g)2773 3446 y Ft(2)2825 3488 y Fg(;)523 3629 y Fv(where)37 b Fg(X)43 b Fv(denotes)37 b(the)g(service)f(time)i (distribution)f(and)g Fg(Q)g Fv(denotes)f(the)i(n)n(um)n(b)r(er)f(in) 523 3729 y(queue.)e(What's)g(imp)r(ortan)n(t)f(to)h(observ)n(e)e(here)i (is)f(that)i(the)f(mean)f(queue)h(length,)g(and)523 3828 y(therefore)23 b(the)i(mean)f(w)n(aiting)f(time)i(and)f(mean)g(slo)n (wdo)n(wn,)e(are)h(all)h(prop)r(ortional)f(to)h(the)523 3928 y(second)g(momen)n(t)h(of)g(the)g(service)f(time)h(distribution,)g (as)g(w)n(as)e(the)j(case)e(for)g(the)h(Random)523 4028 y(and)i(Round-Robin)h(task)f(assignmen)n(t)f(p)r(olicies.)648 4127 y(Using)f(the)i(ab)r(o)n(v)n(e)d(analysis)h(w)n(e)h(can)f(compute) h(the)h(p)r(erformance)e(of)h(the)g(ab)r(o)n(v)n(e)f(task)523 4227 y(assignmen)n(t)33 b(p)r(olicies)i(o)n(v)n(er)d(a)i(range)f(of)i Fg(\013)g Fv(v)-5 b(alues.)34 b(Figure)f(4)h(sho)n(ws)g(the)h (analytically-)523 4327 y(deriv)n(ed)c(mean)h(w)n(aiting)g(time)h(and)f (mean)g(slo)n(wdo)n(wn)e(of)i(the)h(system)f(under)g(eac)n(h)f(p)r(ol-) 523 4426 y(icy)h(o)n(v)n(er)f(the)h(whole)g(range)f(of)h Fg(\013)p Fv(.)h(Figure)e(5)h(again)f(sho)n(ws)g(these)h (analytically-deriv)n(ed)523 4526 y(metrics,)h(but)g(only)g(o)n(v)n(er) e(the)i(range)e(of)i(1)f Fh(\024)f Fg(\013)h Fh(\024)g Fv(2,)g(whic)n(h)h(is)g(the)g(range)f(of)h Fg(\013)g Fv(corre-)523 4625 y(sp)r(onding)26 b(to)g(most)h(empirical)e (measuremen)n(ts)h(of)g(pro)r(cess)f(lifetimes)i(and)g(\014le)f(sizes)g (\(see)523 4725 y(Section)f(2.2\).)g(\(Note)h(that,)g(b)r(ecause)f(of)g (slo)n(w)f(sim)n(ulation)h(con)n(v)n(ergence)e(as)i(describ)r(ed)g(at) 523 4825 y(the)32 b(b)r(eginning)f(of)g(Section)g(4,)g(sim)n(ulation)f (v)-5 b(alues)31 b(are)f(generally)g(lo)n(w)n(er)f(than)j(analytic)523 4924 y(predictions;)27 b(ho)n(w)n(ev)n(er)f(all)h(sim)n(ulation)g (trends)g(agree)f(with)i(analysis\).)p eop %%Page: 10 10 10 9 bop 648 448 a Fv(First)28 b(observ)n(e)g(that)h(the)g(p)r (erformance)f(of)h(the)g(Random)f(and)h(Dynamic)g(p)r(olicies)g(in)523 548 y(b)r(oth)37 b(these)f(\014gures)f(gro)n(ws)f(w)n(orse)h(as)h Fg(\013)g Fv(decreases,)f(where)g(the)i(p)r(erformance)e(curv)n(es)523 648 y(follo)n(w)d(the)h(same)f(shap)r(e)h(as)f(the)h(second)f(momen)n (t)g(of)h(the)g(Bounded)g(P)n(areto)d(distribu-)523 747 y(tion,)c(sho)n(wn)g(in)g(Figure)g(2.)g(This)g(is)g(exp)r(ected)h (since)f(the)h(p)r(erformance)e(of)h(Random)g(and)523 847 y(Dynamic)i(is)g(directly)g(prop)r(ortional)f(to)h(the)g(second)g (momen)n(t)g(of)g(the)h(service)e(time)h(dis-)523 946 y(tribution.)j(By)f(con)n(tast,)g(lo)r(oking)g(at)g(Figure)g(5)g(w)n(e) h(see)f(that)h(in)g(the)g(range)e(1)e Fg(<)h(\013)h(<)e Fv(2,)523 1046 y(the)i(mean)f(w)n(aiting)f(time)i(and)f(esp)r(ecially)f (mean)h(slo)n(wdo)n(wn)f(under)h(the)g(SIT)-7 b(A-E)28 b(p)r(olicy)523 1146 y(is)j(remark)-5 b(ably)29 b(constan)n(t,)i(with)g (mean)g(slo)n(wdo)n(wns)e(around)h(3,)h(whereas)f(Random)g(and)523 1245 y(Dynamic)e(explo)r(de)f(in)h(this)f(range.)g(The)g(insensitivit)n (y)g(of)h(SIT)-7 b(A-E's)27 b(p)r(erformance)f(to)i Fg(\013)523 1345 y Fv(in)g(this)g(range)e(is)i(the)f(most)h(striking)f(prop)r(ert)n (y)f(of)i(our)f(sim)n(ulations)f(and)i(analysis.)491 1568 y 11188078 9397976 4407377 12959006 35916840 39732183 startTexFig 491 1568 a %%BeginDocument: predmeanwait.eps % MathWorks dictionary /MathWorks 150 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 /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % 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 % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack 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 landscapeMode eq {90 rotate} if } 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 /PP { closepath fill } 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 { 2 copy moveto lineto stroke } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 595 91 5749 4884 MR c np 76 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 /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 1 j c1 0 0 6913 5186 PR 6 w DO 4 w SO 6 w c0 898 4614 mt 6255 4614 L 898 388 mt 6255 388 L 898 4614 mt 898 388 L 6255 4614 mt 6255 388 L 898 4614 mt 898 4614 L 6255 4614 mt 6255 4614 L 898 4614 mt 6255 4614 L 898 4614 mt 898 388 L 898 4614 mt 898 4614 L 898 4614 mt 898 4560 L 898 388 mt 898 442 L 798 4783 mt (0.2) s 1493 4614 mt 1493 4560 L 1493 388 mt 1493 442 L 1393 4783 mt (0.4) s 2088 4614 mt 2088 4560 L 2088 388 mt 2088 442 L 1988 4783 mt (0.6) s 2684 4614 mt 2684 4560 L 2684 388 mt 2684 442 L 2584 4783 mt (0.8) s 3279 4614 mt 3279 4560 L 3279 388 mt 3279 442 L 3239 4783 mt (1) s 3874 4614 mt 3874 4560 L 3874 388 mt 3874 442 L 3774 4783 mt (1.2) s 4469 4614 mt 4469 4560 L 4469 388 mt 4469 442 L 4369 4783 mt (1.4) s 5065 4614 mt 5065 4560 L 5065 388 mt 5065 442 L 4965 4783 mt (1.6) s 5660 4614 mt 5660 4560 L 5660 388 mt 5660 442 L 5560 4783 mt (1.8) s 6255 4614 mt 6255 4560 L 6255 388 mt 6255 442 L 6215 4783 mt (2) s 898 4614 mt 925 4614 L 6255 4614 mt 6228 4614 L 898 4614 mt 952 4614 L 6255 4614 mt 6201 4614 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 4667 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 4578 mt (3) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 4432 mt 925 4432 L 6255 4432 mt 6228 4432 L 898 4326 mt 925 4326 L 6255 4326 mt 6228 4326 L 898 4251 mt 925 4251 L 6255 4251 mt 6228 4251 L 898 4192 mt 925 4192 L 6255 4192 mt 6228 4192 L 898 4144 mt 925 4144 L 6255 4144 mt 6228 4144 L 898 4104 mt 925 4104 L 6255 4104 mt 6228 4104 L 898 4069 mt 925 4069 L 6255 4069 mt 6228 4069 L 898 4038 mt 925 4038 L 6255 4038 mt 6228 4038 L 898 4010 mt 925 4010 L 6255 4010 mt 6228 4010 L 898 4010 mt 952 4010 L 6255 4010 mt 6201 4010 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 4063 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 3974 mt (4) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 3829 mt 925 3829 L 6255 3829 mt 6228 3829 L 898 3722 mt 925 3722 L 6255 3722 mt 6228 3722 L 898 3647 mt 925 3647 L 6255 3647 mt 6228 3647 L 898 3588 mt 925 3588 L 6255 3588 mt 6228 3588 L 898 3541 mt 925 3541 L 6255 3541 mt 6228 3541 L 898 3500 mt 925 3500 L 6255 3500 mt 6228 3500 L 898 3465 mt 925 3465 L 6255 3465 mt 6228 3465 L 898 3434 mt 925 3434 L 6255 3434 mt 6228 3434 L 898 3407 mt 925 3407 L 6255 3407 mt 6228 3407 L 898 3407 mt 952 3407 L 6255 3407 mt 6201 3407 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 3460 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 3371 mt (5) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 3225 mt 925 3225 L 6255 3225 mt 6228 3225 L 898 3119 mt 925 3119 L 6255 3119 mt 6228 3119 L 898 3043 mt 925 3043 L 6255 3043 mt 6228 3043 L 898 2985 mt 925 2985 L 6255 2985 mt 6228 2985 L 898 2937 mt 925 2937 L 6255 2937 mt 6228 2937 L 898 2896 mt 925 2896 L 6255 2896 mt 6228 2896 L 898 2861 mt 925 2861 L 6255 2861 mt 6228 2861 L 898 2830 mt 925 2830 L 6255 2830 mt 6228 2830 L 898 2803 mt 925 2803 L 6255 2803 mt 6228 2803 L 898 2803 mt 952 2803 L 6255 2803 mt 6201 2803 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 2856 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 2767 mt (6) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 2621 mt 925 2621 L 6255 2621 mt 6228 2621 L 898 2515 mt 925 2515 L 6255 2515 mt 6228 2515 L 898 2439 mt 925 2439 L 6255 2439 mt 6228 2439 L 898 2381 mt 925 2381 L 6255 2381 mt 6228 2381 L 898 2333 mt 925 2333 L 6255 2333 mt 6228 2333 L 898 2293 mt 925 2293 L 6255 2293 mt 6228 2293 L 898 2258 mt 925 2258 L 6255 2258 mt 6228 2258 L 898 2227 mt 925 2227 L 6255 2227 mt 6228 2227 L 898 2199 mt 925 2199 L 6255 2199 mt 6228 2199 L 898 2199 mt 952 2199 L 6255 2199 mt 6201 2199 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 2252 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 2163 mt (7) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 2017 mt 925 2017 L 6255 2017 mt 6228 2017 L 898 1911 mt 925 1911 L 6255 1911 mt 6228 1911 L 898 1836 mt 925 1836 L 6255 1836 mt 6228 1836 L 898 1777 mt 925 1777 L 6255 1777 mt 6228 1777 L 898 1729 mt 925 1729 L 6255 1729 mt 6228 1729 L 898 1689 mt 925 1689 L 6255 1689 mt 6228 1689 L 898 1654 mt 925 1654 L 6255 1654 mt 6228 1654 L 898 1623 mt 925 1623 L 6255 1623 mt 6228 1623 L 898 1595 mt 925 1595 L 6255 1595 mt 6228 1595 L 898 1595 mt 952 1595 L 6255 1595 mt 6201 1595 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 1648 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 1559 mt (8) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 1414 mt 925 1414 L 6255 1414 mt 6228 1414 L 898 1307 mt 925 1307 L 6255 1307 mt 6228 1307 L 898 1232 mt 925 1232 L 6255 1232 mt 6228 1232 L 898 1173 mt 925 1173 L 6255 1173 mt 6228 1173 L 898 1126 mt 925 1126 L 6255 1126 mt 6228 1126 L 898 1085 mt 925 1085 L 6255 1085 mt 6228 1085 L 898 1050 mt 925 1050 L 6255 1050 mt 6228 1050 L 898 1019 mt 925 1019 L 6255 1019 mt 6228 1019 L 898 992 mt 925 992 L 6255 992 mt 6228 992 L 898 992 mt 952 992 L 6255 992 mt 6201 992 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 1045 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 956 mt (9) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 810 mt 925 810 L 6255 810 mt 6228 810 L 898 704 mt 925 704 L 6255 704 mt 6228 704 L 898 628 mt 925 628 L 6255 628 mt 6228 628 L 898 570 mt 925 570 L 6255 570 mt 6228 570 L 898 522 mt 925 522 L 6255 522 mt 6228 522 L 898 482 mt 925 482 L 6255 482 mt 6228 482 L 898 447 mt 925 447 L 6255 447 mt 6228 447 L 898 416 mt 925 416 L 6255 416 mt 6228 416 L 898 388 mt 925 388 L 6255 388 mt 6228 388 L 898 388 mt 952 388 L 6255 388 mt 6201 388 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 441 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 352 mt (10) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 4614 mt 6255 4614 L 898 388 mt 6255 388 L 898 4614 mt 898 388 L 6255 4614 mt 6255 388 L 898 388 mt 898 388 L 6255 388 mt 6255 388 L gs 898 388 5358 4227 MR c np gr 36 36 898 419 FO 36 36 1196 439 FO 36 36 1493 463 FO 36 36 1791 494 FO 36 36 2088 535 FO 36 36 2386 590 FO 36 36 2684 672 FO 36 36 2981 792 FO 36 36 3279 963 FO 36 36 3577 1182 FO 36 36 3874 1440 FO 36 36 4172 1723 FO 36 36 4469 2019 FO 36 36 4767 2319 FO 36 36 5065 2617 FO 36 36 5362 2906 FO 36 36 5660 3175 FO 36 36 5957 3411 FO 36 36 6255 3604 FO gs 898 388 5358 4227 MR c np 298 193 297 236 298 269 297 289 298 298 298 300 297 296 298 283 297 258 298 219 298 171 297 120 298 82 298 55 297 41 298 31 297 24 298 20 898 419 19 MP stroke gr 36 36 898 1100 FO 36 36 1196 1218 FO 36 36 1493 1372 FO 36 36 1791 1585 FO 36 36 2088 1897 FO 36 36 2386 2399 FO 36 36 2684 3216 FO 36 36 2981 3940 FO 36 36 3279 4039 FO 36 36 3577 3996 FO 36 36 3874 3751 FO 36 36 4172 3465 FO 36 36 4469 3360 FO 36 36 4767 3394 FO 36 36 5065 3506 FO 36 36 5362 3650 FO 36 36 5660 3794 FO 36 36 5957 3915 FO 36 36 6255 4000 FO gs 898 388 5358 4227 MR c np DA 298 85 297 121 298 144 297 144 298 112 298 34 297 -105 298 -286 297 -245 298 -43 298 99 297 724 298 817 298 502 297 312 298 213 297 154 298 118 898 1100 19 MP stroke SO gr 36 36 898 1111 FO 36 36 1196 1131 FO 36 36 1493 1156 FO 36 36 1791 1187 FO 36 36 2088 1227 FO 36 36 2386 1283 FO 36 36 2684 1364 FO 36 36 2981 1485 FO 36 36 3279 1655 FO 36 36 3577 1875 FO 36 36 3874 2133 FO 36 36 4172 2415 FO 36 36 4469 2711 FO 36 36 4767 3012 FO 36 36 5065 3310 FO 36 36 5362 3598 FO 36 36 5660 3867 FO 36 36 5957 4104 FO 36 36 6255 4296 FO gs 898 388 5358 4227 MR c np DO 298 192 297 237 298 269 297 288 298 298 298 301 297 296 298 282 297 258 298 220 298 170 297 121 298 81 298 56 297 40 298 31 297 25 298 20 898 1111 19 MP stroke gr DO 2340 273 mt (Analytically-derived mean waiting time) s 3401 4943 mt (alpha) s 4172 863 mt (- Random) s 4172 1138 mt (-- SITA-E) s 4172 1360 mt (... Dynamic Approximation ) s SO end eplot epage end showpage %%EndDocument endTexFig 2020 1568 a 11188078 9397976 4407377 12959006 35916840 39732183 startTexFig 2020 1568 a %%BeginDocument: predmeanslow.eps % MathWorks dictionary /MathWorks 150 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 /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % 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 % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack 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 landscapeMode eq {90 rotate} if } 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 /PP { closepath fill } 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 { 2 copy moveto lineto stroke } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 595 91 5749 4884 MR c np 76 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 /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 1 j c1 0 0 6913 5186 PR 6 w DO 4 w SO 6 w c0 898 4614 mt 6255 4614 L 898 388 mt 6255 388 L 898 4614 mt 898 388 L 6255 4614 mt 6255 388 L 898 4614 mt 898 4614 L 6255 4614 mt 6255 4614 L 898 4614 mt 6255 4614 L 898 4614 mt 898 388 L 898 4614 mt 898 4614 L 898 4614 mt 898 4560 L 898 388 mt 898 442 L 798 4783 mt (0.2) s 1493 4614 mt 1493 4560 L 1493 388 mt 1493 442 L 1393 4783 mt (0.4) s 2088 4614 mt 2088 4560 L 2088 388 mt 2088 442 L 1988 4783 mt (0.6) s 2684 4614 mt 2684 4560 L 2684 388 mt 2684 442 L 2584 4783 mt (0.8) s 3279 4614 mt 3279 4560 L 3279 388 mt 3279 442 L 3239 4783 mt (1) s 3874 4614 mt 3874 4560 L 3874 388 mt 3874 442 L 3774 4783 mt (1.2) s 4469 4614 mt 4469 4560 L 4469 388 mt 4469 442 L 4369 4783 mt (1.4) s 5065 4614 mt 5065 4560 L 5065 388 mt 5065 442 L 4965 4783 mt (1.6) s 5660 4614 mt 5660 4560 L 5660 388 mt 5660 442 L 5560 4783 mt (1.8) s 6255 4614 mt 6255 4560 L 6255 388 mt 6255 442 L 6215 4783 mt (2) s 898 4614 mt 952 4614 L 6255 4614 mt 6201 4614 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 4667 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 4578 mt (0) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 4473 mt 925 4473 L 6255 4473 mt 6228 4473 L 898 4332 mt 925 4332 L 6255 4332 mt 6228 4332 L 898 4191 mt 925 4191 L 6255 4191 mt 6228 4191 L 898 4051 mt 925 4051 L 6255 4051 mt 6228 4051 L 898 3910 mt 925 3910 L 6255 3910 mt 6228 3910 L 898 3769 mt 925 3769 L 6255 3769 mt 6228 3769 L 898 3628 mt 925 3628 L 6255 3628 mt 6228 3628 L 898 3487 mt 925 3487 L 6255 3487 mt 6228 3487 L 898 3346 mt 925 3346 L 6255 3346 mt 6228 3346 L 898 3910 mt 952 3910 L 6255 3910 mt 6201 3910 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 3963 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 3874 mt (5) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 3769 mt 925 3769 L 6255 3769 mt 6228 3769 L 898 3628 mt 925 3628 L 6255 3628 mt 6228 3628 L 898 3487 mt 925 3487 L 6255 3487 mt 6228 3487 L 898 3346 mt 925 3346 L 6255 3346 mt 6228 3346 L 898 3205 mt 925 3205 L 6255 3205 mt 6228 3205 L 898 3064 mt 925 3064 L 6255 3064 mt 6228 3064 L 898 2924 mt 925 2924 L 6255 2924 mt 6228 2924 L 898 2783 mt 925 2783 L 6255 2783 mt 6228 2783 L 898 2642 mt 925 2642 L 6255 2642 mt 6228 2642 L 898 3205 mt 952 3205 L 6255 3205 mt 6201 3205 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 3258 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 3169 mt (10) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 3064 mt 925 3064 L 6255 3064 mt 6228 3064 L 898 2924 mt 925 2924 L 6255 2924 mt 6228 2924 L 898 2783 mt 925 2783 L 6255 2783 mt 6228 2783 L 898 2642 mt 925 2642 L 6255 2642 mt 6228 2642 L 898 2501 mt 925 2501 L 6255 2501 mt 6228 2501 L 898 2360 mt 925 2360 L 6255 2360 mt 6228 2360 L 898 2219 mt 925 2219 L 6255 2219 mt 6228 2219 L 898 2078 mt 925 2078 L 6255 2078 mt 6228 2078 L 898 1938 mt 925 1938 L 6255 1938 mt 6228 1938 L 898 2501 mt 952 2501 L 6255 2501 mt 6201 2501 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 2554 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 2465 mt (15) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 2360 mt 925 2360 L 6255 2360 mt 6228 2360 L 898 2219 mt 925 2219 L 6255 2219 mt 6228 2219 L 898 2078 mt 925 2078 L 6255 2078 mt 6228 2078 L 898 1938 mt 925 1938 L 6255 1938 mt 6228 1938 L 898 1797 mt 925 1797 L 6255 1797 mt 6228 1797 L 898 1656 mt 925 1656 L 6255 1656 mt 6228 1656 L 898 1515 mt 925 1515 L 6255 1515 mt 6228 1515 L 898 1374 mt 925 1374 L 6255 1374 mt 6228 1374 L 898 1233 mt 925 1233 L 6255 1233 mt 6228 1233 L 898 1797 mt 952 1797 L 6255 1797 mt 6201 1797 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 1850 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 1761 mt (20) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 1656 mt 925 1656 L 6255 1656 mt 6228 1656 L 898 1515 mt 925 1515 L 6255 1515 mt 6228 1515 L 898 1374 mt 925 1374 L 6255 1374 mt 6228 1374 L 898 1233 mt 925 1233 L 6255 1233 mt 6228 1233 L 898 1092 mt 925 1092 L 6255 1092 mt 6228 1092 L 898 951 mt 925 951 L 6255 951 mt 6228 951 L 898 811 mt 925 811 L 6255 811 mt 6228 811 L 898 670 mt 925 670 L 6255 670 mt 6228 670 L 898 529 mt 925 529 L 6255 529 mt 6228 529 L 898 1092 mt 952 1092 L 6255 1092 mt 6201 1092 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 1145 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 1056 mt (25) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 951 mt 925 951 L 6255 951 mt 6228 951 L 898 811 mt 925 811 L 6255 811 mt 6228 811 L 898 670 mt 925 670 L 6255 670 mt 6228 670 L 898 529 mt 925 529 L 6255 529 mt 6228 529 L 898 388 mt 925 388 L 6255 388 mt 6228 388 L 898 388 mt 952 388 L 6255 388 mt 6201 388 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 441 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 352 mt (30) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 4614 mt 6255 4614 L 898 388 mt 6255 388 L 898 4614 mt 898 388 L 6255 4614 mt 6255 388 L 898 388 mt 898 388 L 6255 388 mt 6255 388 L gs 898 388 5358 4227 MR c np gr 36 36 898 722 FO 36 36 1196 1987 FO 36 36 1493 2635 FO 36 36 1791 3030 FO 36 36 2088 3297 FO 36 36 2386 3491 FO 36 36 2684 3638 FO 36 36 2981 3756 FO 36 36 3279 3857 FO 36 36 3577 3947 FO 36 36 3874 4032 FO 36 36 4172 4114 FO 36 36 4469 4194 FO 36 36 4767 4271 FO 36 36 5065 4347 FO 36 36 5362 4418 FO 36 36 5660 4484 FO 36 36 5957 4542 FO 36 36 6255 4589 FO gs 898 388 5358 4227 MR c np DO 298 47 297 58 298 66 297 71 298 76 298 77 297 80 298 82 297 85 298 90 298 101 297 118 298 147 298 194 297 267 298 395 297 648 298 1265 898 722 19 MP stroke SO 298 47 297 58 298 66 297 72 298 75 298 78 297 80 298 82 297 85 298 90 298 100 297 119 298 147 298 194 297 267 298 395 297 647 298 1265 898 561 19 MP stroke gr 36 36 898 561 FO 36 36 1196 1826 FO 36 36 1493 2473 FO 36 36 1791 2868 FO 36 36 2088 3135 FO 36 36 2386 3329 FO 36 36 2684 3476 FO 36 36 2981 3595 FO 36 36 3279 3695 FO 36 36 3577 3785 FO 36 36 3874 3870 FO 36 36 4172 3952 FO 36 36 4469 4032 FO 36 36 4767 4110 FO 36 36 5065 4185 FO 36 36 5362 4257 FO 36 36 5660 4323 FO 36 36 5957 4381 FO 36 36 6255 4428 FO gs 898 388 5358 4227 MR c np DA 298 4 297 5 298 7 297 4 298 -3 298 -11 297 -8 298 0 297 7 298 32 298 144 297 311 298 321 298 298 297 330 298 438 297 678 298 1287 898 720 19 MP stroke SO gr 36 36 898 720 FO 36 36 1196 2007 FO 36 36 1493 2685 FO 36 36 1791 3123 FO 36 36 2088 3453 FO 36 36 2386 3751 FO 36 36 2684 4072 FO 36 36 2981 4383 FO 36 36 3279 4527 FO 36 36 3577 4559 FO 36 36 3874 4566 FO 36 36 4172 4566 FO 36 36 4469 4558 FO 36 36 4767 4547 FO 36 36 5065 4544 FO 36 36 5362 4548 FO 36 36 5660 4555 FO 36 36 5957 4560 FO 36 36 6255 4564 FO gs 898 388 5358 4227 MR c np gr 2405 273 mt (Analytically-derived mean slowdown) s 3401 4943 mt (alpha) s 4172 723 mt (- Random) s 4172 1004 mt (-- SITA-E) s 4172 1286 mt (... Dynamic Approximation) s end eplot epage end showpage %%EndDocument endTexFig 523 2932 a Fo(Fig.)15 b(4.)35 b Fr(Analysis)h(of)g(mean)f(w)n(aiting)i (time)d(and)i(mean)e(slo)n(wdo)n(wn)j(o)n(v)n(er)e(whole)i(range)f(of)g Fi(\013)p Fr(,)523 3024 y(0)22 b Fi(<)f(\013)g Fp(\024)g Fr(2.)491 3454 y 11188078 9397976 4407377 12959006 35916840 39732183 startTexFig 491 3454 a %%BeginDocument: predmeanwaitupclose.eps % MathWorks dictionary /MathWorks 150 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 /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % 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 % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack 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 landscapeMode eq {90 rotate} if } 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 /PP { closepath fill } 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 { 2 copy moveto lineto stroke } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 595 91 5749 4884 MR c np 76 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 /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 1 j c1 0 0 6913 5186 PR 6 w DO 4 w SO 6 w c0 898 4614 mt 6255 4614 L 898 388 mt 6255 388 L 898 4614 mt 898 388 L 6255 4614 mt 6255 388 L 898 4614 mt 898 4614 L 6255 4614 mt 6255 4614 L 898 4614 mt 6255 4614 L 898 4614 mt 898 388 L 898 4614 mt 898 4614 L 898 4614 mt 898 4560 L 898 388 mt 898 442 L 858 4783 mt (1) s 1434 4614 mt 1434 4560 L 1434 388 mt 1434 442 L 1334 4783 mt (1.1) s 1969 4614 mt 1969 4560 L 1969 388 mt 1969 442 L 1869 4783 mt (1.2) s 2505 4614 mt 2505 4560 L 2505 388 mt 2505 442 L 2405 4783 mt (1.3) s 3041 4614 mt 3041 4560 L 3041 388 mt 3041 442 L 2941 4783 mt (1.4) s 3577 4614 mt 3577 4560 L 3577 388 mt 3577 442 L 3477 4783 mt (1.5) s 4112 4614 mt 4112 4560 L 4112 388 mt 4112 442 L 4012 4783 mt (1.6) s 4648 4614 mt 4648 4560 L 4648 388 mt 4648 442 L 4548 4783 mt (1.7) s 5184 4614 mt 5184 4560 L 5184 388 mt 5184 442 L 5084 4783 mt (1.8) s 5719 4614 mt 5719 4560 L 5719 388 mt 5719 442 L 5619 4783 mt (1.9) s 6255 4614 mt 6255 4560 L 6255 388 mt 6255 442 L 6215 4783 mt (2) s 898 4614 mt 925 4614 L 6255 4614 mt 6228 4614 L 898 4614 mt 952 4614 L 6255 4614 mt 6201 4614 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 4667 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 4578 mt (3) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 4432 mt 925 4432 L 6255 4432 mt 6228 4432 L 898 4326 mt 925 4326 L 6255 4326 mt 6228 4326 L 898 4251 mt 925 4251 L 6255 4251 mt 6228 4251 L 898 4192 mt 925 4192 L 6255 4192 mt 6228 4192 L 898 4144 mt 925 4144 L 6255 4144 mt 6228 4144 L 898 4104 mt 925 4104 L 6255 4104 mt 6228 4104 L 898 4069 mt 925 4069 L 6255 4069 mt 6228 4069 L 898 4038 mt 925 4038 L 6255 4038 mt 6228 4038 L 898 4010 mt 925 4010 L 6255 4010 mt 6228 4010 L 898 4010 mt 952 4010 L 6255 4010 mt 6201 4010 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 4063 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 3974 mt (4) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 3829 mt 925 3829 L 6255 3829 mt 6228 3829 L 898 3722 mt 925 3722 L 6255 3722 mt 6228 3722 L 898 3647 mt 925 3647 L 6255 3647 mt 6228 3647 L 898 3588 mt 925 3588 L 6255 3588 mt 6228 3588 L 898 3541 mt 925 3541 L 6255 3541 mt 6228 3541 L 898 3500 mt 925 3500 L 6255 3500 mt 6228 3500 L 898 3465 mt 925 3465 L 6255 3465 mt 6228 3465 L 898 3434 mt 925 3434 L 6255 3434 mt 6228 3434 L 898 3407 mt 925 3407 L 6255 3407 mt 6228 3407 L 898 3407 mt 952 3407 L 6255 3407 mt 6201 3407 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 3460 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 3371 mt (5) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 3225 mt 925 3225 L 6255 3225 mt 6228 3225 L 898 3119 mt 925 3119 L 6255 3119 mt 6228 3119 L 898 3043 mt 925 3043 L 6255 3043 mt 6228 3043 L 898 2985 mt 925 2985 L 6255 2985 mt 6228 2985 L 898 2937 mt 925 2937 L 6255 2937 mt 6228 2937 L 898 2896 mt 925 2896 L 6255 2896 mt 6228 2896 L 898 2861 mt 925 2861 L 6255 2861 mt 6228 2861 L 898 2830 mt 925 2830 L 6255 2830 mt 6228 2830 L 898 2803 mt 925 2803 L 6255 2803 mt 6228 2803 L 898 2803 mt 952 2803 L 6255 2803 mt 6201 2803 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 2856 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 2767 mt (6) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 2621 mt 925 2621 L 6255 2621 mt 6228 2621 L 898 2515 mt 925 2515 L 6255 2515 mt 6228 2515 L 898 2439 mt 925 2439 L 6255 2439 mt 6228 2439 L 898 2381 mt 925 2381 L 6255 2381 mt 6228 2381 L 898 2333 mt 925 2333 L 6255 2333 mt 6228 2333 L 898 2293 mt 925 2293 L 6255 2293 mt 6228 2293 L 898 2258 mt 925 2258 L 6255 2258 mt 6228 2258 L 898 2227 mt 925 2227 L 6255 2227 mt 6228 2227 L 898 2199 mt 925 2199 L 6255 2199 mt 6228 2199 L 898 2199 mt 952 2199 L 6255 2199 mt 6201 2199 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 2252 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 2163 mt (7) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 2017 mt 925 2017 L 6255 2017 mt 6228 2017 L 898 1911 mt 925 1911 L 6255 1911 mt 6228 1911 L 898 1836 mt 925 1836 L 6255 1836 mt 6228 1836 L 898 1777 mt 925 1777 L 6255 1777 mt 6228 1777 L 898 1729 mt 925 1729 L 6255 1729 mt 6228 1729 L 898 1689 mt 925 1689 L 6255 1689 mt 6228 1689 L 898 1654 mt 925 1654 L 6255 1654 mt 6228 1654 L 898 1623 mt 925 1623 L 6255 1623 mt 6228 1623 L 898 1595 mt 925 1595 L 6255 1595 mt 6228 1595 L 898 1595 mt 952 1595 L 6255 1595 mt 6201 1595 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 1648 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 1559 mt (8) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 1414 mt 925 1414 L 6255 1414 mt 6228 1414 L 898 1307 mt 925 1307 L 6255 1307 mt 6228 1307 L 898 1232 mt 925 1232 L 6255 1232 mt 6228 1232 L 898 1173 mt 925 1173 L 6255 1173 mt 6228 1173 L 898 1126 mt 925 1126 L 6255 1126 mt 6228 1126 L 898 1085 mt 925 1085 L 6255 1085 mt 6228 1085 L 898 1050 mt 925 1050 L 6255 1050 mt 6228 1050 L 898 1019 mt 925 1019 L 6255 1019 mt 6228 1019 L 898 992 mt 925 992 L 6255 992 mt 6228 992 L 898 992 mt 952 992 L 6255 992 mt 6201 992 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 1045 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 956 mt (9) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 810 mt 925 810 L 6255 810 mt 6228 810 L 898 704 mt 925 704 L 6255 704 mt 6228 704 L 898 628 mt 925 628 L 6255 628 mt 6228 628 L 898 570 mt 925 570 L 6255 570 mt 6228 570 L 898 522 mt 925 522 L 6255 522 mt 6228 522 L 898 482 mt 925 482 L 6255 482 mt 6228 482 L 898 447 mt 925 447 L 6255 447 mt 6228 447 L 898 416 mt 925 416 L 6255 416 mt 6228 416 L 898 388 mt 925 388 L 6255 388 mt 6228 388 L 898 388 mt 952 388 L 6255 388 mt 6201 388 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 597 441 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 757 352 mt (10) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 4614 mt 6255 4614 L 898 388 mt 6255 388 L 898 4614 mt 898 388 L 6255 4614 mt 6255 388 L 898 388 mt 898 388 L 6255 388 mt 6255 388 L gs 898 388 5358 4227 MR c np gr 36 36 898 963 FO 36 36 1434 1182 FO 36 36 1969 1440 FO 36 36 2505 1723 FO 36 36 3041 2019 FO 36 36 3577 2319 FO 36 36 4112 2617 FO 36 36 4648 2906 FO 36 36 5184 3175 FO 36 36 5719 3411 FO 36 36 6255 3604 FO gs 898 388 5358 4227 MR c np 536 193 535 236 536 269 536 289 535 298 536 300 536 296 536 283 535 258 536 219 898 963 11 MP stroke gr 36 36 898 4039 FO 36 36 1434 3996 FO 36 36 1969 3751 FO 36 36 2505 3465 FO 36 36 3041 3360 FO 36 36 3577 3394 FO 36 36 4112 3506 FO 36 36 4648 3650 FO 36 36 5184 3794 FO 36 36 5719 3915 FO 36 36 6255 4000 FO gs 898 388 5358 4227 MR c np DA 536 85 535 121 536 144 536 144 535 112 536 34 536 -105 536 -286 535 -245 536 -43 898 4039 11 MP stroke SO gr 36 36 898 1655 FO 36 36 1434 1875 FO 36 36 1969 2133 FO 36 36 2505 2415 FO 36 36 3041 2711 FO 36 36 3577 3012 FO 36 36 4112 3310 FO 36 36 4648 3598 FO 36 36 5184 3867 FO 36 36 5719 4104 FO 36 36 6255 4296 FO gs 898 388 5358 4227 MR c np DO 536 192 535 237 536 269 536 288 535 298 536 301 536 296 536 282 535 258 536 220 898 1655 11 MP stroke gr DO 2043 273 mt (Analytically-derived mean waiting time: Upclose) s 3401 4943 mt (alpha) s 4112 681 mt (- Random) s 4112 863 mt (-- SITA-E) s 4112 1045 mt (... Dynamic Approximation) s SO end eplot epage end showpage %%EndDocument endTexFig 2020 3439 a 11188078 9509856 4736286 12959006 35916840 39732183 startTexFig 2020 3439 a %%BeginDocument: predmeanslowupclose.eps % MathWorks dictionary /MathWorks 150 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 /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % 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 % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack 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 landscapeMode eq {90 rotate} if } 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 /PP { closepath fill } 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 { 2 copy moveto lineto stroke } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 648 91 5696 4884 MR c np 76 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 /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 1 j c1 0 0 6913 5186 PR 6 w DO 4 w SO 6 w c0 898 4614 mt 6255 4614 L 898 388 mt 6255 388 L 898 4614 mt 898 388 L 6255 4614 mt 6255 388 L 898 4614 mt 898 4614 L 6255 4614 mt 6255 4614 L 898 4614 mt 6255 4614 L 898 4614 mt 898 388 L 898 4614 mt 898 4614 L 898 4614 mt 898 4560 L 898 388 mt 898 442 L 858 4783 mt (1) s 1434 4614 mt 1434 4560 L 1434 388 mt 1434 442 L 1334 4783 mt (1.1) s 1969 4614 mt 1969 4560 L 1969 388 mt 1969 442 L 1869 4783 mt (1.2) s 2505 4614 mt 2505 4560 L 2505 388 mt 2505 442 L 2405 4783 mt (1.3) s 3041 4614 mt 3041 4560 L 3041 388 mt 3041 442 L 2941 4783 mt (1.4) s 3577 4614 mt 3577 4560 L 3577 388 mt 3577 442 L 3477 4783 mt (1.5) s 4112 4614 mt 4112 4560 L 4112 388 mt 4112 442 L 4012 4783 mt (1.6) s 4648 4614 mt 4648 4560 L 4648 388 mt 4648 442 L 4548 4783 mt (1.7) s 5184 4614 mt 5184 4560 L 5184 388 mt 5184 442 L 5084 4783 mt (1.8) s 5719 4614 mt 5719 4560 L 5719 388 mt 5719 442 L 5619 4783 mt (1.9) s 6255 4614 mt 6255 4560 L 6255 388 mt 6255 442 L 6215 4783 mt (2) s 898 4614 mt 925 4614 L 6255 4614 mt 6228 4614 L 898 4614 mt 952 4614 L 6255 4614 mt 6201 4614 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 650 4667 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 810 4578 mt (0) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 4432 mt 925 4432 L 6255 4432 mt 6228 4432 L 898 4326 mt 925 4326 L 6255 4326 mt 6228 4326 L 898 4251 mt 925 4251 L 6255 4251 mt 6228 4251 L 898 4192 mt 925 4192 L 6255 4192 mt 6228 4192 L 898 4144 mt 925 4144 L 6255 4144 mt 6228 4144 L 898 4104 mt 925 4104 L 6255 4104 mt 6228 4104 L 898 4069 mt 925 4069 L 6255 4069 mt 6228 4069 L 898 4038 mt 925 4038 L 6255 4038 mt 6228 4038 L 898 4010 mt 925 4010 L 6255 4010 mt 6228 4010 L 898 4010 mt 952 4010 L 6255 4010 mt 6201 4010 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 650 4063 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 810 3974 mt (1) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 3829 mt 925 3829 L 6255 3829 mt 6228 3829 L 898 3722 mt 925 3722 L 6255 3722 mt 6228 3722 L 898 3647 mt 925 3647 L 6255 3647 mt 6228 3647 L 898 3588 mt 925 3588 L 6255 3588 mt 6228 3588 L 898 3541 mt 925 3541 L 6255 3541 mt 6228 3541 L 898 3500 mt 925 3500 L 6255 3500 mt 6228 3500 L 898 3465 mt 925 3465 L 6255 3465 mt 6228 3465 L 898 3434 mt 925 3434 L 6255 3434 mt 6228 3434 L 898 3407 mt 925 3407 L 6255 3407 mt 6228 3407 L 898 3407 mt 952 3407 L 6255 3407 mt 6201 3407 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 650 3460 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 810 3371 mt (2) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 3225 mt 925 3225 L 6255 3225 mt 6228 3225 L 898 3119 mt 925 3119 L 6255 3119 mt 6228 3119 L 898 3043 mt 925 3043 L 6255 3043 mt 6228 3043 L 898 2985 mt 925 2985 L 6255 2985 mt 6228 2985 L 898 2937 mt 925 2937 L 6255 2937 mt 6228 2937 L 898 2896 mt 925 2896 L 6255 2896 mt 6228 2896 L 898 2861 mt 925 2861 L 6255 2861 mt 6228 2861 L 898 2830 mt 925 2830 L 6255 2830 mt 6228 2830 L 898 2803 mt 925 2803 L 6255 2803 mt 6228 2803 L 898 2803 mt 952 2803 L 6255 2803 mt 6201 2803 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 650 2856 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 810 2767 mt (3) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 2621 mt 925 2621 L 6255 2621 mt 6228 2621 L 898 2515 mt 925 2515 L 6255 2515 mt 6228 2515 L 898 2439 mt 925 2439 L 6255 2439 mt 6228 2439 L 898 2381 mt 925 2381 L 6255 2381 mt 6228 2381 L 898 2333 mt 925 2333 L 6255 2333 mt 6228 2333 L 898 2293 mt 925 2293 L 6255 2293 mt 6228 2293 L 898 2258 mt 925 2258 L 6255 2258 mt 6228 2258 L 898 2227 mt 925 2227 L 6255 2227 mt 6228 2227 L 898 2199 mt 925 2199 L 6255 2199 mt 6228 2199 L 898 2199 mt 952 2199 L 6255 2199 mt 6201 2199 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 650 2252 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 810 2163 mt (4) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 2017 mt 925 2017 L 6255 2017 mt 6228 2017 L 898 1911 mt 925 1911 L 6255 1911 mt 6228 1911 L 898 1836 mt 925 1836 L 6255 1836 mt 6228 1836 L 898 1777 mt 925 1777 L 6255 1777 mt 6228 1777 L 898 1729 mt 925 1729 L 6255 1729 mt 6228 1729 L 898 1689 mt 925 1689 L 6255 1689 mt 6228 1689 L 898 1654 mt 925 1654 L 6255 1654 mt 6228 1654 L 898 1623 mt 925 1623 L 6255 1623 mt 6228 1623 L 898 1595 mt 925 1595 L 6255 1595 mt 6228 1595 L 898 1595 mt 952 1595 L 6255 1595 mt 6201 1595 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 650 1648 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 810 1559 mt (5) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 1414 mt 925 1414 L 6255 1414 mt 6228 1414 L 898 1307 mt 925 1307 L 6255 1307 mt 6228 1307 L 898 1232 mt 925 1232 L 6255 1232 mt 6228 1232 L 898 1173 mt 925 1173 L 6255 1173 mt 6228 1173 L 898 1126 mt 925 1126 L 6255 1126 mt 6228 1126 L 898 1085 mt 925 1085 L 6255 1085 mt 6228 1085 L 898 1050 mt 925 1050 L 6255 1050 mt 6228 1050 L 898 1019 mt 925 1019 L 6255 1019 mt 6228 1019 L 898 992 mt 925 992 L 6255 992 mt 6228 992 L 898 992 mt 952 992 L 6255 992 mt 6201 992 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 650 1045 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 810 956 mt (6) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 810 mt 925 810 L 6255 810 mt 6228 810 L 898 704 mt 925 704 L 6255 704 mt 6228 704 L 898 628 mt 925 628 L 6255 628 mt 6228 628 L 898 570 mt 925 570 L 6255 570 mt 6228 570 L 898 522 mt 925 522 L 6255 522 mt 6228 522 L 898 482 mt 925 482 L 6255 482 mt 6228 482 L 898 447 mt 925 447 L 6255 447 mt 6228 447 L 898 416 mt 925 416 L 6255 416 mt 6228 416 L 898 388 mt 925 388 L 6255 388 mt 6228 388 L 898 388 mt 952 388 L 6255 388 mt 6201 388 L /Helvetica /ISOLatin1Encoding 96 FMSR /Helvetica /ISOLatin1Encoding 144 FMSR 650 441 mt (10) s /Helvetica /ISOLatin1Encoding 96 FMSR 810 352 mt (7) s /Helvetica /ISOLatin1Encoding 144 FMSR 898 4614 mt 6255 4614 L 898 388 mt 6255 388 L 898 4614 mt 898 388 L 6255 4614 mt 6255 388 L 898 388 mt 898 388 L 6255 388 mt 6255 388 L gs 898 388 5358 4227 MR c np 536 202 535 248 536 284 536 306 535 323 536 333 536 342 536 351 535 364 536 387 898 676 11 MP stroke gr 36 36 898 676 FO 36 36 1434 1063 FO 36 36 1969 1427 FO 36 36 2505 1778 FO 36 36 3041 2120 FO 36 36 3577 2453 FO 36 36 4112 2776 FO 36 36 4648 3082 FO 36 36 5184 3366 FO 36 36 5719 3614 FO 36 36 6255 3816 FO gs 898 388 5358 4227 MR c np gr 36 36 898 4241 FO 36 36 1434 4377 FO 36 36 1969 4408 FO 36 36 2505 4410 FO 36 36 3041 4372 FO 36 36 3577 4327 FO 36 36 4112 4316 FO 36 36 4648 4333 FO 36 36 5184 4360 FO 36 36 5719 4383 FO 36 36 6255 4399 FO gs 898 388 5358 4227 MR c np DA 536 16 535 23 536 27 536 17 535 -11 536 -45 536 -38 536 2 535 31 536 136 898 4241 11 MP stroke DO 536 202 535 249 536 283 536 307 535 322 536 334 536 342 536 351 535 364 536 386 898 1369 11 MP stroke SO gr 36 36 898 1369 FO 36 36 1434 1755 FO 36 36 1969 2119 FO 36 36 2505 2470 FO 36 36 3041 2812 FO 36 36 3577 3146 FO 36 36 4112 3468 FO 36 36 4648 3775 FO 36 36 5184 4058 FO 36 36 5719 4307 FO 36 36 6255 4509 FO gs 898 388 5358 4227 MR c np gr 2108 273 mt (Analytically-derived mean slowdown: Upclose) s 3401 4943 mt (alpha) s 4112 681 mt (- Random) s 4112 1045 mt (-- SITA-E) s 4112 1285 mt (... Dynamic Approximation) s end eplot epage end showpage %%EndDocument endTexFig 523 4818 a Fo(Fig.)15 b(5.)30 b Fr(Analysis)h(of)g(mean)f(w)n(aiting)i (time)d(and)i(mean)e(slo)n(wdo)n(wn)j(o)n(v)n(er)e(empirically)h(relev) l(an)n(t)523 4910 y(range)26 b(of)h Fi(\013)p Fr(,)f(1)c Fp(\024)f Fi(\013)g Fp(\024)g Fr(2.)p eop %%Page: 11 11 11 10 bop 648 448 a Fv(Wh)n(y)24 b(do)r(es)g(SIT)-7 b(A-E)25 b(p)r(erform)f(so)g(w)n(ell)g(in)h(a)f(region)f(of)i(task)f(size)g(v)-5 b(ariabilit)n(y)23 b(wherein)523 548 y(a)j(Dynamic)g(p)r(olicy)g(explo) r(des?)g(A)h(careful)e(analysis)g(of)h(the)h(p)r(erformance)e(of)h(SIT) -7 b(A-E)26 b(at)523 648 y(eac)n(h)h(queue)g(of)h(the)g(system)f(\(see) h([4]\))f(leads)g(us)h(to)f(the)h(follo)n(wing)f(answ)n(ers:)558 842 y(1.)41 b(By)d(limiting)h(the)f(range)f(of)h(task)g(sizes)g(at)g (eac)n(h)f(host,)h(SIT)-7 b(A-E)38 b(greatly)f(reduces)664 942 y(the)24 b(v)-5 b(ariance)22 b(of)h(the)h(task)e(size)h (distribution)h(witnessed)f(b)n(y)g(the)g(lo)n(w)n(ered-n)n(um)n(b)r (ered)664 1041 y(hosts,)d(thereb)n(y)g(impro)n(ving)f(p)r(erformance)g (at)i(these)f(hosts.)g(In)h(fact)f(the)h(p)r(erformance)664 1141 y(at)28 b(most)f(hosts)g(is)h(sup)r(erior)e(to)i(that)f(of)h(an)f (M/M/1)g(queue)g(with)h(utilization)g Fg(\032)p Fv(.)558 1243 y(2.)41 b(When)19 b(load)f(is)g(balanced,)g(the)h(ma)5 b(jorit)n(y)17 b(of)h(tasks)f(are)h(assigned)f(to)h(the)h(lo)n(w-n)n (um)n(b)r(ered)664 1343 y(hosts,)28 b(whic)n(h)g(are)g(the)g(hosts)g (with)h(the)f(b)r(est)h(p)r(erformance.)e(This)i(is)f(in)n(tensi\014ed) g(b)n(y)664 1442 y(the)h(hea)n(vy-tailed)e(prop)r(ert)n(y)h(whic)n(h)g (implies)h(that)f Fl(very)i Fv(few)f(tasks)e(are)h(assigned)f(to)664 1542 y(high)h(n)n(um)n(b)r(ered)f(hosts.)558 1644 y(3.)41 b(F)-7 b(urthermore,)29 b(mean)g(slo)n(wdo)n(wn)f(is)h(impro)n(v)n(ed)f (b)r(ecause)h(small)g(tasks)f(observ)n(e)g(pro-)664 1743 y(p)r(ortionately)f(lo)n(w)n(er)f(w)n(aiting)h(times.)648 1936 y(F)-7 b(or)35 b(the)h(case)f(of)g Fg(\013)i Fh(\024)f Fv(1,)g(sho)n(wn)e(in)i(Figure)g(4,)f(ev)n(en)g(under)h(the)g(SIT)-7 b(A-E)35 b(p)r(olicy)-7 b(,)523 2035 y(system)34 b(p)r(erformance)f(ev) n(en)n(tually)g(deteriorates)f(badly)-7 b(.)34 b(The)g(reason)f(is)g (that)i(as)e(o)n(v)n(er-)523 2135 y(all)i(v)-5 b(ariabilit)n(y)35 b(in)h(task)f(sizes)g(increases,)f(ev)n(en)n(tually)h(ev)n(en)g(host)g (1)h(will)g(witness)f(high)523 2234 y(v)-5 b(ariabilit)n(y)e(.)24 b(F)-7 b(urther)24 b(analysis)f([4])i(indicates)f(that)h(adding)f (hosts)g(can)h(extend)f(the)h(range)523 2334 y(o)n(v)n(er)h(whic)n(h)h (SIT)-7 b(A-E)27 b(sho)n(ws)f(go)r(o)r(d)h(p)r(erformance.)f(F)-7 b(or)27 b(example,)g(when)g(the)h(n)n(um)n(b)r(er)f(of)523 2434 y(hosts)g(is)h(32,)e(SIT)-7 b(A-E's)27 b(p)r(erformance)g(do)r(es) g(not)h(deteriorate)e(un)n(til)i Fg(\013)c Fh(\024)e Fg(:)p Fv(8.)523 2711 y Fm(6)112 b(Conclusion)523 2922 y Fv(In)33 b(this)g(pap)r(er)g(w)n(e)g(ha)n(v)n(e)e(studied)j(ho)n(w)e (the)i(v)-5 b(ariabilit)n(y)32 b(of)g(the)i(task)e(size)h(distribution) 523 3022 y(in\015uences)41 b(whic)n(h)g(task)g(assignmen)n(t)f(p)r (olicy)h(is)g(b)r(est)h(in)f(a)g(distributed)h(system.)f(W)-7 b(e)523 3122 y(consider)21 b(four)g(p)r(olicies:)h(Random,)g (Round-Robin,)f(SIT)-7 b(A-E)22 b(\(a)f(size-based)g(p)r(olicy\),)h (and)523 3221 y(Dynamic)28 b(\(sending)f(the)h(task)f(to)h(the)g(host)f (with)h(the)g(least)f(remaining)g(w)n(ork\).)648 3323 y(W)-7 b(e)38 b(\014nd)h(that)g(the)g(b)r(est)g(c)n(hoice)e(of)h(task)g (assignmen)n(t)g(p)r(olicy)g(dep)r(ends)h(critically)523 3423 y(on)32 b(the)h(v)-5 b(ariabilit)n(y)31 b(of)h(task)g(size)g (distribution.)g(When)h(the)f(task)g(sizes)g(are)f(not)h(highly)523 3522 y(v)-5 b(ariable,)31 b(the)i(Dynamic)g(p)r(olicy)f(is)g (preferable.)g(Ho)n(w)n(ev)n(er,)e(when)j(task)e(sizes)h(sho)n(w)g(the) 523 3622 y(degree)38 b(of)h(v)-5 b(ariabilit)n(y)38 b(more)g(c)n (haracteristic)f(of)i(empirical)g(measuremen)n(ts)f(\()p Fg(\013)k Fh(\031)g Fv(1\),)523 3722 y(SIT)-7 b(A-E)27 b(is)h(b)r(est.)648 3824 y(The)23 b(magnitude)g(of)g(the)h (di\013erence)f(in)g(p)r(erformance)f(of)h(these)g(p)r(olicies)g(can)g (b)r(e)h(quite)523 3923 y(large:)k(Random)h(and)g(Round-Robin)g(are)f (inferior)g(to)h(b)r(oth)h(SIT)-7 b(A-E)29 b(and)g(Dynamic)g(b)n(y)523 4023 y(sev)n(eral)34 b(orders)g(of)i(magnitude.)f(And)i(in)f(the)g (range)e(of)i(task)f(size)g(v)-5 b(ariabilit)n(y)35 b(c)n(harac-)523 4123 y(teristic)28 b(of)h(empirical)f(measuremen)n(ts,)f(SIT)-7 b(A-E)29 b(outp)r(erforms)e(Dynamic)i(b)n(y)f(close)g(to)g(2)523 4222 y(orders)e(of)i(magnitude.)648 4324 y(More)h(imp)r(ortan)n(t)i (than)g(the)g(ab)r(o)n(v)n(e)e(results,)i(though,)f(is)h(the)g(insigh)n (ts)f(ab)r(out)h(these)523 4424 y(four)c(p)r(olicies)h(gleaned)e(from)i (our)e(analysis:)648 4526 y(Our)d(analysis)h(of)g(the)h(Random,)f (Round-Robin)g(and)h(Dynamic)f(p)r(olicies)g(sho)n(ws)g(that)523 4625 y(their)39 b(p)r(erformance)f(is)g(directly)h(prop)r(ortional)e (to)h(the)i(second)e(momen)n(t)h(of)f(the)i(task)523 4725 y(size)27 b(distribution,)g(whic)n(h)f(explains)h(wh)n(y)f(their)h (p)r(erformance)f(deteriorates)f(as)h(the)i(task)523 4825 y(size)f(v)-5 b(ariabilit)n(y)26 b(increases.)g(Th)n(us,)g(ev)n (en)h(the)g(Dynamic)h(p)r(olicy)-7 b(,)27 b(whic)n(h)g(comes)f(closes)g (to)523 4924 y(ac)n(hieving)34 b(instan)n(taneous)g(load)g(balance)g (and)h(directs)g(eac)n(h)f(task)g(to)h(the)h(host)f(where)p eop %%Page: 12 12 12 11 bop 523 448 a Fv(it)38 b(w)n(aits)f(the)h(least,)f(is)h(not)f (capable)g(of)h(comp)r(ensating)f(for)g(the)h(e\013ect)g(of)f (increasing)523 548 y(v)-5 b(ariance)26 b(in)i(the)g(task)f(size)h (distribution.)648 648 y(T)-7 b(o)40 b(understand)h(wh)n(y)g (size-based)f(p)r(olicies)h(are)g(so)f(p)r(o)n(w)n(erful,)h(w)n(e)g(in) n(tro)r(duce)g(the)523 747 y(SIT)-7 b(A-E)29 b(p)r(olicy)h(whic)n(h)f (is)g(a)g(simple)h(formalization)e(of)i(size-based)e(p)r(olicies,)h (de\014ned)h(to)523 847 y(equalize)f(the)h(exp)r(ected)g(load)f(at)h (eac)n(h)e(host.)i(This)g(formalization)e(allo)n(ws)g(us)i(to)f(obtain) 523 946 y(a)k(full)h(analysis)e(of)h(the)h(SIT)-7 b(A-E)33 b(p)r(olicy)-7 b(,)33 b(leading)g(to)g(a)g(3-fold)g(c)n (haracterization)e(of)i(its)523 1046 y(p)r(o)n(w)n(er:)18 b(\(i\))j(By)e(limiting)h(the)g(range)e(of)i(task)f(sizes)g(at)g(eac)n (h)g(host,)g(SIT)-7 b(A-E)19 b(greatly)g(reduces)523 1146 y(the)38 b(v)-5 b(ariabilit)n(y)37 b(of)g(the)h(task)f(size)h (distribution)f(witnessed)h(b)n(y)f(eac)n(h)g(host)g({)h(thereb)n(y)523 1245 y(impro)n(ving)28 b(the)i(p)r(erformance)f(at)g(the)h(host.)g (\(ii\))g(When)g(load)f(is)g(balanced,)g(most)h(tasks)523 1345 y(are)18 b(sen)n(t)h(to)g(the)h(subset)f(the)h(hosts)f(ha)n(ving)f (the)i(b)r(est)f(p)r(erformance.)f(\(iii\))j(Mean)e(slo)n(wdo)n(wn)523 1445 y(is)37 b(impro)n(v)n(ed)f(b)r(ecause)h(small)g(tasks)f(observ)n (e)g(prop)r(ortionately)f(lo)n(w)n(er)h(w)n(aiting)h(times.)523 1544 y(These)c(3)g(prop)r(erties)f(allo)n(w)h(SIT)-7 b(A-E)33 b(to)g(p)r(erform)g(v)n(ery)f(w)n(ell)h(in)g(a)g(region)f(of)i (task)e(size)523 1644 y(v)-5 b(ariabilit)n(y)27 b(in)g(whic)n(h)h(the)g (Dynamic)g(p)r(olicy)f(breaks)f(do)n(wn.)523 1895 y Fm(References)561 2067 y Fr(1.)43 b(M.)27 b(E.)f(Cro)n(v)n(ella)i(and)e(A.)g(Besta)n (vros.)37 b(Self-similarit)n(y)27 b(in)f(World)h(Wide)f(Web)g (tra\016c:)h(Evi-)663 2158 y(dence)20 b(and)g(p)r(ossible)i(causes.)k Fn(IEEE/A)n(CM)d(T)-6 b(r)l(ansactions)25 b(on)e(Networking)p Fr(,)f(5\(6\):835{846,)663 2249 y(Decem)n(b)r(er)i(1997.)561 2337 y(2.)43 b(M.)20 b(E.)h(Cro)n(v)n(ella)g(and)f(L.)g(Lipsky)-6 b(.)24 b(Long-lasting)e(transien)n(t)e(conditions)g(in)g(sim)n (ulations)h(with)663 2428 y(hea)n(vy-tailed)k(w)n(orkloads.)36 b(In)25 b Fn(1997)j(Winter)g(Simulation)f(Confer)l(enc)l(e)p Fr(,)g(1997.)561 2516 y(3.)43 b(M.)31 b(E.)g(Cro)n(v)n(ella,)h(M.)f(S.) g(T)-6 b(aqqu,)30 b(and)g(A.)g(Besta)n(vros.)50 b(Hea)n(vy-tailed)30 b(probabilit)n(y)h(distri-)663 2607 y(butions)26 b(in)f(the)h(w)n(orld) h(wide)f(w)n(eb.)36 b(In)25 b Fn(A)j(Pr)l(actic)l(al)h(Guide)f(T)-6 b(o)28 b(He)l(avy)h(T)-6 b(ails)p Fr(,)26 b(pages)h(1{23.)663 2699 y(Chapman)e(&)g(Hall,)h(New)g(Y)-6 b(ork,)25 b(1998.)561 2786 y(4.)43 b(M.)28 b(Harc)n(hol-Balter,)h(M.)f(E.)f(Cro)n(v)n(ella,)i (and)e(C.)i(D.)e(Murta.)40 b(On)26 b(c)n(ho)r(osing)j(a)f(task)f (assign-)663 2878 y(men)n(t)e(p)r(olicy)j(for)f(a)h(distributed)e(serv) n(er)h(system.)38 b(T)-6 b(ec)n(hnical)27 b(Rep)r(ort)g (MIT-LCS-TR-757,)663 2969 y(MIT)f(Lab)r(oratory)g(for)h(Computer)e (Science,)h(1998.)561 3056 y(5.)43 b(M.)33 b(Harc)n(hol-Balter)h(and)e (A.)g(Do)n(wney)-6 b(.)55 b(Exploiting)33 b(pro)r(cess)h(lifetime)f (distributions)g(for)663 3148 y(dynamic)24 b(load)i(balancing.)36 b Fn(A)n(CM)27 b(T)-6 b(r)l(ansactions)30 b(on)d(Computer)i(Systems)p Fr(,)f(15\(3\),)f(1997.)561 3235 y(6.)43 b(S.)24 b(Hoto)n(vy)-6 b(,)24 b(D.)h(Sc)n(hneider,)f(and)g(T.)i(O'Donnell.)33 b(Analysis)25 b(of)g(the)g(early)g(w)n(orkload)h(on)e(the)663 3327 y(Cornell)j(Theory)f(Cen)n(ter)f(IBM)h(SP2.)35 b(T)-6 b(ec)n(hnical)26 b(Rep)r(ort)g(96TR234,)i(CTC,)f(Jan.)g(1996.)561 3414 y(7.)43 b(G.)21 b(Irlam.)27 b(Unix)20 b(\014le)i(size)g(surv)n(ey) -6 b(.)26 b Fq(http://www.base.com/gordoni/uf)q(s93.)q(html)p Fr(,)i(1994.)561 3502 y(8.)43 b(W.)28 b(E.)h(Leland)f(and)g(T.)h(J.)f (Ott.)42 b(Load-balancing)29 b(heuristics)g(and)e(pro)r(cess)j(b)r(eha) n(vior.)42 b(In)663 3593 y Fn(Pr)l(o)l(c)l(e)l(e)l(dings)29 b(of)f(Performanc)l(e)h(and)e(A)n(CM)h(Sigmetrics)p Fr(,)f(pages)f (54{69,)i(1986.)561 3681 y(9.)43 b(R.)23 b(D.)g(Nelson)h(and)f(T.)h(K.) g(Philips.)32 b(An)22 b(appro)n(ximation)h(for)h(the)f(mean)g(resp)r (onse)h(time)f(for)663 3772 y(shortest)32 b(queue)e(routing)i(with)g (general)h(in)n(terarriv)l(al)f(and)f(service)h(times.)51 b Fn(Performanc)l(e)663 3863 y(Evaluation)p Fr(,)26 b(17:123{139,)k (1998.)523 3951 y(10.)43 b(V.)d(P)n(axson)g(and)g(S.)g(Flo)n(yd.)78 b(Wide-area)40 b(tra\016c:)h(The)g(failure)g(of)g(P)n(oisson)h(mo)r (deling.)663 4042 y Fn(IEEE/A)n(CM)27 b(T)-6 b(r)l(ansactions)29 b(on)f(Networking)p Fr(,)g(pages)e(226{244,)j(June)c(1995.)523 4130 y(11.)43 b(D.)25 b(L.)i(P)n(eterson)f(and)g(D.)g(B.)g(Adams.)34 b(F)-6 b(ractal)27 b(patterns)f(in)f(D)n(ASD)g(I/O)g(tra\016c.)36 b(In)25 b Fn(CMG)663 4221 y(Pr)l(o)l(c)l(e)l(e)l(dings)p Fr(,)j(Decem)n(b)r(er)d(1996.)523 4309 y(12.)43 b(R.)22 b(W.)g(W)-6 b(eb)r(er.)28 b(On)22 b(the)f(optimal)h(assignmen)n(t)g(of) h(customers)f(to)g(parallel)i(serv)n(ers.)29 b Fn(Journal)663 4400 y(of)e(Applie)l(d)g(Pr)l(ob)l(ability)p Fr(,)g(15:406{413,)j (1978.)523 4488 y(13.)43 b(W)-6 b(ard)31 b(Whitt.)51 b(Deciding)32 b(whic)n(h)f(queue)g(to)g(join:)i(Some)d(coun)n (terexamples.)51 b Fn(Op)l(er)l(ations)663 4579 y(R)l(ese)l(ar)l(ch)p Fr(,)28 b(34\(1\):226{244,)i(Jan)n(uary)25 b(1986.)523 4667 y(14.)43 b(W.)25 b(Winston.)34 b(Optimalit)n(y)24 b(of)i(the)f(shortest)g(line)h(discipline.)34 b Fn(Journal)28 b(of)f(Applie)l(d)g(Pr)l(ob)l(a-)663 4758 y(bility)p Fr(,)e(14:181{189,)30 b(1977.)523 4845 y(15.)43 b(R.)25 b(W.)h(W)-6 b(ol\013.)34 b Fn(Sto)l(chastic)c(Mo)l(deling)e(and)g(the)g (The)l(ory)h(of)e(Queues)p Fr(.)35 b(Pren)n(tice)26 b(Hall,)h(1989.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF