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1800 6075 l gs col0 s gr gr % arrowhead 0 slj n 1528 6135 m 1768 6075 l 1528 6015 l 1528 6135 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj [68] 0 sd gs clippath 3015 6135 m 3015 6015 l 2728 6015 l 2968 6075 l 2728 6135 l cp eoclip n 1800 6075 m 3000 6075 l gs col0 s gr gr [] 0 sd % arrowhead 0 slj n 2728 6135 m 2968 6075 l 2728 6015 l 2728 6135 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj [90] 0 sd gs clippath 11565 6135 m 11565 6015 l 11278 6015 l 11518 6075 l 11278 6135 l cp eoclip n 10425 6075 m 11550 6075 l gs col0 s gr gr [] 0 sd % arrowhead 0 slj n 11278 6135 m 11518 6075 l 11278 6015 l 11278 6135 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj gs clippath 12765 6135 m 12765 6015 l 12478 6015 l 12718 6075 l 12478 6135 l cp eoclip n 11625 6075 m 12750 6075 l gs col0 s gr gr % arrowhead 0 slj n 12478 6135 m 12718 6075 l 12478 6015 l 12478 6135 l cp gs 0.00 setgray ef gr col0 s % Polyline 2 slj [90] 0 sd gs clippath 11686 6151 m 11571 6119 l 11494 6396 l 11616 6181 l 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(distribution)f(whose)523 2708 y(underlying)g(Mark)n(o)n(v)h(c)n(hain)h (is)f(of)h(the)f(form)g(in)g(the)f(\014gure,)h(where)h(0)22 b Fw(\024)g Ft(p)2730 2716 y Fn(i)2778 2708 y Fw(\024)g Fx(1)k(and)g Ft(\025)3119 2716 y Fn(i)3167 2708 y Ft(>)c Fx(0)k(for)523 2799 y(all)h(0)21 b Fw(\024)g Ft(i)h Fw(\024)f Ft(n)p Fx(.)523 3095 y Fz(to)h(b)r(e)g(e\013ectiv)n(e)g(in)g (predicting)g(mean)g(p)r(erformance)f(for)g(v)-5 b(ariet)n(y)21 b(of)h(man)n(y)f(computer)h(sys-)523 3195 y(tem)k(mo)r(dels)g([7,)13 b(10,)g(23,)g(29,)g(33].)25 b(Clearly)-7 b(,)25 b(ho)n(w)n(ev)n(er,)f (three)h(momen)n(ts)h(migh)n(t)f(not)h(alw)n(a)n(ys)523 3294 y(su\016ce)h(for)g(ev)n(ery)f(problem,)g(and)h(w)n(e)g(lea)n(v)n (e)f(the)i(problem)e(of)h(matc)n(hing)g(more)f(momen)n(ts)523 3394 y(to)i(future)g(w)n(ork.)648 3499 y(Most)g(existing)h(algorithms)e (for)i(\014tting)g(a)g(general)f(distribution)h Fq(G)g Fz(to)g(a)g(PH)f(distri-)523 3599 y(bution,)d(restrict)g(their)g(atten) n(tion)g(to)g(a)g(subset)g(of)g(PH)g(distributions,)g(since)f(general)g (PH)523 3699 y(distributions)f(ha)n(v)n(e)e(so)h(man)n(y)g(parameters)f (that)i(it)g(is)g(di\016cult)g(to)g(\014nd)g(time-e\016cien)n(t)g(al-) 523 3798 y(gorithms)j(for)g(\014tting)h(to)g(the)g(general)e(PH)h (distributions)h([14,)13 b(15,)g(20,)g(27,)g(32].)25 b(The)i(most)523 3898 y(commonly)e(c)n(hosen)h(subset)g(is)g(the)g (class)f(of)h(Co)n(xian)f(distributions,)h(sho)n(wn)g(in)g(Figure)f(2.) 523 3998 y(Co)n(xian)e(distributions)g(ha)n(v)n(e)g(the)h(adv)-5 b(an)n(tage)23 b(of)h(b)r(eing)f(m)n(uc)n(h)h(simpler)g(than)g(general) e(PH)523 4097 y(distributions,)h(while)h(including)f(a)g(large)f (subset)i(of)f(PH)g(distributions)g(without)h(needing)523 4197 y(additional)i(phases.)g(F)-7 b(or)26 b(example,)g(for)h(an)n(y)e (acyclic)h(PH)h(distribution)g Fq(P)2897 4209 y Fl(n)2942 4197 y Fz(,)g(there)f(exists)523 4296 y(a)36 b(Co)n(xian)g (distribution)h Fq(C)1413 4308 y Fl(n)1495 4296 y Fz(with)g(the)g(same) f(n)n(um)n(b)r(er)h(of)g(phases)f(suc)n(h)g(that)h Fq(P)3189 4308 y Fl(n)3271 4296 y Fz(and)523 4396 y Fq(C)582 4408 y Fl(n)663 4396 y Fz(ha)n(v)n(e)e(the)h(same)f(distribution)h(function) h([5].)e(In)h(this)g(pap)r(er)f(w)n(e)h(will)g(restrict)f(our)523 4496 y(atten)n(tion)27 b(to)h(Co)n(xian)e(distributions.)523 4725 y Fr(Motivation)49 b(and)h(Go)-5 b(al)60 b Fz(When)43 b(\014nding)f(a)f(Co)n(xian)g(distribution)h Fq(C)48 b Fz(whic)n(h)42 b(w)n(ell-)523 4825 y(represen)n(ts)35 b(a)h(giv)n(en)g(distribution)h Fq(G)p Fz(,)g(it)g(is)f(desirable)g (that)h Fq(C)43 b Fz(b)r(e)37 b Fo(minimal)p Fz(,)h(i.e.,)f(the)523 4924 y(n)n(um)n(b)r(er)g(of)g(phases)f(in)i Fq(C)43 b Fz(b)r(e)38 b(as)f(small)f(as)h(p)r(ossible.)g(This)g(is)g(imp)r(ortan) n(t)g(b)r(ecause)g(it)p eop %%Page: 3 3 3 2 bop 523 448 a Fz(minimizes)29 b(the)g(additional)f(states)g (necessary)e(in)j(the)g(resulting)f(Mark)n(o)n(v)e(c)n(hain)i(for)g (the)523 548 y(queueing)d(system.)h(Unfortunately)-7 b(,)26 b(it)g(is)g(not)g(kno)n(wn)f(what)h(is)f(the)h(minimal)h(n)n(um) n(b)r(er)e(of)523 648 y(phases)j(necessary)f(to)i(w)n(ell-represen)n(t) e(a)i(giv)n(en)f(distribution)h Fq(G)g Fz(b)n(y)g(a)g(Co)n(xian)e (distribu-)523 747 y(tion.)i(This)g(mak)n(es)g(it)g(di\016cult)h(to)f (ev)-5 b(aluate)29 b(the)g(e\013ectiv)n(eness)g(of)g(di\013eren)n(t)g (algorithms)523 847 y(and)e(also)g(mak)n(es)g(the)g(design)h(of)f (\014tting)h(algorithms)e(op)r(en-ended.)648 948 y(The)20 b Fo(primary)k(go)l(al)e Fz(of)e(this)g(pap)r(er)g(is)g(to)g(c)n (haracterize)e(the)j(set)f(of)g(distributions)g(whic)n(h)523 1048 y(are)27 b(w)n(ell-represen)n(ted)e(b)n(y)j(an)f Fq(n)p Fz(-phase)g(Co)n(xian)f(distribution,)i(for)f(eac)n(h)g Fq(n)22 b Fz(=)h(1)p Fq(;)14 b Fz(2)p Fq(;)g Fz(3)p Fq(;)g(:)g(:)g(:)n Fz(.)523 1213 y Fp(De\014nition)31 b(2.)41 b Fo(L)l(et)29 b Fk(S)1284 1183 y Fy(\()p Fl(n)p Fy(\))1412 1213 y Fo(denote)h(the)g (set)g(of)g(distributions)h(that)f(ar)l(e)g(wel)t(l-r)l(epr)l(esente)l (d)523 1312 y(by)g(an)g Fq(n)p Fo(-phase)h(Coxian)f(distribution)h(for) f(p)l(ositive)i(inte)l(ger)d Fq(n)p Fo(.)523 1476 y Fz(Our)i(c)n (haracterization)e(of)i Fk(fS)1505 1445 y Fy(\()p Fl(n)p Fy(\))1602 1476 y Fq(;)14 b(n)30 b Fk(\025)f Fz(1)p Fk(g)i Fz(will)h(allo)n(w)e(one)h(to)h(determine,)f(for)h(an)n(y)e(dis-)523 1575 y(tribution)36 b Fq(G)p Fz(,)h(the)f(minimal)g(n)n(um)n(b)r(er)g (of)g(phases)f(that)i(are)e(needed)h(to)g(w)n(ell-represen)n(t)523 1675 y Fq(G)31 b Fz(b)n(y)f(a)g(Co)n(xian)f(distribution.)1541 1645 y Fy(1)1609 1675 y Fz(Suc)n(h)h(a)g(c)n(haracterization)e(will)j (b)r(e)f(a)g(useful)h(guideline)523 1774 y(for)c(designing)g (algorithms)f(whic)n(h)i(\014t)g(general)e(distributions)h(to)h(Co)n (xian)e(distributions.)523 1874 y(Another)34 b(application)f(of)g(this) h(c)n(haracterization)d(is)j(that)g(some)f(existing)g(\014tting)i (algo-)523 1974 y(rithms,)28 b(suc)n(h)g(as)f(Johnson)g(and)h(T)-7 b(aa\013e's)27 b(nonlinear)g(programming)f(approac)n(h)g([15],)h(re-) 523 2073 y(quire)g(kno)n(wing)g(the)h(n)n(um)n(b)r(er)f(of)h(phases)e Fq(n)i Fz(in)g(the)g(minimal)g(Co)n(xian)e(distribution.)i(The)523 2173 y(curren)n(t)34 b(approac)n(h)f(in)n(v)n(olv)n(es)f(simply)j (iterating)f(o)n(v)n(er)f(all)h(c)n(hoices)g(for)g Fq(n)g Fz([15],)g(whereas)523 2273 y(our)27 b(c)n(haracterization)e(w)n(ould)i (immediately)h(sp)r(ecify)g Fq(n)p Fz(.)648 2374 y(Pro)n(viding)c (su\016cien)n(t)j(and)f(necessary)f(conditions)h(for)g(a)g (distribution)h(to)g(b)r(e)g(in)f Fk(S)3307 2344 y Fy(\()p Fl(n)p Fy(\))523 2474 y Fz(do)r(es)d(not)f(alw)n(a)n(ys)f(immediately)i (giv)n(e)f(one)h(a)f(sense)h(of)g Fo(which)h Fz(distributions)f (satisfy)f(those)523 2573 y(conditions,)40 b(or)f(of)i(the)f(magnitude) h(of)f(the)h(set)f(of)g(distributions)g(whic)n(h)h(satisfy)f(the)523 2673 y(condition.)e(A)h Fo(se)l(c)l(ondary)i(go)l(al)e Fz(of)g(this)g(pap)r(er)f(is)g(to)g(pro)n(vide)f(examples)h(of)h (common)523 2772 y(distributions)27 b(whic)n(h)h(are)f(included)h(in)g Fk(S)1870 2742 y Fy(\()p Fl(n)p Fy(\))1995 2772 y Fz(for)f(particular)f (in)n(tegers)h Fq(n)p Fz(.)648 2874 y(In)f(\014nding)h Fo(simple)g Fz(c)n(haracterizations)d(of)i Fk(S)2069 2844 y Fy(\()p Fl(n)p Fy(\))2166 2874 y Fz(,)h(it)g(will)f(b)r(e)h(v)n (ery)e(helpful)i(to)g(start)e(b)n(y)523 2974 y(de\014ning)c(an)g (alternativ)n(e)f(to)h(the)h(standard)e(momen)n(ts,)h(whic)n(h)g(w)n(e) g(refer)g(to)g(as)f Fo(normalize)l(d)523 3073 y(moments)p Fz(.)523 3238 y Fp(De\014nition)31 b(3.)41 b Fo(L)l(et)35 b Fq(\026)1284 3208 y Fl(F)1284 3262 y(k)1374 3238 y Fo(b)l(e)h(the)g Fq(k)s Fo(-th)f(moment)g(of)h(a)g(distribution)h Fq(F)47 b Fo(for)37 b Fq(k)f Fz(=)d(1)p Fq(;)14 b Fz(2)p Fq(;)g Fz(3)p Fo(.)523 3338 y(The)31 b Fp(normalized)f Fq(k)s Fp(-th)i(momen)m(t)26 b Fq(m)1823 3308 y Fl(F)1823 3361 y(k)1908 3338 y Fo(of)31 b Fq(F)41 b Fo(for)31 b Fq(k)26 b Fz(=)d(2)p Fq(;)14 b Fz(3)28 b Fo(is)i(de\014ne)l(d)g(to)g(b) l(e)1318 3582 y Fq(m)1391 3548 y Fl(F)1391 3603 y Fy(2)1470 3582 y Fz(=)1618 3526 y Fq(\026)1668 3496 y Fl(F)1668 3547 y Fy(2)p 1567 3563 208 4 v 1567 3641 a Fz(\()p Fq(\026)1649 3613 y Fl(F)1649 3663 y Fy(1)1705 3641 y Fz(\))1737 3617 y Fy(2)1899 3582 y Fz(and)84 b Fq(m)2190 3548 y Fl(F)2190 3603 y Fy(3)2268 3582 y Fz(=)2419 3526 y Fq(\026)2469 3496 y Fl(F)2469 3547 y Fy(3)p 2366 3563 211 4 v 2366 3641 a Fq(\026)2416 3613 y Fl(F)2416 3663 y Fy(1)2471 3641 y Fq(\026)2521 3613 y Fl(F)2521 3663 y Fy(2)2586 3582 y Fq(:)523 3818 y Fz(Notice)30 b(the)h(corresp)r(ondence)d(to)i (the)h(co)r(e\016cien)n(t)f(of)g(v)-5 b(ariabilit)n(y)29 b Fq(C)2707 3830 y Fl(F)2793 3818 y Fz(and)h(sk)n(ewness)f Fq(\015)3350 3830 y Fl(F)523 3938 y Fz(of)40 b Fq(F)12 b Fz(:)39 b Fq(m)830 3908 y Fl(F)830 3959 y Fy(2)929 3938 y Fz(=)j Fq(C)1101 3908 y Fy(2)1095 3961 y Fl(F)1177 3938 y Fz(+)26 b(1)40 b(and)f Fq(m)1596 3908 y Fl(F)1596 3959 y Fy(3)1694 3938 y Fz(=)k Fq(\027)1843 3950 y Fl(F)1898 3865 y Fj(p)p 1981 3865 129 4 v 73 x Fq(m)2054 3909 y Fl(F)2054 3960 y Fy(2)2109 3938 y Fz(,)d(where)f Fq(\027)2465 3950 y Fl(F)2563 3938 y Fz(=)2753 3896 y Fl(\026)2793 3871 y Fi(F)2793 3912 y Fh(3)p 2681 3919 231 4 v 2681 3971 a Fy(\()p Fl(\026)2747 3951 y Fi(F)2747 3991 y Fh(2)2795 3971 y Fy(\))2821 3954 y Fh(3)p Fi(=)p Fh(2)2922 3938 y Fz(.)h(\(Notice)g(the)523 4095 y(corresp)r(ondence)25 b(b)r(et)n(w)n(een)h Fq(\027)1462 4107 y Fl(F)1543 4095 y Fz(and)g(the)h(sk)n(ewness)e(of)h Fq(F)12 b Fz(,)27 b Fq(\015)2442 4107 y Fl(F)2497 4095 y Fz(,)f(where)g Fq(\015)2828 4107 y Fl(F)2906 4095 y Fz(=)3081 4052 y Fy(\026)-38 b Fl(\026)3116 4027 y Fi(F)3116 4069 y Fh(3)p 3004 4076 V 3004 4127 a Fy(\()5 b(\026)-38 b Fl(\026)3070 4107 y Fi(F)3070 4147 y Fh(2)3118 4127 y Fy(\))3144 4111 y Fh(3)p Fi(=)p Fh(2)3271 4095 y Fz(and)530 4225 y(\026)-49 b Fq(\026)573 4195 y Fl(F)573 4249 y(k)656 4225 y Fz(is)27 b(the)h(cen)n(tralized)f Fq(k)s Fz(-th)h(momen)n(t)f(of)h Fq(F)39 b Fz(for)27 b Fq(k)f Fz(=)d(2)p Fq(;)14 b Fz(3.\))p 523 4286 473 4 v 546 4345 a Fm(1)606 4376 y Fx(One)k(migh)n(t)g (initially)h(argue)g(that)g Fw(S)1665 4345 y Fm(\(2\))1746 4376 y Fx(,)g(the)f(set)h(of)h(distributions)e(w)n(ell-represen)n(ted)h (b)n(y)f(a)h(t)n(w)n(o-)606 4468 y(phase)29 b(Co)n(xian)h (distribution,)g(should)f(include)g(all)h(distributions,)g(since)f(a)h (2-phase)f(Co)n(xian)606 4559 y(distribution)j(has)g(four)g(parameters) g(\()p Ft(p)1813 4567 y Fm(1)1847 4559 y Fx(,)g Ft(p)1939 4567 y Fm(2)1974 4559 y Fx(,)g Ft(\025)2072 4567 y Fm(1)2106 4559 y Fx(,)g Ft(\025)2204 4567 y Fm(2)2239 4559 y Fx(\),)g(whereas)h (w)n(e)f(only)g(need)f(to)h(matc)n(h)606 4650 y(three)h(momen)n(ts)f (of)i Ft(G)p Fx(.)g(A)f(simple)g(coun)n(ter)g(example)g(sho)n(ws)h (this)g(argumen)n(t)e(to)i(b)r(e)f(false.)606 4742 y(Let)28 b Ft(G)g Fx(b)r(e)f(a)h(distribution)g(whose)h(\014rst)e(three)h(momen) n(ts)e(are)i(1,)h(2,)f(and)f(12.)i(The)f(system)f(of)606 4833 y(equations)k(for)h(matc)n(hing)e Ft(G)h Fx(to)h(a)f(2-phase)g(Co) n(xian)h(distribution)f(with)g(three)g(parameters)606 4924 y(\()p Ft(\025)681 4932 y Fm(1)715 4924 y Fx(,)26 b Ft(\025)807 4932 y Fm(2)842 4924 y Fx(,)g Ft(p)p Fx(\))f(results)h (in)f(either)h Ft(\025)1579 4932 y Fm(1)1639 4924 y Fx(or)g Ft(\025)1778 4932 y Fm(2)1838 4924 y Fx(b)r(eing)g(negativ)n(e.)p eop %%Page: 4 4 4 3 bop 523 448 a Fr(R)-5 b(elevant)38 b(Pr)-5 b(evious)38 b(Work)46 b Fz(All)31 b(prior)f(w)n(ork)f(on)h(c)n(haracterizing)e Fk(S)2857 418 y Fy(\()p Fl(n)p Fy(\))2986 448 y Fz(has)i(fo)r(cused)523 548 y(on)j(c)n(haracterizing)f Fk(S)1240 518 y Fy(\(2\))1325 493 y Fg(\003)1364 548 y Fz(,)i(where)f Fk(S)1723 518 y Fy(\(2\))1808 493 y Fg(\003)1881 548 y Fz(is)h(the)g(set)g(of)g (distributions)f(whic)n(h)h(are)f(w)n(ell-)523 648 y(represen)n(ted)i (b)n(y)g(a)g(2-phase)g(Co)n(xian)1737 617 y Fy(+)1826 648 y Fz(distribution,)h(where)f(a)g(Co)n(xian)2890 617 y Fy(+)2980 648 y Fz(distribution)523 747 y(is)g(simply)f(a)h(Co)n (xian)e(distribution)i(with)g(no)f(mass)g(probabilit)n(y)g(at)h(zero,)e (i.e.)i Fq(p)3169 759 y Fy(1)3241 747 y Fz(=)f(1.)523 847 y(Observ)n(e)19 b Fk(S)891 817 y Fy(\(2\))976 792 y Fg(\003)1038 847 y Fk(\032)k(S)1182 817 y Fy(\(2\))1272 847 y Fz(.)e(Altiok)f([2])h(sho)n(w)n(ed)e(a)h(su\016cien)n(t)h (condition)g(for)f(a)g(distribution)h Fq(G)523 946 y Fz(to)k(b)r(e)h(in)f Fk(S)883 916 y Fy(\(2\))968 891 y Fg(\003)1008 946 y Fz(.)g(More)g(recen)n(tly)-7 b(,)24 b(T)-7 b(elek)25 b(and)h(Heindl)g([31])e(expanded)h(Altiok's)g (condition)523 1046 y(and)31 b(pro)n(v)n(ed)e(the)i(necessary)e(and)h (su\016cien)n(t)h(condition)f(for)g(a)g(distribution)h Fq(G)g Fz(to)g(b)r(e)g(in)523 1146 y Fk(S)579 1116 y Fy(\(2\))664 1091 y Fg(\003)703 1146 y Fz(.)f(While)h(neither)f(Altiok) g(nor)f(T)-7 b(elek)30 b(and)g(Heindl)g(expressed)f(these)h(conditions) g(in)523 1245 y(terms)21 b(of)h(normalized)e(momen)n(ts,)i(the)g (results)f(can)g(b)r(e)h(expressed)e(more)h(simply)h(with)g(our)523 1345 y(normalized)h(momen)n(ts,)g(as)h(sho)n(wn)f(in)h(Theorem)f(1.)g (In)h(this)g(pap)r(er,)g(w)n(e)f(will)h(c)n(haracterize)523 1445 y Fk(S)579 1414 y Fy(\(2\))669 1445 y Fz(,)j(as)g(w)n(ell)h(as)f (c)n(haracterizing)e Fk(S)1681 1414 y Fy(\()p Fl(n)p Fy(\))1778 1445 y Fz(,)j(for)f(all)h(in)n(tegers)e Fq(n)d Fk(\025)f Fz(2.)523 1664 y Fr(Our)38 b(R)-5 b(esults)47 b Fz(While)32 b(the)g(goal)e(of)h(the)h(pap)r(er)f(is)h(to)f(c)n (haracterize)e(the)j(set)g Fk(S)3119 1634 y Fy(\()p Fl(n)p Fy(\))3216 1664 y Fz(,)g(this)523 1764 y(c)n(haracterization)26 b(turns)j(out)g(to)f(b)r(e)h(ugly)-7 b(.)29 b(One)f(of)h(the)g(k)n(ey)f (ideas)h(in)g(the)g(pap)r(er)f(is)h(that)523 1864 y(there)23 b(is)f(a)g(set)h Fk(S)1048 1876 y Ff(V)1099 1827 y Fy(\()p Fl(n)p Fy(\))1219 1864 y Fk(\032)g(S)1363 1833 y Fy(\()p Fl(n)p Fy(\))1483 1864 y Fz(whic)n(h)g(is)g(v)n(ery)e(close)h(to)h Fk(S)2319 1833 y Fy(\()p Fl(n)p Fy(\))2439 1864 y Fz(in)g(size,)f(suc)n (h)h(that)g Fk(S)3114 1876 y Ff(V)3165 1827 y Fy(\()p Fl(n)p Fy(\))3285 1864 y Fz(has)523 1963 y(a)31 b(v)n(ery)g(simple)h (sp)r(eci\014cation)g(via)f(normalized)g(momen)n(ts.)g(Th)n(us,)h(m)n (uc)n(h)g(of)g(the)g(pro)r(ofs)523 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y(b)r(et)n(w)n(een)26 b Fk(S)892 2899 y Ff(V)943 2851 y Fy(\()p Fl(n)p Fy(\))1066 2887 y Fz(and)f Fk(S)1281 2857 y Fy(\()p Fl(n)p Fy(\))1404 2887 y Fz(for)g(all)h Fq(n)d Fk(\025)f Fz(2.)j(This)h(relationship)f(is)g(illustrated)h(in)f (Figure)g(3)523 2986 y(and)k(pro)n(v)n(en)f(in)h(Section)g(3.)g(There)f (are)g(three)h(p)r(oin)n(ts)g(to)g(observ)n(e:)f(\(i\))i Fk(S)2885 2956 y Fy(\()p Fl(n)p Fy(\))3011 2986 y Fz(is)f(a)g(prop)r (er)523 3095 y(subset)35 b(of)h Fk(S)944 3065 y Fy(\()p Fl(n)p Fy(+1\))1161 3095 y Fz(for)f(all)g(in)n(tegers)f Fq(n)i Fk(\025)g Fz(2,)f(and)g(lik)n(ewise)g Fk(S)2553 3107 y Ff(V)2604 3059 y Fy(\()p Fl(n)p Fy(\))2736 3095 y Fz(is)h(a)f(prop)r(er)f(subset)523 3195 y(of)g Fk(S)674 3207 y Ff(V)725 3159 y Fy(\()p Fl(n)p Fy(+1\))906 3195 y Fz(;)h(\(ii\))f Fk(S)1158 3207 y Ff(V)1210 3159 y Fy(\()p Fl(n)p Fy(\))1341 3195 y Fz(is)g(con)n(tained)f(in)i Fk(S)1973 3165 y Fy(\()p Fl(n)p Fy(\))2104 3195 y Fz(and)f(close)f(to)h Fk(S)2641 3165 y Fy(\()p Fl(n)p Fy(\))2773 3195 y Fz(in)h(size;)e(pro)n (viding)523 3304 y(a)e(simple)g(c)n(haracterization)d(for)i Fk(S)1654 3274 y Fy(\()p Fl(n)p Fy(\))1752 3304 y Fz(;)h(\(iii\))g Fk(S)2026 3274 y Fy(\()p Fl(n)p Fy(\))2155 3304 y Fz(is)g(almost)f(con) n(tained)g(in)h Fk(S)3042 3316 y Ff(V)3093 3267 y Fy(\()p Fl(n)p Fy(+1\))3306 3304 y Fz(for)523 3413 y(all)i(in)n(tegers)f Fq(n)h Fk(\025)f Fz(2)h(\(more)g(precisely)-7 b(,)33 b(w)n(e)g(will)g(sho)n(w)g Fk(S)2378 3382 y Fy(\()p Fl(n)p Fy(\))2508 3413 y Fk(\032)f(S)2655 3425 y Ff(V)2707 3376 y Fy(\()p Fl(n)p Fy(+1\))2910 3413 y Fk([)23 b(E)3039 3382 y Fy(\()p Fl(n)p Fy(\))3136 3413 y Fz(,)34 b(where)523 3512 y Fk(E)574 3482 y Fy(\()p Fl(n)p Fy(\))709 3512 y Fz(is)k(the)g(set)g(of)g(distributions)g(w)n(ell-represen)n(ted)e(b)n (y)h(an)h(Erlang-)p Fq(n)e Fz(distribution\).)523 3612 y(This)e(result)f(yields)h(a)f(necessary)f(n)n(um)n(b)r(er)h(and)h(a)f (su\016cien)n(t)h(n)n(um)n(b)r(er)g(of)f(phases)g(for)g(a)523 3711 y(giv)n(en)h(distribution)g(to)g(b)r(e)h(w)n(ell-represen)n(ted)e (b)n(y)h(a)g(Co)n(xian)f(distribution.)i(Additional)523 3811 y(con)n(tributions)27 b(of)g(the)h(pap)r(er)f(are)g(describ)r(ed)g (b)r(elo)n(w.)648 3915 y(With)37 b(resp)r(ect)g(to)f(the)h(set)g Fk(S)1622 3885 y Fy(\(2\))1711 3915 y Fz(,)g(w)n(e)g(deriv)n(e)e(the)i (exact)g(necessary)d(and)j(su\016cien)n(t)523 4014 y(condition)i(for)f (a)h(distribution)g Fq(G)h Fz(to)e(b)r(e)i(in)f Fk(S)2088 3984 y Fy(\(2\))2217 4014 y Fz(as)f(a)h(function)h(of)f(the)g (normalized)523 4114 y(momen)n(ts)25 b(of)g Fq(G)p Fz(.)g(This)g (complemen)n(ts)g(the)g(results)f(of)h(T)-7 b(elek)25 b(and)g(Heindl,)g(who)g(analyzed)523 4214 y Fk(S)579 4184 y Fy(\(2\))664 4158 y Fg(\003)703 4214 y Fz(,)j(whic)n(h)g(is)f(a) g(subset)h(of)f Fk(S)1549 4184 y Fy(\(2\))1639 4214 y Fz(.)h(\(See)g(Section)f(2\).)648 4317 y(Lastly)-7 b(,)18 b(w)n(e)g(pro)n(vide)g(a)h(few)g(examples)f(of)h(common,)f(practical)g (distributions)h(included)523 4417 y(in)28 b(the)h(set)e Fk(S)943 4429 y Ff(V)995 4381 y Fy(\()p Fl(n)p Fy(\))1115 4417 y Fk(\032)c(S)1259 4387 y Fy(\()p Fl(n)p Fy(\))1357 4417 y Fz(.)28 b(All)g(distributions)g(w)n(e)g(consider)f(ha)n(v)n(e)f (\014nite)j(third)f(momen)n(t.)523 4517 y(The)e(P)n(areto)e (distribution)i(and)g(the)g(Bounded)g(P)n(areto)e(distribution)i(\(as)f (de\014ned)i(in)f([8]\))523 4616 y(ha)n(v)n(e)31 b(b)r(een)i(sho)n(wn)e (to)h(\014t)h(man)n(y)f(recen)n(t)f(measuremen)n(ts)g(of)i(job)f (service)f(requiremen)n(ts)523 4716 y(in)c(computing)f(systems,)g (including)g(the)h(\014le)f(size)g(requested)g(b)n(y)f(HTTP)h(requests) g([3,)13 b(4],)523 4816 y(the)30 b(CPU)f(requiremen)n(ts)f(of)h(UNIX)h (jobs)f([9,)14 b(19],)28 b(and)i(the)f(duration)g(of)g(FTP)g(transfers) 523 4924 y([24].)i(W)-7 b(e)33 b(sho)n(w)e(that)h(a)g(large)e(subset)i (of)g(Bounded)g(P)n(areto)e(distributions)i(is)f(in)i Fk(S)3242 4936 y Ff(V)3293 4888 y Fy(\(2\))3382 4924 y Fz(.)p eop %%Page: 5 5 5 4 bop 1454 365 a 8051672 8051672 0 0 11840716 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4452 y Fz(.)523 4551 y(F)-7 b(rom)29 b(Figure)h(5,)f(w)n(e)h(see)f(that)h Fk(S)1595 4563 y Ff(V)1646 4515 y Fy(\()p Fl(n)p Fy(\))1773 4551 y Fz(and)g Fk(S)1987 4563 y Ff(V)2038 4515 y Fy(\()p Fl(n)p Fy(+1\))2249 4551 y Fz(are)f(quite)h(close)f(for)h(high)f Fq(n)p Fz(.)h(Th)n(us)523 4651 y(w)n(e)24 b(ha)n(v)n(e)e(a)i(v)n(ery)e (accurate)h(represen)n(tation)f(of)i Fk(S)2088 4621 y Fy(\()p Fl(n)p Fy(\))2186 4651 y Fz(.)g(Theorem)f(3)g(follo)n(ws)g (from)h(the)g(next)523 4751 y(t)n(w)n(o)j(lemmas:)523 4912 y Fp(Lemma)j(1.)40 b Fk(S)1033 4882 y Fy(\()p Fl(n)p Fy(\))1154 4912 y Fk(\032)23 b(S)1292 4924 y Ff(V)1343 4876 y Fy(\()p Fl(n)p Fy(+1\))1542 4912 y Fk([)c(E)1667 4882 y Fy(\()p Fl(n)p Fy(\))1764 4912 y Fo(.)p eop %%Page: 8 8 8 7 bop 914 365 a 16577000 11840716 0 0 69531320 49665228 startTexFig 914 365 a %%BeginDocument: Sn2.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: Sn2.eps %%Creator: fig2dev Version 3.2 Patchlevel 3d %%CreationDate: Thu Jun 19 09:25:17 2003 %%For: osogami@gs57.sp.cs.cmu.edu 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(lines,)g(whic)n(h)f(includes)h(the)e(b)r(order,)i(and)f(dashed)g (lines,)h(whic)n(h)f(do)r(es)523 2405 y(not)f(include)f(the)g(b)r (order.)523 2690 y Fp(Lemma)30 b(2.)40 b Fk(S)1027 2702 y Ff(V)1078 2653 y Fy(\()p Fl(n)p Fy(\))1199 2690 y Fk(\032)22 b(S)1342 2660 y Fy(\()p Fl(n)p Fy(\))1440 2690 y Fo(.)523 2890 y(Pr)l(o)l(of)43 b(\(L)l(emma)35 b(1\).)h Fz(The)d(pro)r(of)g(pro) r(ceeds)f(b)n(y)h(induction.)h(When)g Fq(n)e Fz(=)g(2,)h(the)h(lemma) 523 2998 y(follo)n(ws)28 b(from)g(\(1\),)h(\(5\),)g(and)f(Theorem)g(2.) g(Next,)h(assume)f(that)h Fk(S)2669 2968 y Fy(\()p Fl(n)p Fy(\))2791 2998 y Fk(\032)c(S)2931 3010 y Ff(V)2982 2962 y Fy(\()p Fl(n)p Fy(+1\))3182 2998 y Fk([)20 b(E)3308 2968 y Fy(\()p Fl(n)p Fy(\))523 3098 y Fz(for)29 b Fq(n)c Fk(\024)h Fq(k)c Fk(\000)d Fz(1.)29 b(Consider)f(an)h(arbitrary)e (distribution)i Fq(G)d Fk(2)g(S)2563 3068 y Fy(\()p Fl(k)q Fy(\))2657 3098 y Fz(.)j(Let)g Fq(Z)6 b Fz(\()p Fk(\001)p Fz(\))26 b(=)g(\()p Fq(X)7 b Fz(\()p Fk(\001)p Fz(\))19 b Fk(\012)523 3198 y Fq(Y)g Fz(\()p Fk(\001)p Fz(\)\))p Fq(p)e Fz(+)g(1)g Fk(\000)f Fq(p)p Fz(,)27 b(where)f Fq(X)34 b Fz(is)27 b(an)f(exp)r(onen)n(tial)h(distribution)g(and)f Fq(Y)46 b Fz(is)27 b(a)f(\()p Fq(k)20 b Fk(\000)d Fz(1\)-phase)523 3297 y(Co)n(xian)k(distribution.)1247 3267 y Fy(4)1306 3297 y Fz(Observ)n(e)f(that)i(for)f(an)n(y)g(arbitrary)f(distribution)h Fq(G)j Fk(2)f(S)3082 3267 y Fy(\()p Fl(k)q Fy(\))3175 3297 y Fz(,)f(there)523 3397 y(exists)35 b(some)f(suc)n(h)h Fq(Z)41 b Fz(whic)n(h)35 b(w)n(ell-represen)n(ts)e Fq(G)p Fz(.)j(By)e(the)i(assumption)e(of)i(induction,)523 3496 y Fq(Y)42 b Fk(2)23 b(S)741 3508 y Ff(V)792 3460 y Fy(\()p Fl(k)q Fy(\))902 3496 y Fk([)18 b(E)1026 3466 y Fy(\()p Fl(k)q Ff(\000)p Fy(1\))1204 3496 y Fz(.)27 b(W)-7 b(e)27 b(pro)n(v)n(e)e(that)i(\(i\))h(if)g Fq(Y)41 b Fk(2)24 b(S)2207 3508 y Ff(V)2258 3460 y Fy(\()p Fl(k)q Fy(\))2350 3496 y Fz(,)k(then)f Fq(Z)i Fk(2)23 b(S)2803 3508 y Ff(V)2854 3460 y Fy(\()p Fl(k)q Fy(+1\))3058 3496 y Fz(and)k(\(ii\))g(if)523 3605 y Fq(Y)43 b Fk(2)24 b(E)744 3575 y Fy(\()p Fl(k)q Ff(\000)p Fy(1\))922 3605 y Fz(,)k(then)h Fq(Z)h Fk(2)24 b(S)1379 3617 y Ff(V)1430 3569 y Fy(\()p Fl(k)q Fy(+1\))1626 3605 y Fk([)19 b(E)1751 3575 y Fy(\()p Fl(k)q Fy(\))1844 3605 y Fz(.)29 b(Without)f(loss)g(of)g(generalit)n(y)-7 b(,)27 b(w)n(e)h(can)f(set)i(the)523 3705 y(\014rst)d(momen)n(t)f(of)h Fq(X)32 b Fz(to)26 b(1.)g(T)-7 b(o)25 b(see)g(wh)n(y)h(this)g(is)g(p)r (ossible,)f(observ)n(e)g(that)h Fq(Z)31 b Fz(is)26 b(comprised)523 3805 y(of)c Fq(k)j Fz(exp)r(onen)n(tial)d(phases,)f(and)h(the)g (normalized)f(second)h(and)g(third)g(momen)n(ts)g(of)g Fq(Z)6 b Fz(,)22 b Fq(m)3352 3774 y Fl(Z)3352 3825 y Fy(2)523 3904 y Fz(and)28 b Fq(m)758 3874 y Fl(Z)758 3925 y Fy(3)839 3904 y Fz(are)f(b)r(oth)i(in)n(v)-5 b(arian)n(t)26 b(to)i(m)n(ultiplying)g(all)g(the)g(rates)f(of)h(exp)r(onen)n(tial)g (phases)f(in)523 4004 y Fq(Z)j Fz(b)n(y)24 b(the)h(same)f(constan)n(t.) g(Th)n(us,)g(if)h(the)g(\014rst)f(momen)n(t)h(of)f Fq(X)31 b Fz(equals)24 b Fq(\026)2841 3974 y Fl(X)2841 4024 y Fy(1)2927 4004 y Fk(6)p Fz(=)f(1,)h(then)h(the)523 4103 y(rates)i(of)h(all)g(the)g(phases)f(in)h Fq(Z)34 b Fz(ma)n(y)27 b(b)r(e)h(m)n(ultiplied)h(b)n(y)f Fq(\026)2387 4073 y Fl(X)2387 4124 y Fy(1)2478 4103 y Fz(to)g(bring)f(the)h(\014rst)g (momen)n(t)523 4203 y(of)g Fq(X)34 b Fz(do)n(wn)27 b(to)g(1.)p 523 4282 473 4 v 546 4335 a Fm(4)606 4367 y Fx(T)-6 b(o)29 b(shed)f(ligh)n(t)h(on)g(this)g(expression,)g(consider)g(random)f(v)l (ariables)h Ft(V)2690 4375 y Fn(X)2777 4367 y Fx(and)f Ft(V)2974 4375 y Fn(Y)3056 4367 y Fx(whose)h(dis-)606 4459 y(tributions)d(are)g Ft(X)32 b Fx(and)25 b Ft(Y)17 b Fx(,)26 b(resp)r(ectiv)n(ely)-6 b(.)25 b(Then)h(random)f(v)l(ariable) 1279 4689 y Ft(V)1324 4697 y Fn(Z)1395 4689 y Fx(=)1476 4574 y Fj(\032)1550 4643 y Ft(V)1595 4651 y Fn(X)1669 4643 y Fx(+)17 b Ft(V)1791 4651 y Fn(Y)1867 4643 y Fx(with)26 b(probabilit)n(y)g Ft(p)1550 4735 y Fx(0)279 b(with)26 b(probabilit)n(y)g(1)17 b Fw(\000)g Ft(p;)606 4924 y Fx(has)26 b(distribution)g Ft(Z)5 b Fx(,)26 b(since)g(Pr)q(\()p Ft(V)1612 4932 y Fn(Z)1682 4924 y Ft(<)21 b(t)p Fx(\))g(=)g Ft(p)12 b Fx(Pr)q(\()p Ft(V)2132 4932 y Fn(X)2206 4924 y Fx(+)17 b Ft(V)2328 4932 y Fn(Y)2402 4924 y Ft(<)22 b(t)p Fx(\))16 b(+)h(\(1)g Fw(\000)g Ft(p)p Fx(\).)p eop %%Page: 9 9 9 8 bop 523 448 a Fz(\(i\))28 b(Supp)r(ose)g Fq(Y)42 b Fk(2)23 b(S)1182 460 y Ff(V)1233 412 y Fy(\()p Fl(k)q Fy(\))p 638 478 688 4 v 1326 448 a Fz(:)28 b(W)-7 b(e)28 b(\014rst)f(pro)n(v)n(e)f(that)i Fq(m)2168 418 y Fl(Z)2168 469 y Fy(2)2244 448 y Fq(>)2342 416 y Fl(k)q Fy(+1)p 2342 430 121 4 v 2384 477 a Fl(k)2473 448 y Fz(.)f(Observ)n(e)f(that) 1243 667 y Ft(m)1311 631 y Fn(Z)1311 680 y Fm(2)1381 667 y Fx(=)1472 618 y(2)18 b(+)e(2)p Ft(\026)1688 587 y Fn(Y)1688 632 y Fm(1)1759 618 y Fx(+)h Ft(\026)1882 587 y Fn(Y)1882 632 y Fm(2)p 1472 650 464 4 v 1522 719 a Ft(p)p Fx(\(1)f(+)h Ft(\026)1768 694 y Fn(Y)1768 739 y Fm(1)1822 719 y Fx(\))1852 698 y Fm(2)1967 667 y Ft(>)2058 604 y Fx(2)g(+)g(2)p Ft(\026)2274 572 y Fn(Y)2274 618 y Fm(1)2345 604 y Fx(+)2471 573 y Fn(k)p 2432 587 113 4 v 2432 629 a(k)q Fe(\000)p Fm(1)2554 604 y Ft(\026)2600 572 y Fn(Y)2600 618 y Fm(2)p 2058 650 596 4 v 2174 719 a Ft(p)p Fx(\(1)f(+)h Ft(\026)2420 694 y Fn(Y)2420 739 y Fm(1)2474 719 y Fx(\))2504 698 y Fm(2)2664 667 y Ft(;)551 876 y Fz(where)27 b(the)h(inequalit)n(y)f(follo)n(ws)g(from)g Fq(Y)42 b Fk(2)24 b(S)2007 888 y Ff(V)2058 840 y Fy(\()p Fl(k)q Fy(\))2151 876 y Fz(.)k(The)f(righ)n(t)g(hand)h(side)g(is)f (minimized)523 996 y(when)i Fq(\026)791 966 y Fl(Y)791 1017 y Fy(1)875 996 y Fz(=)c Fq(k)d Fk(\000)e Fz(1.)28 b(Th)n(us,)h Fq(m)1516 966 y Fl(Z)1516 1017 y Fy(2)1596 996 y Fq(>)1696 963 y Fl(k)q Fy(+1)p 1696 977 121 4 v 1721 1025 a Fl(pk)1852 996 y Fk(\025)1953 963 y Fl(k)q Fy(+1)p 1953 977 V 1995 1025 a Fl(k)2083 996 y Fz(.)h(Next,)f(w)n(e)g (pro)n(v)n(e)f(that)2904 954 y Fl(m)2963 929 y Fi(Z)2963 970 y Fh(3)p 2904 977 106 4 v 2904 1029 a Fl(m)2963 1009 y Fi(Z)2963 1049 y Fh(2)3045 996 y Fk(\025)3146 963 y Fl(k)q Fy(+3)p 3146 977 121 4 v 3146 1025 a Fl(k)q Fy(+2)3306 996 y Fz(for)523 1153 y(all)f Fq(m)711 1123 y Fl(Z)711 1173 y Fy(2)788 1153 y Fq(>)885 1120 y Fl(k)q Fy(+1)p 885 1134 V 927 1181 a Fl(k)1016 1153 y Fz(.)h(Notice)f(that)1517 1111 y Fl(m)1576 1085 y Fi(Z)1576 1127 y Fh(3)p 1517 1134 106 4 v 1517 1185 a Fl(m)1576 1165 y Fi(Z)1576 1205 y Fh(2)1660 1153 y Fz(is)g(indep)r(enden)n(t)i(of)e Fq(p)p Fz(:)1310 1329 y Ft(m)1378 1297 y Fn(Z)1378 1343 y Fm(3)p 1310 1360 117 4 v 1310 1430 a Ft(m)1378 1404 y Fn(Z)1378 1450 y Fm(2)1459 1377 y Fx(=)1550 1328 y(\(6)17 b(+)g(6)p Ft(\026)1796 1297 y Fn(Y)1796 1342 y Fm(1)1867 1328 y Fx(+)g(3)p Ft(\026)2028 1297 y Fn(Y)2028 1342 y Fm(2)2099 1328 y Fx(+)f Ft(\026)2221 1297 y Fn(Y)2221 1342 y Fm(3)2275 1328 y Fx(\)\(1)h(+)g Ft(\026)2513 1297 y Fn(Y)2513 1342 y Fm(1)2566 1328 y Fx(\))p 1550 1360 1047 4 v 1794 1430 a(\(2)g(+)g(2)p Ft(\026)2040 1404 y Fn(Y)2040 1450 y Fm(1)2111 1430 y Fx(+)g Ft(\026)2234 1404 y Fn(Y)2234 1450 y Fm(2)2287 1430 y Fx(\))2317 1409 y Fm(2)2606 1377 y Ft(:)555 1603 y Fz(Since)787 1560 y Fl(m)846 1535 y Fi(Z)846 1577 y Fh(3)p 787 1584 106 4 v 787 1635 a Fl(m)846 1615 y Fi(Z)846 1655 y Fh(2)934 1603 y Fz(is)32 b(an)g(increasing)e (function)j(of)f Fq(\026)2013 1573 y Fl(Y)2013 1623 y Fy(3)2071 1603 y Fz(,)g(it)g(is)h(minimized)f(at)g Fq(\026)2863 1573 y Fl(Y)2863 1623 y Fy(3)2951 1603 y Fz(=)3057 1570 y Fl(k)q Fy(+2)p 3057 1584 121 4 v 3057 1631 a Fl(k)q Fy(+1)3197 1560 y(\()p Fl(\026)3263 1535 y Fi(Y)3263 1577 y Fh(2)3314 1560 y Fy(\))3340 1535 y Fh(2)p 3197 1584 175 4 v 3239 1635 a Fl(\026)3279 1615 y Fi(Y)3279 1655 y Fh(1)3382 1603 y Fz(,)523 1743 y(since)27 b Fq(Y)42 b Fk(2)24 b(S)945 1755 y Ff(V)996 1707 y Fy(\()p Fl(k)q Fy(\))1088 1743 y Fz(.)k(Th)n(us,)648 1883 y Ft(m)716 1852 y Fn(Z)716 1897 y Fm(3)p 648 1915 117 4 v 648 1984 a Ft(m)716 1959 y Fn(Z)716 2004 y Fm(2)797 1932 y Fw(\025)888 1883 y Fx(\(1)17 b(+)g Ft(\026)1096 1851 y Fn(Y)1096 1896 y Fm(1)1149 1883 y Fx(\)\(6\()p Ft(k)i Fx(+)e(1\))p Ft(\026)1527 1851 y Fn(Y)1527 1896 y Fm(1)1598 1883 y Fx(+)g(6\()p Ft(k)i Fx(+)e(1\)\()p Ft(\026)2023 1851 y Fn(Y)2023 1896 y Fm(1)2077 1883 y Fx(\))2107 1851 y Fm(2)2158 1883 y Fx(+)g(3\()p Ft(k)j Fx(+)c(1\))p Ft(\026)2553 1851 y Fn(Y)2553 1896 y Fm(1)2607 1883 y Ft(\026)2653 1851 y Fn(Y)2653 1896 y Fm(2)2724 1883 y Fx(+)h(\()p Ft(k)i Fx(+)d(2\)\()p Ft(\026)3110 1851 y Fn(Y)3110 1896 y Fm(2)3164 1883 y Fx(\))3194 1851 y Fm(2)3229 1883 y Fx(\))p 888 1915 2371 4 v 1627 1984 a(\()p Ft(k)j Fx(+)e(1\))p Ft(\026)1907 1959 y Fn(Y)1907 2004 y Fm(1)1961 1984 y Fx(\(2)g(+)g(2)p Ft(\026)2207 1959 y Fn(Y)2207 2004 y Fm(1)2278 1984 y Fx(+)g Ft(\026)2401 1959 y Fn(Y)2401 2004 y Fm(2)2454 1984 y Fx(\))2484 1963 y Fm(2)3268 1932 y Ft(:)551 2119 y Fz(The)27 b(in\014m)n(um)i(of)e(the)h(righ)n(t)f (hand)g(side)h(o)r(ccurs)f(at:)972 2325 y Fq(\026)1022 2291 y Fl(Y)1022 2346 y Fy(2)1102 2325 y Fz(=)c(max)1358 2208 y Fj(\032)1553 2269 y Fz(6\()p Fq(k)e Fz(+)d(1\))p Fq(\026)1898 2239 y Fl(Y)1898 2290 y Fy(1)1955 2269 y Fz(\(1)g(+)g Fq(\026)2180 2239 y Fl(Y)2180 2290 y Fy(1)2238 2269 y Fz(\))p 1430 2306 962 4 v 1430 2384 a(4)g(+)g(4)p Fq(\026)1665 2356 y Fl(Y)1665 2406 y Fy(1)1741 2384 y Fz(+)g(\()p Fq(k)j Fz(+)d(1\)\(4)g(+)g Fq(\026)2302 2356 y Fl(Y)2302 2406 y Fy(1)2360 2384 y Fz(\))2402 2325 y Fq(;)2521 2269 y(k)p 2449 2306 189 4 v 2449 2382 a(k)j Fk(\000)d Fz(1)2648 2325 y(\()p Fq(\026)2730 2291 y Fl(Y)2730 2346 y Fy(1)2788 2325 y Fz(\))2820 2291 y Fy(2)2857 2208 y Fj(\033)2933 2325 y Fq(:)523 2566 y Fz(By)27 b(ev)-5 b(aluating)1062 2524 y Fl(m)1121 2499 y Fi(Z)1121 2541 y Fh(3)p 1062 2547 106 4 v 1062 2599 a Fl(m)1121 2579 y Fi(Z)1121 2619 y Fh(2)1205 2566 y Fz(at)27 b Fq(\026)1356 2536 y Fl(Y)1356 2587 y Fy(2)1437 2566 y Fz(=)1577 2534 y Fl(k)p 1535 2548 122 4 v 1535 2595 a(k)q Ff(\000)p Fy(1)1666 2566 y Fz(\()p Fq(\026)1748 2536 y Fl(Y)1748 2587 y Fy(1)1806 2566 y Fz(\))1838 2536 y Fy(2)1876 2566 y Fz(,)g(w)n(e)g(ha)n(v)n(e)596 2758 y Ft(m)664 2726 y Fn(Z)664 2772 y Fm(3)p 596 2789 117 4 v 596 2859 a Ft(m)664 2833 y Fn(Z)664 2879 y Fm(2)744 2806 y Fw(\025)835 2745 y Fx(\(1)17 b(+)g Ft(\026)1043 2713 y Fn(Y)1043 2758 y Fm(1)1097 2745 y Fx(\))1139 2680 y Fj(\002)1174 2745 y Fx(6\()p Ft(k)i Fx(+)e(1\)\()p Ft(k)i Fw(\000)e Fx(1\))1680 2713 y Fm(2)1715 2745 y Fx(\(1)g(+)g Ft(\026)1923 2713 y Fn(Y)1923 2758 y Fm(1)1977 2745 y Fx(\))g(+)f(3)p Ft(k)r Fx(\()p Ft(k)2252 2713 y Fm(2)2305 2745 y Fw(\000)h Fx(1\)\()p Ft(\026)2526 2713 y Fn(Y)2526 2758 y Fm(1)2579 2745 y Fx(\))2609 2713 y Fm(2)2661 2745 y Fx(+)g Ft(k)2780 2713 y Fm(2)2814 2745 y Fx(\()p Ft(k)i Fx(+)e(2\)\()p Ft(\026)3124 2713 y Fn(Y)3124 2758 y Fm(1)3178 2745 y Fx(\))3208 2713 y Fm(3)3242 2680 y Fj(\003)p 835 2789 2442 4 v 1360 2871 a Fx(\()p Ft(k)i Fx(+)e(1\))c([2\()p Ft(k)19 b Fw(\000)e Fx(1\))h(+)e(2\()p Ft(k)k Fw(\000)d Fx(1\))p Ft(\026)2313 2845 y Fn(Y)2313 2890 y Fm(1)2383 2871 y Fx(+)g Ft(k)r Fx(\()p Ft(\026)2578 2845 y Fn(Y)2578 2890 y Fm(1)2632 2871 y Fx(\))2662 2849 y Fm(2)2696 2871 y Fx(])2718 2831 y Fm(2)3304 2806 y Fw(\001)551 3043 y Fz(By)27 b(Lemma)g(4)h(in)f(App)r(endix)i(A,)1647 3001 y Fl(m)1706 2976 y Fi(Z)1706 3017 y Fh(3)p 1647 3024 106 4 v 1647 3076 a Fl(m)1706 3056 y Fi(Z)1706 3096 y Fh(2)1785 3043 y Fk(\025)1883 3010 y Fl(k)q Fy(+3)p 1883 3024 121 4 v 1883 3072 a Fl(k)q Fy(+2)2013 3043 y Fz(.)f(By)f(ev)-5 b(aluating)2603 3001 y Fl(m)2662 2976 y Fi(Z)2662 3017 y Fh(3)p 2603 3024 106 4 v 2603 3076 a Fl(m)2662 3056 y Fi(Z)2662 3096 y Fh(2)2746 3043 y Fz(at)1398 3268 y Ft(\026)1444 3232 y Fn(Y)1444 3281 y Fm(2)1518 3268 y Fx(=)1722 3219 y(6\()p Ft(k)20 b Fx(+)d(1\))p Ft(\026)2041 3187 y Fn(Y)2041 3232 y Fm(1)2095 3219 y Fx(\(1)g(+)g Ft(\026)2303 3187 y Fn(Y)2303 3232 y Fm(1)2356 3219 y Fx(\))p 1609 3251 890 4 v 1609 3320 a(4)h(+)f(4)p Ft(\026)1826 3295 y Fn(Y)1826 3340 y Fm(1)1897 3320 y Fx(+)f(\()p Ft(k)k Fx(+)c(1\)\(4)i(+)f Ft(\026)2416 3295 y Fn(Y)2416 3340 y Fm(1)2469 3320 y Fx(\))2509 3268 y Ft(;)551 3455 y Fz(w)n(e)27 b(ha)n(v)n(e)1131 3545 y Ft(m)1199 3513 y Fn(Z)1199 3558 y Fm(3)p 1131 3576 117 4 v 1131 3646 a Ft(m)1199 3620 y Fn(Z)1199 3666 y Fm(2)1279 3593 y Fw(\025)1370 3532 y Fx(3)1422 3467 y Fj(\002)1456 3532 y Fx(8\(1)18 b(+)f Ft(\026)1703 3500 y Fn(Y)1703 3545 y Fm(1)1756 3532 y Fx(\))g(+)g(\()p Ft(k)i Fx(+)e(1\)\(8)h(+)e(5)p Ft(\026)2360 3500 y Fn(Y)2360 3545 y Fm(1)2414 3532 y Fx(\))2444 3467 y Fj(\003)p 1370 3576 1109 4 v 1623 3646 a Fx(16\(2)i(+)f Ft(k)r Fx(\)\(1)h(+)e Ft(\026)2142 3620 y Fn(Y)2142 3666 y Fm(1)2196 3646 y Fx(\))2510 3593 y Fw(\025)2601 3545 y Ft(k)j Fx(+)e(3)p 2601 3576 175 4 v 2601 3644 a Ft(k)i Fx(+)e(2)2785 3593 y Ft(;)551 3769 y Fz(where)27 b(the)h(last)f(inequalit)n(y)g(holds)g(i\013)i Fq(\026)1843 3739 y Fl(Y)1843 3790 y Fy(1)1923 3769 y Fk(\024)2046 3737 y Fy(8)p Fl(k)p 2021 3751 121 4 v 2021 3798 a(k)q Fy(+9)2151 3769 y Fz(.)f(Ho)n(w)n(ev)n(er,)e Fq(\026)2610 3739 y Fl(Y)2610 3790 y Fy(1)2690 3769 y Fk(\024)2814 3737 y Fy(8)p Fl(k)p 2788 3751 V 2788 3798 a(k)q Fy(+9)2946 3769 y Fz(holds)i(if)1329 3939 y(6\()p Fq(k)21 b Fz(+)d(1\))p Fq(\026)1674 3909 y Fl(Y)1674 3959 y Fy(1)1732 3939 y Fz(\(1)g(+)g Fq(\026)1957 3909 y Fl(Y)1957 3959 y Fy(1)2015 3939 y Fz(\))p 1207 3976 962 4 v 1207 4054 a(4)g(+)g(4)p Fq(\026)1442 4026 y Fl(Y)1442 4076 y Fy(1)1518 4054 y Fz(+)g(\()p Fq(k)j Fz(+)d(1\)\(4)g(+)g Fq(\026)2079 4026 y Fl(Y)2079 4076 y Fy(1)2137 4054 y Fz(\))2202 3995 y Fq(>)2371 3939 y(k)p 2300 3976 189 4 v 2300 4052 a(k)j Fk(\000)d Fz(1)2498 3995 y(\()p Fq(\026)2580 3961 y Fl(Y)2580 4016 y Fy(1)2638 3995 y Fz(\))2670 3961 y Fy(2)2708 3995 y Fq(:)648 4250 y Fz(\(ii\))31 b(Supp)r(ose)d Fq(Y)41 b Fk(2)24 b(E)1334 4220 y Fy(\()p Fl(k)q Ff(\000)p Fy(1\))p 789 4280 723 4 v 1512 4250 a Fz(:)30 b(W)-7 b(e)31 b(will)g(pro)n(v)n(e)e(that)i(\(a\))g(if)g Fq(\026)2547 4220 y Fl(Y)2547 4271 y Fy(1)2633 4250 y Fz(=)c(\()p Fq(k)d Fk(\000)c Fz(1\))30 b(and)h Fq(p)d Fz(=)f(1,)523 4371 y(then)33 b Fq(Z)k Fk(2)32 b(E)949 4341 y Fy(\()p Fl(k)q Fy(\))1042 4371 y Fz(,)h(and)f(\(b\))i(if)f Fq(\026)1539 4341 y Fl(Y)1539 4392 y Fy(1)1628 4371 y Fk(6)p Fz(=)e(\()p Fq(k)25 b Fk(\000)c Fz(1\))33 b(or)e Fq(p)g(<)h Fz(1,)g(then)h Fq(Z)k Fk(2)32 b(S)2815 4383 y Ff(V)2866 4335 y Fy(\()p Fl(k)q Fy(+1\))3043 4371 y Fz(.)g(F)-7 b(or)32 b(part)523 4471 y(\(a\),)d(observ)n(e)e(that)i(if)h Fq(Y)44 b Fk(2)25 b(E)1460 4441 y Fy(\()p Fl(k)q Ff(\000)p Fy(1\))1638 4471 y Fz(,)k Fq(\026)1740 4441 y Fl(Y)1740 4491 y Fy(1)1823 4471 y Fz(=)c(\()p Fq(k)d Fk(\000)d Fz(1\),)29 b(and)g Fq(p)c Fz(=)f(1,)29 b(then)g(w)n(e)g(ha)n(v)n(e)e(already)523 4570 y(seen)37 b(that)h Fq(m)976 4540 y Fl(Z)976 4591 y Fy(2)1068 4570 y Fz(=)1182 4538 y Fl(k)q Fy(+1)p 1182 4552 121 4 v 1224 4599 a Fl(k)1350 4570 y Fz(in)g(part)f(\(i\).)h(It)f (is)g(also)g(easy)f(to)h(see)g(that)h Fq(m)2874 4540 y Fl(Z)2874 4591 y Fy(3)2966 4570 y Fz(=)3080 4538 y Fl(k)q Fy(+2)p 3080 4552 V 3122 4599 a Fl(k)3211 4570 y Fz(,)f(and)523 4681 y(hence)c Fq(Z)k Fk(2)c(E)992 4651 y Fy(\()p Fl(k)q Fy(\))1085 4681 y Fz(.)g(F)-7 b(or)32 b(part)g(\(b\),)i(if)f Fq(\026)1778 4651 y Fl(Y)1778 4702 y Fy(1)1868 4681 y Fk(6)p Fz(=)e(\()p Fq(k)25 b Fk(\000)d Fz(1\))32 b(or)g Fq(p)g(<)f Fz(1,)i(then)g(\014rst)g(notice)g (that)523 4781 y Fq(m)596 4750 y Fl(Z)596 4801 y Fy(2)680 4781 y Fq(>)786 4748 y Fl(k)q Fy(+1)p 786 4762 V 828 4809 a Fl(k)917 4781 y Fz(,)f(since)h Fq(m)1254 4750 y Fl(Z)1254 4801 y Fy(2)1340 4781 y Fz(is)f(minimized)h(when)g Fq(\026)2106 4750 y Fl(Y)2106 4801 y Fy(1)2194 4781 y Fz(=)e(\()p Fq(k)25 b Fk(\000)c Fz(1\))32 b(and)g Fq(p)f Fz(=)g(1.)h(Also,)g(since)523 4913 y Fq(m)596 4883 y Fl(Y)596 4934 y Fy(3)676 4913 y Fz(=)775 4881 y Fl(k)q Fy(+1)p 774 4895 122 4 v 774 4942 a Fl(k)q Ff(\000)p Fy(1)929 4913 y Fq(>)1027 4881 y Fl(k)q Fy(+2)p 1026 4895 V 1026 4942 a Fl(k)q Ff(\000)p Fy(1)1158 4913 y Fz(,)1219 4871 y Fl(m)1278 4846 y Fi(Z)1278 4888 y Fh(3)p 1219 4894 106 4 v 1219 4946 a Fl(m)1278 4926 y Fi(Z)1278 4966 y Fh(2)1357 4913 y Fk(\025)1454 4881 y Fl(k)q Fy(+3)p 1454 4895 121 4 v 1454 4942 a Fl(k)q Fy(+2)1613 4913 y Fz(b)n(y)27 b(part)g(\(i\),)i(and)e(hence)h Fq(Z)g Fk(2)c(S)2653 4925 y Ff(V)2704 4877 y Fy(\()p Fl(k)q Fy(+1\))2881 4913 y Fz(.)446 b Fk(u)-55 b(t)p eop %%Page: 10 10 10 9 bop 523 448 a Fo(Pr)l(o)l(of)43 b(\(L)l(emma)36 b(2\).)h Fz(When)f Fq(n)f Fz(=)g(2,)f(the)h(lemma)g(follo)n(ws)f(from)g (Theorem)g(2.)h(The)g(re-)523 548 y(mainder)24 b(of)h(the)g(pro)r(of)f (assumes)f Fq(n)g Fk(\025)g Fz(3.)h(W)-7 b(e)25 b(pro)n(v)n(e)e(that)i (for)f(an)g(arbitrary)f(distribution)523 648 y Fq(G)42 b Fk(2)f(S)776 660 y Ff(V)827 611 y Fy(\()p Fl(n)p Fy(\))924 648 y Fz(,)e(there)f(exists)g(an)g Fq(n)p Fz(-phase)g(Co)n(xian)f Fq(Z)44 b Fz(suc)n(h)38 b(that)h(the)g(normalized)f(mo-)523 747 y(men)n(ts)33 b(of)g Fq(G)g Fz(and)g Fq(Z)39 b Fz(agree.)31 b(Notice)i(that)h(the)f(\014rst)g(momen)n(t)g(of)g Fq(Z)38 b Fz(is)33 b(easily)f(matc)n(hed)523 847 y(to)k Fq(G)g Fz(b)n(y)f(normalization)f(without)i(c)n(hanging)f(the)h(normalizing)e (momen)n(ts)i(of)f Fq(Z)6 b Fz(.)36 b(The)523 946 y(pro)r(of)26 b(consists)f(of)h(t)n(w)n(o)f(parts:)g(\(i\))i(the)g(case)e(when)h(the) h(normalized)e(momen)n(ts)g(of)h Fq(G)h Fz(sat-)523 1046 y(isfy)35 b Fq(m)756 1016 y Fl(G)756 1067 y Fy(3)847 1046 y Fq(>)f Fz(2)p Fq(m)1061 1016 y Fl(G)1061 1067 y Fy(2)1140 1046 y Fk(\000)22 b Fz(1;)35 b(\(ii\))g(the)g(case)f(when)h (the)g(normalized)f(momen)n(ts)h(of)f Fq(G)h Fz(satisfy)523 1146 y Fq(m)596 1116 y Fl(G)596 1166 y Fy(3)675 1146 y Fk(\024)22 b Fz(2)p Fq(m)877 1116 y Fl(G)877 1166 y Fy(2)951 1146 y Fk(\000)c Fz(1.)648 1254 y(\(i\))27 b(Supp)r(ose)g Fq(G)d Fk(2)f(S)1304 1266 y Ff(V)1355 1218 y Fy(\()p Fl(n)p Fy(\))1480 1254 y Fz(and)28 b Fq(m)1715 1224 y Fl(G)1715 1275 y Fy(3)1793 1254 y Fq(>)23 b Fz(2)p Fq(m)1996 1224 y Fl(G)1996 1275 y Fy(2)2070 1254 y Fk(\000)18 b Fz(1)p 762 1288 1433 4 v -1 w(:)27 b(W)-7 b(e)27 b(need)f(to)h(sho)n(w) e(that)i Fq(G)g Fz(is)g(w)n(ell-)523 1358 y(represen)n(ted)17 b(b)n(y)h(some)g Fq(n)p Fz(-phase)f(Co)n(xian)h(distribution.)g(W)-7 b(e)19 b(will)f(pro)n(v)n(e)f(something)h(stronger)523 1449 y(that)33 b Fq(G)g Fz(is)g(w)n(ell-represen)n(ted)e(b)n(y)h(a)h (distribution)g Fq(Z)38 b Fz(where)32 b Fq(Z)38 b Fz(=)31 b Fq(X)d Fz(+)21 b Fq(Y)e Fz(,)33 b(and)g Fq(X)39 b Fz(is)32 b(a)523 1540 y(particular)e(t)n(w)n(o-phase)f(Co)n(xian)g(distribution) j(with)f(no)g(mass)f(probabilit)n(y)g(at)g(zero)g(and)523 1632 y Fq(Y)45 b Fz(is)25 b(a)h(particular)f(Erlang-\()p Fq(n)13 b Fk(\000)h Fz(2\))26 b(distribution.)g(\(F)-7 b(or)26 b(the)g(in)n(tuition)g(b)r(ehind)h(this)f(par-)523 1723 y(ticular)i(w)n(a)n(y)e(of)i(represen)n(ting)f Fq(G)p Fz(,)h(please)g(refer)f(to)h([22]\).)f(The)h(normalized)f(momen)n(ts)h (of)523 1814 y Fq(X)34 b Fz(are)27 b(c)n(hosen)f(as)h(follo)n(ws:)667 2005 y Ft(m)735 1970 y Fn(X)735 2019 y Fm(2)815 2005 y Fx(=)908 1956 y Ft(m)976 1925 y Fn(G)976 1970 y Fm(2)1028 1956 y Fx(\()p Ft(n)17 b Fw(\000)g Fx(3\))g Fw(\000)g Fx(\()p Ft(n)g Fw(\000)g Fx(2\))p 908 1988 690 4 v 908 2058 a Ft(m)976 2032 y Fn(G)976 2078 y Fm(2)1028 2058 y Fx(\()p Ft(n)g Fw(\000)g Fx(2\))g Fw(\000)g Fx(\()p Ft(n)g Fw(\000)g Fx(1\))1608 2005 y(;)667 2225 y Ft(m)735 2189 y Fn(X)735 2238 y Fm(3)815 2225 y Fx(=)898 2160 y Fj(\000)936 2225 y Fx(\()p Ft(n)h Fw(\000)e Fx(1\))p Ft(m)1242 2189 y Fn(X)1242 2238 y Fm(2)1317 2225 y Fw(\000)h Fx(\()p Ft(n)g Fw(\000)g Fx(2\))1632 2160 y Fj(\001)c(\000)1721 2225 y Fx(\()p Ft(n)18 b Fw(\000)e Fx(2\))p Ft(m)2027 2189 y Fn(X)2027 2238 y Fm(2)2102 2225 y Fw(\000)h Fx(\()p Ft(n)g Fw(\000)g Fx(3\))2417 2160 y Fj(\001)2455 2176 y Fm(2)2516 2177 y Ft(m)2584 2145 y Fn(G)2584 2190 y Fm(3)p 2513 2208 126 4 v 2513 2278 a Ft(m)2581 2252 y Fn(X)2581 2298 y Fm(2)898 2461 y Fw(\000)968 2399 y Fx(\()p Ft(n)g Fw(\000)g Fx(2\)\()p Ft(m)1304 2367 y Fn(X)1304 2413 y Fm(2)1379 2399 y Fw(\000)g Fx(1\))1537 2334 y Fj(\000)1575 2399 y Ft(n)p Fx(\()p Ft(n)g Fw(\000)g Fx(1\)\()p Ft(m)1957 2367 y Fn(X)1957 2413 y Fm(2)2015 2399 y Fx(\))2045 2367 y Fm(2)2096 2399 y Fw(\000)g Ft(n)p Fx(\(2)p Ft(n)h Fw(\000)f Fx(5\))p Ft(m)2564 2367 y Fn(X)2564 2413 y Fm(2)2639 2399 y Fx(+)f(\()p Ft(n)i Fw(\000)f Fx(1\)\()p Ft(n)g Fw(\000)g Fx(3\))3192 2334 y Fj(\001)p 968 2444 2263 4 v 2036 2513 a Ft(m)2104 2488 y Fn(X)2104 2533 y Fm(2)3240 2461 y Ft(:)548 2679 y Fz(The)25 b(\014rst)g(momen)n(t)g (of)g Fq(Y)43 b Fz(is)25 b(c)n(hosen)f(as)h(follo)n(ws:)f Fq(\026)2173 2649 y Fl(Y)2173 2700 y Fy(1)2253 2679 y Fz(=)f(\()p Fq(n)13 b Fk(\000)g Fz(2\)\()p Fq(m)2693 2649 y Fl(X)2693 2700 y Fy(2)2769 2679 y Fk(\000)g Fz(1\))p Fq(\026)2971 2649 y Fl(X)2971 2700 y Fy(1)3034 2679 y Fz(.)25 b(It)g(is)g(easy)523 2771 y(to)j(see)f(that)h(the)g(normalized) e(momen)n(ts)h(of)h Fq(G)g Fz(and)f Fq(Z)34 b Fz(agree:)1077 2977 y Ft(m)1145 2942 y Fn(Z)1145 2991 y Fm(2)1217 2977 y Fx(=)1310 2929 y Ft(m)1378 2897 y Fn(X)1378 2943 y Fm(2)1452 2929 y Fx(+)17 b(2)p Ft(y)j Fx(+)d Ft(m)1770 2897 y Fn(Y)1770 2943 y Fm(2)1823 2929 y Ft(y)1864 2897 y Fm(2)p 1310 2960 589 4 v 1470 3028 a Fx(\(1)h(+)f Ft(y)s Fx(\))1704 3007 y Fm(2)1929 2977 y Fx(=)k Ft(m)2078 2942 y Fn(G)2078 2991 y Fm(2)2130 2977 y Fx(;)1077 3194 y Ft(m)1145 3158 y Fn(Z)1145 3207 y Fm(3)1217 3194 y Fx(=)1310 3146 y Ft(m)1378 3114 y Fn(X)1378 3159 y Fm(2)1435 3146 y Ft(m)1503 3114 y Fn(X)1503 3159 y Fm(3)1578 3146 y Fx(+)c(3)p Ft(m)1761 3114 y Fn(X)1761 3159 y Fm(2)1819 3146 y Ft(y)i Fx(+)e(3)p Ft(m)2059 3114 y Fn(Y)2059 3159 y Fm(2)2112 3146 y Ft(y)2153 3114 y Fm(2)2205 3146 y Fx(+)f Ft(m)2349 3114 y Fn(Y)2349 3159 y Fm(2)2402 3146 y Ft(m)2470 3114 y Fn(Y)2470 3159 y Fm(3)2523 3146 y Ft(y)2564 3114 y Fm(3)p 1310 3177 1289 4 v 1514 3247 a Fx(\()p Ft(m)1612 3221 y Fn(X)1612 3267 y Fm(2)1686 3247 y Fx(+)h(2)p Ft(y)j Fx(+)d Ft(m)2004 3221 y Fn(Y)2004 3267 y Fm(2)2057 3247 y Ft(y)2098 3225 y Fm(2)2132 3247 y Fx(\)\(1)g(+)g Ft(y)s Fx(\))2630 3194 y(=)k Ft(m)2779 3158 y Fn(G)2779 3207 y Fm(3)2830 3194 y Fx(;)542 3439 y Fz(where)e Fq(m)847 3409 y Fl(Y)847 3459 y Fy(2)927 3439 y Fz(=)1025 3406 y Fl(n)p Ff(\000)p Fy(1)p 1025 3420 127 4 v 1025 3467 a Fl(n)p Ff(\000)p Fy(2)1180 3439 y Fz(and)g Fq(m)1406 3409 y Fl(Y)1406 3459 y Fy(3)1487 3439 y Fz(=)1627 3406 y Fl(n)p 1584 3420 V 1584 3467 a(n)p Ff(\000)p Fy(2)1740 3439 y Fz(are)f(the)h(normalized)g(momen)n (ts)f(of)i Fq(Y)e Fz(,)i(and)f Fq(y)25 b Fz(=)3280 3396 y Fl(\026)3320 3371 y Fi(Y)3320 3413 y Fh(1)p 3278 3420 94 4 v 3278 3471 a Fl(\026)3318 3451 y Fi(X)3318 3491 y Fh(1)3382 3439 y Fz(.)523 3557 y(Finally)-7 b(,)28 b(w)n(e)g(will)h(sho)n(w)e(that)i(there)f(exists)g(a)g(t)n(w)n(o-phase) f(Co)n(xian)g(distribution)h(with)h(no)523 3657 y(mass)g(probabilit)n (y)f(at)i(zero,)e(with)i(normalized)f(momen)n(ts)g Fq(m)2505 3627 y Fl(X)2505 3678 y Fy(2)2598 3657 y Fz(and)g Fq(m)2834 3627 y Fl(X)2834 3678 y Fy(3)2897 3657 y Fz(.)h(By)f(Theorem)523 3757 y(1,)22 b(it)g(su\016ces)g(to)g(sho)n(w)f(that)h Fq(m)1507 3727 y Fl(X)1507 3777 y Fy(2)1593 3757 y Fq(>)h Fz(2)e(and)h Fq(m)1973 3727 y Fl(X)1973 3777 y Fy(3)2059 3757 y Fq(>)2157 3724 y Fy(3)p 2157 3738 34 4 v 2157 3785 a(2)2200 3757 y Fq(m)2273 3727 y Fl(X)2273 3777 y Fy(2)2336 3757 y Fz(.)g(The)g(\014rst)g(condition,)g Fq(m)3167 3727 y Fl(X)3167 3777 y Fy(2)3253 3757 y Fq(>)g Fz(2,)523 3873 y(can)32 b(b)r(e)h(sho)n(wn)f(using)1330 3841 y Fl(n)p 1288 3855 127 4 v 1288 3902 a(n)p Ff(\000)p Fy(1)1455 3873 y Fq(<)f(m)1624 3843 y Fl(G)1624 3894 y Fy(2)1680 3873 y Fz(,)i(whic)n(h)g(follo)n(ws)e(from)i Fq(G)f Fk(2)g(S)2692 3885 y Ff(V)2743 3837 y Fy(\()p Fl(n)p Fy(\))2840 3873 y Fz(.)h(It)g(can)f(also)g(b)r(e)523 3987 y(sho)n(wn)24 b(that)g Fq(m)1021 3957 y Fl(X)1021 4008 y Fy(3)1107 3987 y Fq(>)f Fz(2)p Fq(m)1310 3957 y Fl(X)1310 4008 y Fy(2)1384 3987 y Fk(\000)12 b Fz(1)22 b Fk(\025)1623 3955 y Fy(3)p 1623 3969 34 4 v 1623 4016 a(2)1666 3987 y Fq(m)1739 3957 y Fl(X)1739 4008 y Fy(2)1826 3987 y Fz(using)2093 3955 y Fl(n)p 2050 3969 127 4 v 2050 4016 a(n)p Ff(\000)p Fy(1)2209 3987 y Fq(<)h(m)2370 3957 y Fl(G)2370 4008 y Fy(2)2450 3987 y Fz(and)h Fq(m)2681 3957 y Fl(G)2681 4008 y Fy(3)2760 3987 y Fq(>)f Fz(2)p Fq(m)2963 3957 y Fl(G)2963 4008 y Fy(2)3030 3987 y Fk(\000)12 b Fz(1,)23 b(whic)n(h)523 4087 y(is)28 b(the)g(assumption)f(that)g(w)n (e)h(made)f(at)h(the)g(b)r(eginning)f(of)h(\(i\).)648 4196 y(\(ii\))k(Supp)r(ose)c Fq(G)23 b Fk(2)g(S)1332 4208 y Ff(V)1383 4160 y Fy(\()p Fl(n)p Fy(\))1508 4196 y Fz(and)28 b Fq(m)1743 4166 y Fl(G)1743 4216 y Fy(3)1821 4196 y Fk(\024)23 b Fz(2)p Fq(m)2024 4166 y Fl(G)2024 4216 y Fy(2)2098 4196 y Fk(\000)18 b Fz(1)p 790 4230 1433 4 v -1 w(:)32 b(W)-7 b(e)32 b(again)e(m)n(ust)i(sho)n(w)e(that)i Fq(G)g Fz(is)523 4295 y(w)n(ell-represen)n(ted)39 b(b)n(y)h(an)g Fq(n)p Fz(-phase)g(Co)n(xian)f(distribution.)h(W)-7 b(e)41 b(will)g(sho)n(w)f(that)g Fq(G)h Fz(is)523 4395 y(w)n(ell-represen)n (ted)33 b(b)n(y)i(a)f(distribution)h Fq(Z)6 b Fz(\()p Fk(\001)p Fz(\))36 b(=)f Fq(U)9 b Fz(\()p Fk(\001)p Fz(\))p Fq(p)24 b Fz(+)e(1)h Fk(\000)g Fq(p)35 b Fz(\(See)g(Section)g(2)g(for)f (an)523 4495 y(explanation)22 b(of)h Fq(Z)6 b 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1319 2496 a(3)1367 2446 y Ft(m)1435 2410 y Fn(X)1435 2459 y Fm(2)1514 2446 y Fw(\024)21 b Ft(m)1663 2410 y Fn(X)1663 2459 y Fm(3)1742 2446 y Fx(=)g(2)p Ft(m)1929 2410 y Fn(X)1929 2459 y Fm(2)2004 2446 y Fw(\000)c Fx(1)22 b Fw(\024)2232 2397 y Fx(6\()p Ft(m)2368 2365 y Fn(X)2368 2410 y Fm(2)2443 2397 y Fw(\000)16 b Fx(1\))p 2232 2429 356 4 v 2347 2498 a Ft(m)2415 2473 y Fn(X)2415 2518 y Fm(2)2598 2446 y Ft(;)551 2628 y Fz(where)28 b(the)g(\014rst)g(inequalit)n(y)f(holds)h(when)g Fq(m)2000 2598 y Fl(X)2000 2649 y Fy(2)2087 2628 y Fk(\025)2185 2595 y Fy(3)p 2185 2609 34 4 v 2185 2657 a(2)2256 2628 y Fz(and)g(the)g(second)g(inequalit)n(y)f(holds)523 2761 y(when)743 2728 y Fy(3)p 743 2742 V 743 2789 a(2)809 2761 y Fk(\024)c Fq(m)970 2731 y Fl(X)970 2781 y Fy(2)1056 2761 y Fk(\024)f Fz(2.)f(Since)g Fq(G)i Fk(2)h(S)1656 2773 y Ff(V)1707 2725 y Fy(\()p Fl(n)p Fy(\))1804 2761 y Fz(,)d Fq(m)1921 2731 y Fl(G)1921 2781 y Fy(3)1999 2761 y Fk(\025)2097 2728 y Fl(n)p Fy(+2)p 2097 2742 126 4 v 2097 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Fz(In)d(this)g(section,)g(w)n(e)f(giv)n(e)g(examples)h (of)f(distributions)h(that)g(are)f(w)n(ell-represen)n(ted)f(b)n(y)523 3463 y(an)23 b Fq(n)p Fz(-phase)f(Co)n(xian)g(distribution.)i(In)f (particular,)f(w)n(e)h(discuss)g(Bounded)g(P)n(areto)e(distri-)523 3563 y(butions,)34 b(uniform)f(distributions,)h(symmetric)f(triangular) f(distributions,)i(and)f(P)n(areto)523 3662 y(distributions,)d(and)g (deriv)n(e)f(the)i(necessary)d(and)i(su\016cien)n(t)g(condition)g(for)f (these)h(distri-)523 3762 y(butions)e(to)f(b)r(e)h(in)g Fk(S)1180 3774 y Ff(V)1231 3726 y Fy(\()p Fl(n)p Fy(\))1351 3762 y Fk(\032)23 b(S)1495 3732 y Fy(\()p Fl(n)p Fy(\))1593 3762 y Fz(.)k(A)h(summary)f(is)g(sho)n(wn)g(in)h(Figure)f(6.)648 3862 y(W)-7 b(e)27 b(\014rst)g(discuss)g(the)g(set)g(of)g(Bounded)g(P)n (areto)f(distributions.)h(A)g(Bounded)g(P)n(areto)523 3961 y(distribution)h(has)f(a)g(densit)n(y)g(function)1513 4179 y Fq(f)9 b Fz(\()p Fq(x)p Fz(\))23 b(=)g Fq(\013x)1885 4145 y Ff(\000)p Fl(\013)p Ff(\000)p Fy(1)2206 4123 y Fq(l)2233 4093 y Fl(\013)p 2080 4160 326 4 v 2080 4247 a Fz(1)18 b Fk(\000)2223 4180 y Fj(\000)2280 4215 y Fl(l)p 2271 4229 40 4 v 2271 4276 a(u)2320 4180 y Fj(\001)2358 4195 y Fl(\013)523 4426 y Fz(for)h Fq(l)25 b Fk(\024)e Fq(x)g Fk(\024)g Fq(u)c Fz(and)h(0)f(elsewhere,)h(where)f(0)k Fq(<)f(\013)i(<)e Fz(2)e([8].)g(Bounded)f(P)n(areto)f(distributions)523 4526 y(ha)n(v)n(e)38 b(b)r(een)i(empirically)f(sho)n(wn)g(to)g(\014t)h (man)n(y)f(recen)n(t)g(measuremen)n(ts)g(of)g(computing)523 4625 y(w)n(orkloads.)22 b(These)h(include)i(Unix)f(pro)r(cess)f(CPU)g (requiremen)n(ts)g(measured)g(at)h(Bellcore:)523 4725 y(1)29 b Fk(\024)h Fq(\013)g Fk(\024)f Fz(1)p Fq(:)p Fz(25)h([19],)h(Unix)h(pro)r(cess)f(CPU)g(requiremen)n(ts)f(measured)h (at)h(UC)f(Berk)n(eley:)523 4825 y Fq(\013)24 b Fk(\031)e Fz(1)k([9],)f(sizes)g(of)h(\014les)g(transferred)e(through)h(the)i(W)-7 b(eb:)26 b(1)p Fq(:)p Fz(1)c Fk(\024)h Fq(\013)g Fk(\024)g Fz(1)p Fq(:)p Fz(3)i([3,)13 b(4],)26 b(sizes)f(of)523 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stroke} bdef /PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef /FAn {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef /PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } 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get 7 -1 roll astore } bdef /compute_transform { dmat dtri tri_to_matrix tmat1 invertmatrix smat stri tri_to_matrix tmat2 concatmatrix } bdef /ds {stri astore pop} bdef /dt {dtri astore pop} bdef /db {2 copy /cols xdef /rows xdef mul dup string currentfile exch readhexstring pop /bmap xdef pop pop} bdef /it {gs np dtri aload pop moveto lineto lineto cp c cols rows 8 compute_transform {bmap} image gr}bdef /il {newpath moveto lineto stroke}bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 26 179 548 594 MathWorks begin bpage %%EndPageSetup %%BeginObject: obj1 bplot /dpi2point 12 def portraitMode 0216 7344 csm 107 207 6264 4978 MR c np 91 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef c0 1 j 1 sg 0 0 6913 5186 PR 6 w 0 4226 5356 0 0 -4226 899 4615 4 MP 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5 27 4 27 5 27 5 26 5 27 5 27 6 27 5 27 6 26 6 27 6 27 7 27 7 26 7 27 7 27 8 27 8 27 8 26 9 27 8 27 10 27 10 26 10 27 10 27 11 27 12 27 12 26 13 27 13 27 14 27 15 26 15 27 16 27 17 27 18 27 19 26 20 27 21 27 23 952 3690 99 MP stroke DA 27 0 27 0 27 0 26 0 27 0 27 0 27 0 27 0 26 0 27 0 27 0 27 0 26 0 27 0 27 0 27 0 27 0 26 0 27 0 27 1 27 0 26 0 27 0 27 0 27 0 27 0 26 0 27 0 27 1 27 0 26 0 27 0 27 0 27 0 27 1 26 0 27 0 27 0 27 0 26 0 27 1 27 0 27 0 27 0 26 0 27 1 27 0 27 0 26 0 27 1 27 0 27 0 27 0 26 1 27 0 27 0 27 0 27 1 26 0 27 0 27 1 27 0 26 0 27 1 27 0 27 0 27 1 26 0 27 1 27 0 27 0 26 1 27 0 27 1 27 0 27 0 26 1 27 0 27 1 27 0 26 1 27 0 27 1 27 0 27 1 26 0 27 1 27 0 27 1 26 1 27 0 27 1 27 1 27 0 26 1 27 1 27 0 27 1 26 1 3577 4447 100 MP stroke 27 0 27 1 27 1 27 1 26 0 27 1 27 1 27 1 27 1 26 1 27 1 27 1 27 1 26 1 27 1 27 1 27 1 27 1 26 1 27 1 27 1 27 2 26 1 27 1 27 2 27 1 27 1 26 2 27 1 27 2 27 1 26 2 27 2 27 1 27 2 27 2 26 2 27 2 27 2 27 2 26 2 27 2 27 2 27 3 27 2 26 3 27 2 27 3 27 3 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Fq(F)35 b Fk(2)24 b(S)3257 1242 y Ff(V)3308 1194 y Fy(\()p Fl(n)p Fy(\))523 1339 y Fo(if)j(and)g(only)h(if)f Fq(n)c Fk(\025)1183 1306 y Fy(7+14)p Fl(r)r Fy(+30)p Fl(r)1516 1281 y Fh(2)1548 1306 y Fy(+14)p Fl(r)1698 1281 y Fh(3)1731 1306 y Fy(+7)p Fl(r)1848 1281 y Fh(4)p 1183 1320 697 4 v 1272 1367 a Fy(\(1)p Ff(\000)p Fl(r)r Fy(\))1442 1351 y Fh(2)1474 1367 y Fy(\(1+4)p Fl(r)r Fy(+)p Fl(r)1734 1351 y Fh(2)1765 1367 y Fy(\))1890 1339 y Fo(,)k(wher)l(e)g Fq(r)f Fz(=)2333 1306 y Fl(u)p 2333 1320 40 4 v 2342 1367 a(l)2382 1339 y Fo(.)h(In)f(p)l(articular,)i(for)g(al)t(l)f(values)523 1448 y(of)k Fq(u)p Fo(,)e Fq(n)23 b Fz(=)g(7)29 b Fo(if)i Fq(l)24 b Fz(=)f(0)p Fo(,)29 b(and)i Fq(n)23 b(>)f Fz(7)29 b Fo(whenever)i Fq(l)25 b(>)d Fz(0)p Fo(.)648 1547 y(L)l(et)29 b Fq(F)35 b Fk(2)24 b(T)d(RI)6 b(AN)12 b(G)5 b(U)j(LAR)33 b Fo(with)d(p)l(ar)l(ameters)h Fq(l)g Fo(and)f Fq(u)p Fo(.)g(Then,)h Fq(F)k Fk(2)24 b(S)3015 1559 y Ff(V)3066 1511 y Fy(\()p Fl(n)p Fy(\))3193 1547 y Fo(if)30 b(and)523 1664 y(only)k(if)g Fq(n)29 b Fk(\025)974 1624 y Fy(4\(11+34)p Fl(r)r Fy(+54)p Fl(r)1399 1599 y Fh(2)1431 1624 y Fy(+34)p Fl(r)1581 1599 y Fh(3)1613 1624 y Fy(+11)p Fl(r)1763 1599 y Fh(4)1795 1624 y Fy(\))p 974 1645 848 4 v 1104 1693 a(\(1)p Ff(\000)p Fl(r)r Fy(\))1274 1676 y Fh(2)1306 1693 y Fy(\(5+14)p Fl(r)r Fy(+5)p Fl(r)1632 1676 y Fh(2)1664 1693 y Fy(\))1831 1664 y Fo(,)34 b(wher)l(e)f Fq(r)f Fz(=)2300 1631 y Fl(u)p 2300 1645 40 4 v 2309 1693 a(l)2349 1664 y Fo(.)i(In)f(p)l(articular,)h(for)g(al)t(l)g(values)523 1773 y(of)d Fq(l)g Fo(and)f Fq(u)p Fo(,)g Fq(n)23 b Fk(\025)f Fz(9)p Fo(.)648 1873 y(L)l(et)32 b Fq(F)40 b Fk(2)28 b(P)7 b(ARE)g(T)22 b(O)35 b Fo(with)e(p)l(ar)l(ameters)h Fq(\013)f Fo(and)g Fq(k)s Fo(.)g(Then,)g Fq(F)40 b Fk(2)29 b(S)2826 1885 y Ff(V)2877 1836 y Fy(\()p Fl(n)p Fy(\))3007 1873 y Fo(if)k(and)g(only)523 1972 y(if)e Fq(n)23 b(>)f Fz(\()p Fq(\013)d Fk(\000)f Fz(1\))1025 1942 y Fy(2)1092 1972 y Fo(for)31 b(al)t(l)g(values)f(of)g Fq(k)s Fo(.)g(In)g(p)l (articular,)h Fq(n)23 b(>)g Fz(4)29 b Fo(for)h(al)t(l)h Fq(\013)24 b(>)e Fz(3)29 b Fo(and)i Fq(k)s Fo(.)523 2236 y Fs(5)112 b(Conclusion)523 2434 y Fz(The)19 b(con)n(tribution)g(of)f (this)i(pap)r(er)e(is)h(a)g(c)n(haracterization)d(of)j(the)h(set)f Fk(S)2745 2404 y Fy(\()p Fl(n)p Fy(\))2861 2434 y Fz(of)g (distributions)523 2533 y Fq(G)30 b Fz(whic)n(h)g(are)f(w)n (ell-represen)n(ted)f(b)n(y)h(an)h Fq(n)p Fz(-phase)f(Co)n(xian)f (distribution.)i(W)-7 b(e)30 b(in)n(tro)r(duce)523 2633 y(sev)n(eral)j(ideas)i(whic)n(h)g(help)g(in)g(creating)f(a)h(simple)g (form)n(ulation)f(of)h Fk(S)2837 2603 y Fy(\()p Fl(n)p Fy(\))2934 2633 y Fz(.)g(The)g(\014rst)g(is)523 2733 y(the)26 b(concept)g(of)g(normalized)f(momen)n(ts.)h(The)g(second)f(is) h(the)g(notion)g(of)g Fk(S)2916 2745 y Ff(V)2967 2696 y Fy(\()p Fl(n)p Fy(\))3064 2733 y Fz(,)g(a)g(nearly)523 2832 y(complete)c(subset)f(of)h Fk(S)1261 2802 y Fy(\()p Fl(n)p Fy(\))1381 2832 y Fz(with)g(an)f(extremely)h(simple)f(represen)n (tation.)g(The)g(argumen)n(ts)523 2932 y(required)i(in)g(pro)n(ving)f (the)i(ab)r(o)n(v)n(e)e(results)h(ha)n(v)n(e)f(an)i(elegan)n(t)e (structure)h(whic)n(h)g(rep)r(eatedly)523 3031 y(mak)n(es)k(use)g(of)h (the)g(recursiv)n(e)d(nature)j(of)f(the)h(Co)n(xian)e(distributions.) 648 3131 y(Our)33 b(c)n(haracterization)f(of)i Fk(S)1596 3101 y Fy(\()p Fl(n)p Fy(\))1728 3131 y Fz(pro)n(vides)f(a)h(necessary) f(n)n(um)n(b)r(er)h(of)h(phases)e(and)h(a)523 3231 y(su\016cien)n(t)i (n)n(um)n(b)r(er)g(of)h(phases)e(for)h(a)f(giv)n(en)h(distribution)g (to)g(b)r(e)h(w)n(ell-represen)n(ted)d(b)n(y)523 3330 y(a)f(Co)n(xian)f(distribution,)i(and)f(these)g(b)r(ounds)h(are)e (nearly)g(tigh)n(t.)i(This)f(result)g(has)g(sev-)523 3430 y(eral)26 b(practical)g(uses.)h(First,)f(in)i(designing)e (algorithms)f(whic)n(h)i(\014t)h(general)d(distributions)523 3530 y(to)38 b(Co)n(xian)f(distributions)i(\(\014tting)g(algorithms\),) e(it)i(is)f(desirable)g(to)g(\014nd)h(a)f Fo(minimal)523 3629 y Fz(\(few)n(est)31 b(n)n(um)n(b)r(er)g(of)h(phases\))e(Co)n(xian) g(distribution.)i(Our)e(c)n(haracterization)f(allo)n(ws)h(al-)523 3729 y(gorithm)38 b(designers)f(to)h(determine)g(ho)n(w)g(close)f (their)i(Co)n(xian)e(distribution)h(is)g(to)h(the)523 3828 y(minimal)i(Co)n(xian)f(distribution,)g(and)h(pro)n(vides)e(in)n (tuition)i(for)g(coming)f(up)h(with)g(im-)523 3928 y(pro)n(v)n(ed)33 b(algorithms.)g(W)-7 b(e)34 b(ha)n(v)n(e)f(ourselv)n(es)f(b)r (ene\014tted)k(from)e(exactly)f(this)i(p)r(oin)n(t.)f(In)g(a)523 4028 y(companion)g(pap)r(er)h([22],)g(w)n(e)g(dev)n(elop)f(an)h (algorithm)f(for)h(\014nding)g(a)g(minimal)g(Co)n(xian)523 4127 y(distribution)f(that)h(w)n(ell-represen)n(ts)d(a)h(giv)n(en)h (distribution.)g(W)-7 b(e)34 b(\014nd)h(that)f(the)h(simple)523 4227 y(c)n(haracterization)28 b(of)j Fk(S)1287 4197 y Fy(\()p Fl(n)p Fy(\))1415 4227 y Fz(pro)n(vided)f(herein)g(is)h(v)n (ery)e(useful)i(in)g(this)g(task.)g(Our)f(results)523 4327 y(are)e(also)f(useful)i(as)f(an)h(input)g(to)g(some)f(existing)g (\014tting)h(algorithms,)f(suc)n(h)g(as)g(Johnson)523 4426 y(and)23 b(T)-7 b(aa\013e's)23 b(nonlinear)f(programming)g (approac)n(h)f([15],)i(whic)n(h)g(require)f(kno)n(wing)h(a)g(pri-)523 4526 y(ori)g(the)h(n)n(um)n(b)r(er)g(of)g(phases)f Fq(n)h Fz(in)g(the)h(minimal)f(Co)n(xian)e(distribution.)49 b(F)-7 b(urthermore)22 b(w)n(e)523 4625 y(classify)29 b(a)f(few)i(examples)e(of)h(common)g(and)g(practical)f(distributions)h (as)g(b)r(eing)g(subsets)523 4725 y(of)f Fk(S)674 4695 y Fy(\()p Fl(n)p Fy(\))799 4725 y Fz(for)f(some)g Fq(n)p Fz(.)648 4825 y(F)-7 b(uture)21 b(w)n(ork)f(includes)i(a)f(simple)h(c)n (haracterization)c(of)k(the)g(set)f(of)g(distributions)h(that)523 4924 y(are)32 b(w)n(ell-represen)n(ted)f(b)n(y)i(general)e Fq(n)p Fz(-phase)h(PH)h(distributions.)f(If)i(w)n(e)e(w)n(ere)g(to)h (follo)n(w)p eop %%Page: 15 15 15 14 bop 523 448 a Fz(the)32 b(approac)n(h)e(in)i(this)g(pap)r(er,)f (w)n(e)h(w)n(ould)f(start)g(b)n(y)h(sp)r(ecifying)f(the)i(lo)n(w)n(er)d (b)r(ounds)i(for)523 548 y(the)d(second)e(and)i(third)f(normalized)f (momen)n(ts)h(of)h(general)e Fq(n)p Fz(-phase)g(PH)h(distributions.)523 648 y(Ho)n(w)n(ev)n(er,)17 b(this)i(seems)f(to)h(b)r(e)g(non)n (trivial:)e(although)h(the)h(lo)n(w)n(er)e(b)r(ound)j(on)e(the)h (normalized)523 747 y(second)28 b(momen)n(t)g(is)g(kno)n(wn)g([1],)g (the)h(lo)n(w)n(er)e(b)r(ound)h(on)g(the)h(normalized)f(third)g(momen)n (t)523 847 y(of)g Fq(n)p Fz(-phase)e(PH)i(distributions)f(is)h(not)f (kno)n(wn.)523 1133 y Fs(Ac)m(kno)m(wledgemen)m(t)523 1354 y Fz(W)-7 b(e)28 b(w)n(ould)f(lik)n(e)h(to)f(thank)h(Miklos)f(T)-7 b(elek)28 b(for)f(his)h(help)g(in)g(impro)n(ving)e(the)i(presen)n (tation)523 1453 y(and)f(qualit)n(y)h(of)f(this)h(pap)r(er.)523 1740 y Fs(References)561 1952 y Fx(1.)43 b(D.)34 b(Aldous)g(and)g(L.)h (Shepp.)59 b(The)34 b(least)i(v)l(ariable)e(phase)h(t)n(yp)r(e)e (distribution)i(is)g(Erlang.)663 2043 y Fb(Communic)l(ations)28 b(in)f(Statistics)i(-)f(Sto)l(chastic)i(Mo)l(dels)p Fx(,)c(3:467)i({)e (473,)h(1987.)561 2139 y(2.)43 b(T.)26 b(Altiok.)34 b(On)25 b(the)g(phase-t)n(yp)r(e)f(appro)n(ximations)i(of)g(general)g (distributions.)35 b Fb(IIE)26 b(T)-6 b(r)l(ans-)663 2230 y(actions)p Fx(,)27 b(17:110)h({)e(116,)h(1985.)561 2325 y(3.)43 b(M.)28 b(E.)g(Cro)n(v)n(ella)i(and)d(A.)h(Besta)n(vros.) 42 b(Self-similarit)n(y)28 b(in)g(World)g(Wide)g(Web)f(tra\016c:)i(Ev-) 663 2417 y(idence)h(and)f(p)r(ossible)i(causes.)49 b Fb(IEEE/A)n(CM)30 b(T)-6 b(r)l(ansactions)34 b(on)e(Networking)p Fx(,)g(5\(6\):835)g({)663 2508 y(846,)27 b(Decem)n(b)r(er)d(1997.)561 2604 y(4.)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 2695 y(butions)e(in)g(the)g(w)n(orld)g (wide)h(w)n(eb.)45 b(In)29 b Fb(A)h(Pr)l(actic)l(al)i(Guide)f(T)-6 b(o)31 b(He)l(avy)h(T)-6 b(ails)p Fx(,)29 b(c)n(hapter)g(1,)663 2786 y(pages)d(1)g({)g(23.)h(Chapman)e(&)g(Hall,)i(New)e(Y)-6 b(ork,)26 b(1998.)561 2882 y(5.)43 b(A.)32 b(Cumani.)54 b(On)32 b(the)g(canonical)i(represen)n(tation)f(of)g(homogeneous)g (Mark)n(o)n(v)f(pro)r(cesses)663 2973 y(mo)r(deling)24 b(failure-time)g(distributions.)31 b Fb(Micr)l(o)l(ele)l(ctr)l(onics)d (and)f(R)l(eliability)p Fx(,)c(22:583)j({)e(602,)663 3064 y(1982.)561 3160 y(6.)43 b(A.)29 b(F)-6 b(eldmann)27 b(and)i(W.)g(Whitt.)44 b(Fitting)30 b(mixtures)e(of)i(exp)r(onen)n (tials)f(to)g(long-tail)i(distri-)663 3251 y(butions)26 b(to)g(analyze)h(net)n(w)n(ork)g(p)r(erformance)f(mo)r(dels.)37 b Fb(Performanc)l(e)29 b(Evaluation)p Fx(,)f(32:245)663 3343 y({)e(279,)h(1998.)561 3438 y(7.)43 b(H.)20 b(F)-6 b(rank)n(e,)20 b(J.)h(Jann,)f(J.)h(Moreira,)h(P)-6 b(.)21 b(P)n(attnaik,)f(and)g(M.)h(Jette.)26 b(An)20 b(ev)l(aluation)g(of)h (parallel)663 3529 y(job)k(sc)n(heduling)h(for)g(ASCI)f (blue-paci\014c.)33 b(In)25 b Fb(Pr)l(o)l(c)l(e)l(e)l(dings)k(of)e(Sup) l(er)l(c)l(omputing)j('99)p Fx(,)25 b(pages)663 3621 y(679)h({)g(691,)i(No)n(v)n(em)n(b)r(er)23 b(1999.)561 3716 y(8.)43 b(M.)22 b(Harc)n(hol-Balter.)28 b(T)-6 b(ask)21 b(assignmen)n(t)g(with)h(unkno)n(wn)e(duration.)27 b Fb(Journal)d(of)g(the)g(A)n(CM)p Fx(,)663 3808 y(49\(2\),)j(2002.)561 3903 y(9.)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 3994 y(dynamic)20 b(load)j(balancing.)29 b(In)21 b Fb(Pr)l(o)l(c)l(e)l(e)l(dings)27 b(of)c(SIGMETRICS)g('96)p Fx(,)g(pages)f(13)h({)f(24,)h(1996.)523 4090 y(10.)43 b(M.)28 b(Harc)n(hol-Balter,)g(C.)g(Li,)g(T.)g(Osogami,)g(A.)g(Sc)n (heller-W)-6 b(olf,)28 b(and)f(M.)h(Squillan)n(te.)39 b(T)-6 b(ask)663 4181 y(assignmen)n(t)24 b(with)i(cycle)f(stealing)h (under)e(cen)n(tral)h(queue.)33 b(In)24 b Fb(Pr)l(o)l(c)l(e)l(e)l (dings)29 b(of)e(ICDCS)f('03)p Fx(,)663 4273 y(pages)g(628{637,)j(Ma)n (y)d(2003.)523 4368 y(11.)43 b(A.)27 b(Horv\023)-38 b(ath)26 b(and)h(M.)h(T)-6 b(elek.)39 b(Appro)n(ximating)26 b(hea)n(vy)g(tailed) i(b)r(eha)n(vior)f(with)g(phase)h(t)n(yp)r(e)663 4459 y(distributions.)23 b(In)18 b Fb(A)l(dvanc)l(es)k(in)f(Matrix-A)n (nalytic)h(Metho)l(ds)h(for)e(Sto)l(chastic)i(Mo)l(dels)p Fx(,)c(pages)663 4551 y(191)26 b({)g(214.)i(Notable)e(Publications,)h (July)e(2000.)523 4646 y(12.)43 b(A.)24 b(Horv\023)-38 b(ath)25 b(and)f(M.)i(T)-6 b(elek.)33 b(Ph\014t:)25 b(A)g(general)h (phase-t)n(yp)r(e)d(\014tting)i(to)r(ol.)34 b(In)25 b Fb(Pr)l(o)l(c)l(e)l(e)l(dings)663 4738 y(of)i(Performanc)l(e)i(TOOLS)e (2002)p Fx(,)g(pages)f(82)h({)f(91,)g(April)g(2002.)523 4833 y(13.)43 b(G.)19 b(Irlam.)k(Unix)18 b(\014le)h(size)h(surv)n(ey)d (-)i(1993.)25 b(Av)l(ailable)19 b(at)g Fv(http://www.base.com/gordoni)p Fx(-)663 4924 y Fv(/ufs93.html)p Fx(,)28 b(Septem)n(b)r(er)c(1994.)p eop %%Page: 16 16 16 15 bop 523 448 a Fx(14.)43 b(M.)25 b(A.)f(Johnson)h(and)f(M.)g(R.)h (T)-6 b(aa\013e.)33 b(Matc)n(hing)24 b(momen)n(ts)f(to)h(phase)h (distributions:)g(Den-)663 540 y(sit)n(y)i(function)h(shap)r(es.)42 b Fb(Communic)l(ations)30 b(in)f(Statistics)i(|)f(Sto)l(chastic)h(Mo)l (dels)p Fx(,)e(6:283)h({)663 631 y(306,)d(1990.)523 731 y(15.)43 b(M.)18 b(A.)f(Johnson)h(and)e(M.)i(R.)g(T)-6 b(aa\013e.)21 b(Matc)n(hing)d(momen)n(ts)d(to)j(phase)f(distributions:) h(Nonlin-)663 823 y(ear)24 b(programming)f(approac)n(hes.)32 b Fb(Communic)l(ations)26 b(in)g(Statistics)h(|)f(Sto)l(chastic)i(Mo)l (dels)p Fx(,)663 914 y(6:259)f({)f(281,)h(1990.)523 1014 y(16.)43 b(S.)31 b(Karlin)g(and)f(W.)h(Studden.)49 b Fb(Tchebyche\013)34 b(Systems:)f(With)f(Applic)l(ations)i(in)d(A)n (nalysis)663 1105 y(and)d(Statistics)p Fx(.)36 b(John)26 b(Wiley)g(and)f(Sons,)h(1966.)523 1206 y(17.)43 b(R.)32 b(E.)h(A.)g(Kha)n(y)n(ari,)f(R.)h(Sadre,)f(and)h(B.)g(Ha)n(v)n(erk)n (ort.)54 b(Fitting)33 b(w)n(orld-wide)h(w)n(eb)e(request)663 1297 y(traces)26 b(with)g(the)f(EM-algorithm.)35 b Fb(Performanc)l(e)29 b(Evalutation)p Fx(,)f(52:175)f({)f(191,)i(2003.)523 1397 y(18.)43 b(G.)37 b(Latouc)n(he)h(and)f(V.)g(Ramasw)n(ami.)68 b Fb(Intr)l(o)l(duction)40 b(to)f(Matrix)g(A)n(nalytic)g(Metho)l(ds)g (in)663 1489 y(Sto)l(chastic)29 b(Mo)l(deling)p Fx(.)35 b(ASA-SIAM,)24 b(Philadelphia,)j(1999.)523 1589 y(19.)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 1680 y Fb(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 Fx(,)f(pages)f(54)g({)g(69,)h(1986.)523 1780 y(20.)43 b(R.)29 b(Marie.)45 b(Calculating)31 b(equilibrium)d (probabilities)j(for)e Ft(\025)p Fx(\()p Ft(n)p Fx(\))p Ft(=c)2639 1789 y Fn(k)2678 1780 y Ft(=)p Fx(1)p Ft(=n)i Fx(queues.)43 b(In)29 b Fb(Pr)l(o-)663 1872 y(c)l(e)l(e)l(dings)g(of)e Fx(P)n(erformance)f(1980,)i(pages)e(117)h({)f(125,)h(1980.)523 1972 y(21.)43 b(M.)33 b(F.)f(Neuts.)53 b Fb(Matrix-Ge)l(ometric)37 b(Solutions)d(in)f(Sto)l(chastic)j(Mo)l(dels:)d(A)n(n)h(A)n(lgorithmic) 663 2063 y(Appr)l(o)l(ach)p Fx(.)i(The)26 b(Johns)g(Hopkins)f(Univ)n (ersit)n(y)g(Press,)i(1981.)523 2163 y(22.)43 b(T.)18 b(Osogami)g(and)g(M.)g(Harc)n(hol-Balter.)23 b(A)17 b(closed-form)h (solution)h(for)f(mapping)f(general)i(dis-)663 2255 y(tributions)g(to)h (minimal)e(PH)i(distributions.)k(In)19 b Fb(Pr)l(o)l(c)l(e)l(e)l(dings) 24 b(of)e(TOOLS)f(2003)p Fx(,)g(Septem)n(b)r(er)663 2346 y(2003.)523 2446 y(23.)43 b(T.)31 b(Osogami,)f(M.)h(Harc)n(hol-Balter,) h(and)e(A.)f(Sc)n(heller-W)-6 b(olf.)49 b(Analysis)30 b(of)h(cycle)f(stealing)663 2538 y(with)j(switc)n(hing)i(cost.)58 b(In)33 b Fb(Pr)l(o)l(c)l(e)l(e)l(dings)k(of)d(SIGMETRICS)g('03)p Fx(,)g(pages)g(184{195,)j(June)663 2629 y(2003.)523 2729 y(24.)43 b(V.)g(P)n(axson)g(and)g(S.)g(Flo)n(yd.)86 b(Wide-are)43 b(tra\016c:)h(The)g(failure)g(of)g(P)n(oisson)h(mo)r(deling.)663 2821 y Fb(IEEE/A)n(CM)27 b(T)-6 b(r)l(ansactions)29 b(on)f(Networking)p Fx(,)g(pages)e(226)h({)f(244,)h(June)f(1995.)523 2921 y(25.)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 Fb(CMG)663 3012 y(Pr)l(o)l(c)l(e)l(e)l(dings)p Fx(,)j(Decem)n(b)r(er)d(1995.)523 3112 y(26.)43 b(A.)24 b(Risk)l(a,)h(V.)g(Diev,)f(and)h(E.)g(Smirni.)32 b(E\016cien)n(t)24 b(\014tting)h(of)g(long-tailed)h(data)f(sets)g(in)n(to)g(PH)663 3204 y(distributions.)34 b Fb(Performanc)l(e)29 b(Evaluation)p Fx(,)e(2003)g(\(to)f(app)r(ear\).)523 3304 y(27.)43 b(C.)33 b(Sauer)f(and)g(K.)h(Chandy)-6 b(.)54 b(Appro)n(ximate)31 b(analysis)i(of)g(cen)n(tral)g(serv)n(er)g(mo)r(dels.)55 b Fb(IBM)663 3395 y(Journal)28 b(of)f(R)l(ese)l(ar)l(ch)j(and)e (Development)p Fx(,)f(19:301)h({)e(313,)h(1975.)523 3495 y(28.)43 b(B.)32 b(Sc)n(hro)r(eder)g(and)f(M.)h(Harc)n(hol-Balter.)54 b(Ev)l(aluation)32 b(of)g(task)g(assignmen)n(t)g(p)r(olicies)h(for)663 3587 y(sup)r(ercomputing)19 b(serv)n(ers:)h(The)g(case)h(for)g(load)g (un)n(balancing)f(and)g(fairness.)26 b(In)20 b Fb(Pr)l(o)l(c)l(e)l(e)l (dings)663 3678 y(of)27 b Fx(HPDC)e(2000,)j(pages)e(211{219,)j(2000.) 523 3778 y(29.)43 b(M.)36 b(Squillan)n(te.)64 b(Matrix-analytic)36 b(metho)r(ds)f(in)h(sto)r(c)n(hastic)h(parallel-serv)n(er)g(sc)n (heduling)663 3870 y(mo)r(dels.)47 b(In)30 b Fb(A)l(dvanc)l(es)j(in)e (Matrix-A)n(nalytic)i(Metho)l(ds)g(for)f(Sto)l(chastic)i(Mo)l(dels)p Fx(.)d(Notable)663 3961 y(Publications,)c(July)f(1998.)523 4061 y(30.)43 b(D.)26 b(Starobinski)h(and)f(M.)h(Sidi.)37 b(Mo)r(deling)27 b(and)g(analysis)g(of)g(p)r(o)n(w)n(er-tail)h (distributions)f(via)663 4153 y(classical)h(teletra\016c)e(metho)r(ds.) 34 b Fb(Queueing)28 b(Systems)p Fx(,)g(36:243)g({)e(267,)h(2000.)523 4253 y(31.)43 b(M.)31 b(T)-6 b(elek)31 b(and)f(A.)h(Heindl.)49 b(Matc)n(hing)31 b(momen)n(ts)e(for)j(acyclic)f(discrete)h(and)e(con)n (tin)n(uous)663 4344 y(phase-t)n(yp)r(e)20 b(distributions)h(of)h (second)g(order.)27 b Fb(International)e(Journal)f(of)f(Simulation)p Fx(,)e(3:47)663 4435 y({)26 b(57,)g(2003.)523 4536 y(32.)43 b(W.)25 b(Whitt.)33 b(Appro)n(ximating)24 b(a)i(p)r(oin)n(t)f(pro)r (cess)h(b)n(y)e(a)i(renew)n(al)g(pro)r(cess:)g(Tw)n(o)h(basic)f(meth-) 663 4627 y(o)r(ds.)35 b Fb(Op)l(er)l(ations)29 b(R)l(ese)l(ar)l(ch)p Fx(,)f(30:125)g({)e(147,)h(1982.)523 4727 y(33.)43 b(Y.)17 b(Zhang,)i(H.)e(F)-6 b(rank)n(e,)18 b(J.)h(Moreira,)h(and)d(A.)h(Siv)l (asubramaniam.)i(An)d(in)n(tegrated)i(approac)n(h)663 4819 y(to)37 b(parallel)j(sc)n(heduling)d(using)h(gang-sc)n(heduling,)h (bac)n(k\014lling,)f(and)f(migration.)70 b Fb(IEEE)663 4910 y(T)-6 b(r)l(ansactions)29 b(on)f(Par)l(al)t(lel)g(and)g (Distribute)l(d)h(Systems)p Fx(,)e(14:236)h({)e(247,)h(2003.)p eop %%Page: 17 17 17 16 bop 523 448 a Fs(A)112 b(T)-9 b(ec)m(hnical)36 b(Lemmas)523 641 y Fp(Lemma)30 b(3.)40 b Fo(The)29 b(set)e(de\014ne)l (d)h(in)f(\(3\))h(and)g(the)f(set)g(de\014ne)l(d)h(in)g(\(4\))f(ar)l(e) h(e)l(quivalent)g(sets.)523 900 y(Pr)l(o)l(of.)43 b Fz(Recall)26 b(that)h(the)h(set)e(de\014ned)h(in)g(\(3\))g(is)g(the)g(union)f(of)h (the)g(follo)n(wing)f(three)g(sets:)523 1120 y Ft(A)581 1128 y Fm(1)638 1120 y Fx(=)721 1005 y Fj(\032)784 1120 y Ft(F)844 1027 y Fj(\014)844 1077 y(\014)844 1126 y(\014)871 1120 y Fw(9)p Ft(p;)996 1071 y Fx(9)p Ft(pm)1141 1039 y Fn(F)1141 1085 y Fm(2)1209 1071 y Fw(\000)17 b Fx(12)h(+)e(3)1494 1009 y Fw(p)p 1559 1009 39 4 v 1559 1071 a Fx(2\(2)h Fw(\000)g Ft(pm)1866 1039 y Fn(F)1866 1085 y Fm(2)1916 1071 y Fx(\))1956 1020 y Fh(3)p 1956 1029 29 4 v 1956 1061 a(2)p 996 1103 1003 4 v 1402 1173 a Ft(p)1441 1151 y Fm(2)1475 1173 y Ft(m)1543 1147 y Fn(F)1543 1192 y Fm(2)2030 1120 y Fw(\024)k Ft(m)2179 1084 y Fn(F)2179 1133 y Fm(3)2251 1120 y Fw(\024)2342 1071 y Fx(6\()p Ft(pm)2517 1039 y Fn(F)2517 1085 y Fm(2)2584 1071 y Fw(\000)c Fx(1\))p 2342 1103 388 4 v 2440 1173 a Ft(p)2479 1151 y Fm(2)2513 1173 y Ft(m)2581 1147 y Fn(F)2581 1192 y Fm(2)2765 1120 y Fx(and)2943 1072 y(3)p 2924 1103 77 4 v 2924 1171 a(2)p Ft(p)3032 1120 y Fw(\024)k Ft(m)3181 1084 y Fn(F)3181 1133 y Fm(2)3253 1120 y Ft(<)3344 1072 y Fx(2)p 3344 1103 39 4 v 3344 1171 a Ft(p)3393 1005 y Fj(\033)3468 1120 y Ft(;)523 1353 y(A)581 1361 y Fm(2)638 1353 y Fx(=)721 1238 y Fj(\032)784 1353 y Ft(F)844 1259 y Fj(\014)844 1309 y(\014)844 1359 y(\014)871 1353 y Fw(9)p Ft(p;)12 b(m)1054 1317 y Fn(F)1054 1366 y Fm(3)1126 1353 y Fx(=)1217 1304 y(3)p 1217 1336 V 1217 1403 a Ft(p)1291 1353 y Fx(and)26 b Ft(m)1509 1317 y Fn(F)1509 1366 y Fm(2)1581 1353 y Fx(=)1672 1304 y(2)p 1672 1336 V 1672 1403 a Ft(p)1720 1238 y Fj(\033)1795 1353 y Ft(;)90 b(A)1964 1361 y Fm(3)2020 1353 y Fx(=)2101 1238 y Fj(\032)2163 1353 y Ft(F)2223 1259 y Fj(\014)2223 1309 y(\014)2223 1359 y(\014)2250 1353 y Fw(9)p Ft(p;)2376 1304 y Fx(3)p 2376 1336 V 2376 1403 a(2)2424 1353 y Ft(m)2492 1317 y Fn(F)2492 1366 y Fm(2)2564 1353 y Ft(<)21 b(m)2713 1317 y Fn(F)2713 1366 y Fm(3)2789 1353 y Fx(and)2948 1304 y(2)p 2948 1336 V 2948 1403 a Ft(p)3018 1353 y(<)g(m)3167 1317 y Fn(F)3167 1366 y Fm(2)3218 1238 y Fj(\033)3293 1353 y Fx(;)551 1569 y Fz(the)28 b(set)f(de\014ned)h(in)g(\(4\))g(is)f (the)h(union)g(of)g(the)g(follo)n(wing)e(three)h(sets:)527 1780 y Ft(B)585 1788 y Fm(1)643 1780 y Fx(=)726 1665 y Fj(\032)788 1780 y Ft(F)848 1687 y Fj(\014)848 1737 y(\014)848 1787 y(\014)885 1732 y Fx(4)p 885 1763 V 885 1831 a(3)933 1780 y Ft(m)1001 1745 y Fn(F)1001 1794 y Fm(2)1073 1780 y Fw(\024)21 b Ft(m)1222 1745 y Fn(F)1222 1794 y Fm(3)1294 1780 y Fw(\024)1385 1731 y Fx(6\()p Ft(m)1521 1700 y Fn(F)1521 1745 y Fm(2)1589 1731 y Fw(\000)c Fx(1\))p 1385 1763 349 4 v 1500 1833 a Ft(m)1568 1807 y Fn(F)1568 1853 y Fm(2)1770 1780 y Fx(and)1929 1732 y(3)p 1929 1763 39 4 v 1929 1831 a(2)1999 1780 y Fw(\024)k Ft(m)2148 1745 y Fn(F)2148 1794 y Fm(2)2220 1780 y Fw(\024)g Fx(2)2339 1665 y Fj(\033)2414 1780 y Ft(;)527 1991 y(B)585 1999 y Fm(2)643 1991 y Fx(=)726 1901 y Fj(n)781 1991 y Ft(F)841 1898 y Fj(\014)841 1947 y(\014)841 1997 y(\014)878 1943 y Fx(4)p 878 1974 V 878 2042 a(3)927 1991 y Ft(m)995 1955 y Fn(F)995 2004 y Fm(2)1066 1991 y Fw(\024)g Ft(m)1215 1955 y Fn(F)1215 2004 y Fm(3)1287 1991 y Fw(\024)1378 1943 y Fx(3)p 1378 1974 V 1378 2042 a(2)1427 1991 y Ft(m)1495 1955 y Fn(F)1495 2004 y Fm(2)1571 1991 y Fx(and)k(2)d Ft(<)f(m)1929 1955 y Fn(F)1929 2004 y Fm(2)1979 1901 y Fj(o)2047 1991 y Ft(;)90 b(B)2216 1999 y Fm(3)2272 1991 y Fx(=)2354 1901 y Fj(n)2409 1991 y Ft(F)2469 1898 y Fj(\014)2469 1947 y(\014)2469 1997 y(\014)2506 1943 y Fx(3)p 2506 1974 V 2506 2042 a(2)2555 1991 y Ft(m)2623 1955 y Fn(F)2623 2004 y Fm(2)2694 1991 y Ft(<)21 b(m)2843 1999 y Fm(3)2903 1991 y Fx(and)26 b(2)21 b Ft(<)g(m)3261 1955 y Fn(F)3261 2004 y Fm(2)3312 1901 y Fj(o)3380 1991 y Ft(:)648 2255 y Fz(It)39 b(su\016ces)g(to)g(pro)n(v)n(e)e(that)j (\(i\))g Fq(A)1772 2267 y Fy(1)1852 2255 y Fz(=)i Fq(B)2022 2267 y Fy(1)2085 2255 y Fk([)26 b Fq(B)2229 2267 y Fy(2)2267 2255 y Fz(,)39 b(\(ii\))h Fq(A)2541 2267 y Fy(2)2621 2255 y Fk(\032)i Fq(B)2791 2267 y Fy(1)2854 2255 y Fk([)27 b Fq(B)2999 2267 y Fy(2)3036 2255 y Fz(,)39 b(and)g(\(iii\))523 2354 y Fq(A)585 2366 y Fy(3)646 2354 y Fz(=)22 b Fq(B)796 2366 y Fy(3)833 2354 y Fz(.)27 b(\(ii\))g(and)f(\(iii\))g(are)g (immediate)g(from)g(the)g(de\014nition.)h(T)-7 b(o)26 b(pro)n(v)n(e)e(\(i\),)j(w)n(e)f(pro)n(v)n(e)523 2454 y(that)i Fq(A)765 2466 y Fy(1)825 2454 y Fk(\032)23 b Fq(B)976 2466 y Fy(1)1032 2454 y Fk([)c Fq(B)1169 2466 y Fy(2)1233 2454 y Fz(and)28 b Fq(B)1458 2466 y Fy(1)1514 2454 y Fk([)18 b Fq(B)1650 2466 y Fy(2)1711 2454 y Fk(\032)k Fq(A)1860 2466 y Fy(1)1898 2454 y Fz(.)648 2545 y(Consider)30 b(a)g(distribution)h Fq(F)41 b Fk(2)29 b Fq(A)1766 2557 y Fy(1)1803 2545 y Fz(.)i(W)-7 b(e)31 b(\014rst)g(sho)n(w)f(that)h Fq(F)41 b Fk(2)29 b Fq(B)2811 2557 y Fy(1)2869 2545 y Fk([)21 b Fq(B)3008 2557 y Fy(2)3045 2545 y Fz(.)31 b(Let)g Fq(u)p Fz(\()p Fq(p)p Fz(\))523 2637 y(and)c Fq(l)r Fz(\()p Fq(p)p Fz(\))h(b)r(e)g(the)g(upp)r(er)g(and)f(lo)n(w)n(er)f(b)r(ound)i (of)g Fq(m)2141 2607 y Fl(F)2141 2657 y Fy(3)2196 2637 y Fz(,)f(resp)r(ectiv)n(ely:)912 2910 y Ft(l)q Fx(\()p Ft(p)p Fx(\))20 b(=)1146 2824 y(3)1198 2734 y Fj(\020)1247 2824 y Fx(3)p Ft(pm)1392 2792 y Fn(F)1392 2837 y Fm(2)1460 2824 y Fw(\000)c Fx(4)i(+)1669 2762 y Fw(p)p 1733 2762 V 62 x Fx(2\(2)f Fw(\000)g Ft(pm)2040 2792 y Fn(F)2040 2837 y Fm(2)2090 2824 y Fx(\))2130 2772 y Fh(3)p 2130 2781 29 4 v 2130 2813 a(2)2172 2734 y Fj(\021)p 1146 2893 1076 4 v 1588 2963 a Ft(p)1627 2941 y Fm(2)1661 2963 y Ft(m)1729 2937 y Fn(F)1729 2982 y Fm(2)2232 2910 y Fx(;)90 b Ft(u)p Fx(\()p Ft(p)p Fx(\))21 b(=)2598 2861 y(6\()p Ft(pm)2773 2829 y Fn(F)2773 2875 y Fm(2)2840 2861 y Fw(\000)c Fx(1\))p 2598 2893 388 4 v 2696 2963 a Ft(p)2735 2941 y Fm(2)2769 2963 y Ft(m)2837 2937 y Fn(F)2837 2982 y Fm(2)2995 2910 y Ft(:)545 3125 y Fz(Then,)k Fq(u)p Fz(\()p Fq(p)p Fz(\))h(and)f Fq(l)r Fz(\()p Fq(p)p Fz(\))h(are)e(b)r(oth)i(con)n(tin)n(uous)f(and)g(increasing)f (functions)i(of)g Fq(p)f Fz(for)3221 3093 y Fy(3)p 3168 3107 140 4 v 3168 3158 a(2)p Fl(m)3260 3138 y Fi(F)3260 3178 y Fh(2)3340 3125 y Fk(\024)523 3256 y Fq(p)i Fk(\024)722 3224 y Fy(2)p 685 3238 107 4 v 685 3289 a Fl(m)744 3269 y Fi(F)744 3309 y Fh(2)802 3256 y Fz(.)k(When)i Fq(m)1168 3226 y Fl(F)1168 3277 y Fy(2)1246 3256 y Fk(\024)22 b Fz(2,)28 b(the)g(range)e(of)h Fq(p)h Fz(is)2105 3224 y Fy(3)p 2052 3238 140 4 v 2052 3289 a(2)p Fl(m)2144 3269 y Fi(F)2144 3309 y Fh(2)2225 3256 y Fk(\024)22 b Fq(p)h Fk(\024)g Fz(1.)k(Th)n(us,)1187 3452 y Fx(4)p 1187 3483 39 4 v 1187 3551 a(3)1236 3500 y Ft(m)1304 3464 y Fn(F)1304 3513 y Fm(2)1375 3500 y Fx(=)22 b Ft(l)1493 3385 y Fj(\022)1624 3452 y Fx(3)p 1565 3483 157 4 v 1565 3553 a(2)p Ft(m)1671 3527 y Fn(F)1671 3572 y Fm(2)1731 3385 y Fj(\023)1814 3500 y Fw(\024)f Ft(m)1963 3464 y Fn(F)1963 3513 y Fm(3)2035 3500 y Fw(\024)g Ft(u)p Fx(\(1\))g(=)2370 3451 y(6\()p Ft(m)2506 3419 y Fn(F)2506 3465 y Fm(2)2574 3451 y Fw(\000)c Fx(1\))p 2370 3483 349 4 v 2486 3553 a Ft(m)2554 3527 y Fn(F)2554 3572 y Fm(2)2729 3500 y Ft(;)551 3724 y Fz(and)27 b(hence)h Fq(F)35 b Fk(2)23 b Fq(B)1172 3736 y Fy(1)1209 3724 y Fz(.)28 b(When)g(2)23 b Fq(<)f(m)1727 3694 y Fl(F)1727 3745 y Fy(2)1783 3724 y Fz(,)27 b(the)h(range)e(of)i Fq(p)f Fz(is)2513 3691 y Fy(3)p 2460 3705 140 4 v 2460 3757 a(2)p Fl(m)2552 3737 y Fi(F)2552 3777 y Fh(2)2632 3724 y Fk(\024)c Fq(p)g Fk(\024)2919 3691 y Fy(2)p 2882 3705 107 4 v 2882 3757 a Fl(m)2941 3737 y Fi(F)2941 3777 y Fh(2)2999 3724 y Fz(.)k(Th)n(us,)1196 3919 y Fx(4)p 1196 3951 39 4 v 1196 4018 a(3)1244 3968 y Ft(m)1312 3932 y Fn(F)1312 3981 y Fm(2)1384 3968 y Fx(=)21 b Ft(l)1502 3853 y Fj(\022)1632 3919 y Fx(3)p 1573 3951 157 4 v 1573 4020 a(2)p Ft(m)1679 3995 y Fn(F)1679 4040 y Fm(2)1740 3853 y Fj(\023)1822 3968 y Fw(\024)g Ft(m)1971 3932 y Fn(F)1971 3981 y Fm(3)2043 3968 y Fw(\024)g Ft(u)2181 3853 y Fj(\022)2292 3919 y Fx(2)p 2252 3951 119 4 v 2252 4020 a Ft(m)2320 3995 y Fn(F)2320 4040 y Fm(2)2381 3853 y Fj(\023)2463 3968 y Fx(=)2554 3919 y(3)p 2554 3951 39 4 v 2554 4018 a(2)2602 3968 y Ft(m)2670 3932 y Fn(F)2670 3981 y Fm(2)2721 3968 y Ft(;)552 4184 y Fz(and)28 b(hence)g Fq(F)37 b Fk(2)25 b Fq(B)1178 4196 y Fy(2)1215 4184 y Fz(.)j(Therefore,)g Fq(A)1729 4196 y Fy(1)1791 4184 y Fk(\032)c Fq(B)1943 4196 y Fy(1)1999 4184 y Fk([)19 b Fq(B)2136 4196 y Fy(2)2174 4184 y Fz(.)28 b(Ho)n(w)n(ev)n(er,)f(since)h Fq(u)p Fz(\()p Fq(p)p Fz(\))g(and)h Fq(l)r Fz(\()p Fq(p)p Fz(\))f(are)523 4283 y(con)n(tin)n(uous)f(functions)i(of)f Fq(p)p Fz(,)h Fq(m)1560 4253 y Fl(F)1560 4304 y Fy(3)1643 4283 y Fz(can)f(tak)n(e)g (an)n(y)f(v)-5 b(alue)29 b(b)r(et)n(w)n(een)f(the)h(lo)n(w)n(er)e(and)h (upp)r(er)523 4383 y(b)r(ounds.)g(Therefore,)e Fq(B)1297 4395 y Fy(1)1353 4383 y Fk([)19 b Fq(B)1490 4395 y Fy(2)1550 4383 y Fk(\032)k Fq(A)1700 4395 y Fy(1)1737 4383 y Fz(.)1590 b Fk(u)-55 b(t)523 4541 y Fp(Lemma)30 b(4.)40 b Fo(L)l(et)30 b Fq(y)25 b Fk(\025)e Fz(0)29 b Fo(and)h Fq(k)c Fk(\025)d Fz(1)p Fo(.)30 b(Then,)762 4699 y Fx(\(1)17 b(+)g Ft(y)s Fx(\))1007 4634 y Fj(\002)1042 4699 y Fx(6\()p Ft(k)i Fx(+)e(1\)\()p Ft(k)i Fw(\000)e Fx(1\))1548 4667 y Fm(2)1583 4699 y Fx(\(1)g(+)g Ft(y)s Fx(\))f(+)h(3)p Ft(k)r Fx(\()p Ft(k)2061 4667 y Fm(2)2114 4699 y Fw(\000)f Fx(1\))p Ft(y)2299 4667 y Fm(2)2351 4699 y Fx(+)h Ft(k)2470 4667 y Fm(2)2504 4699 y Fx(\()p Ft(k)i Fx(+)e(2\))p Ft(y)2779 4667 y Fm(3)2814 4634 y Fj(\003)p 762 4743 2087 4 v 1198 4821 a Fx(\()p Ft(k)i Fx(+)d(1\))d([)q(2\()p Ft(k)19 b Fw(\000)e Fx(1\))g(+)g(2\()p Ft(k)j Fw(\000)d Fx(1\))p Ft(y)i Fx(+)e Ft(k)r(y)2322 4799 y Fm(2)2356 4821 y Fx(])2378 4785 y Fm(2)2879 4760 y Fw(\025)2970 4712 y Ft(k)j Fx(+)d(3)p 2970 4743 175 4 v 2970 4811 a Ft(k)j Fx(+)d(2)3155 4760 y Ft(:)p eop %%Page: 18 18 18 17 bop 523 448 a Fo(Pr)l(o)l(of.)43 b Fz(Let)623 610 y Ft(g)s Fx(\()p Ft(y)s(;)12 b(k)r Fx(\))23 b(=)g(\(1)18 b(+)e Ft(y)s Fx(\))1191 545 y Fj(\002)1225 610 y Fx(6\()p Ft(k)k Fx(+)c(1\)\()p Ft(k)k Fw(\000)d Fx(1\))1732 574 y Fm(2)1766 610 y Fx(\(1)h(+)f Ft(y)s Fx(\))f(+)h(3)p Ft(k)r Fx(\()p Ft(k)2245 574 y Fm(2)2297 610 y Fw(\000)g Fx(1\))p Ft(y)2483 574 y Fm(2)2534 610 y Fx(+)g Ft(k)2653 574 y Fm(2)2688 610 y Fx(\()p Ft(k)i Fx(+)e(2\))p Ft(y)2963 574 y Fm(3)2997 545 y Fj(\003)3044 610 y Fx(\()p Ft(k)j Fx(+)c(2\))945 755 y Fw(\000)p Fx(\()p Ft(k)j Fx(+)e(1\))1252 689 y Fj(\002)1286 755 y Fx(2\()p Ft(k)j Fw(\000)d Fx(1\))g(+)g(2\()p Ft(k)i Fw(\000)e Fx(1\))p Ft(y)j Fx(+)d Ft(k)r(y)2143 719 y Fm(2)2177 689 y Fj(\003)2212 707 y Fm(2)2259 755 y Fx(\()p Ft(k)i Fx(+)e(3\))862 887 y(=)23 b(\(2)18 b(+)e(4)p Ft(y)k Fx(+)d Ft(y)1321 851 y Fm(2)1355 887 y Fx(\))p Ft(k)1427 851 y Fm(4)1479 887 y Fw(\000)g Fx(2\(1)g(+)g(2)p Ft(y)j Fx(+)d(4)p Ft(y)2008 851 y Fm(2)2060 887 y Fx(+)f Ft(y)2177 851 y Fm(3)2211 887 y Fx(\))p Ft(k)2283 851 y Fm(3)2335 887 y Fw(\000)h Fx(\(2)g(+)g(4)p Ft(y)j Fx(+)d Ft(y)2788 851 y Fm(2)2839 887 y Fw(\000)g Fx(5)p Ft(y)2995 851 y Fm(3)3046 887 y Fw(\000)g Ft(y)3164 851 y Fm(4)3198 887 y Fx(\))p Ft(k)3270 851 y Fm(2)945 1003 y Fx(+2\(1)h(+)e Ft(y)s Fx(\)\(1)h(+)g Ft(y)j Fx(+)c(3)p Ft(y)1651 967 y Fm(2)1686 1003 y Fx(\))p Ft(k)r(:)551 1185 y Fz(W)-7 b(e)29 b(pro)n(v)n(e)d(that)j Fq(g)s Fz(\()p Fq(y)s(;)14 b(k)s Fz(\))23 b Fk(\025)h Fz(0.)k(Let)g Fq(h)p Fz(\()p Fq(y)s(;)14 b(k)s Fz(\))24 b(=)2049 1144 y Fl(g)r Fy(\()p Fl(y)r(;k)q Fy(\))p 2049 1166 179 4 v 2120 1213 a Fl(k)2238 1185 y Fz(.)k(It)h(su\016ces)e(to)h(pro)n(v)n(e)f Fq(h)p Fz(\()p Fq(y)s(;)14 b(k)s Fz(\))24 b Fk(\025)f Fz(0.)523 1334 y(Observ)n(e)j(that)1032 1293 y Fl(@)t(h)p Fy(\()p Fl(y)r(;k)q Fy(\))p 1032 1315 222 4 v 1105 1362 a Fl(@)t(k)1287 1334 y Fz(=)c(0)28 b(i\013)g Fq(k)e Fz(=)1709 1285 y Fy(2+4)p Fl(y)r Fy(+8)p Fl(y)1982 1260 y Fh(2)2014 1285 y Fy(+2)p Fl(y)2134 1260 y Fh(3)2167 1285 y Ff(\006)2219 1228 y Fk(p)p 2288 1228 123 4 v 57 x Fl(d)p Fy(\()p Fl(y)r Fy(\))p 1709 1315 701 4 v 1881 1362 a(3\(2+4)p Fl(y)r Fy(+)p Fl(y)2180 1346 y Fh(2)2212 1362 y Fy(\))2420 1334 y Fz(,)i(where)1044 1540 y Fq(d)p Fz(\()p Fq(y)s Fz(\))23 b(=)g(16)17 b(+)h(64)p Fq(y)j Fz(+)d(108)p Fq(y)1889 1506 y Fy(2)1942 1540 y Fz(+)g(66)p Fq(y)2153 1506 y Fy(3)2208 1540 y Fz(+)g(17)p Fq(y)2419 1506 y Fy(4)2473 1540 y Fz(+)g(5)p Fq(y)2642 1506 y Fy(5)2697 1540 y Fz(+)g Fq(y)2824 1506 y Fy(6)2861 1540 y Fq(:)523 1714 y Fz(Notice)28 b(that)g Fq(d)p Fz(\()p Fq(y)s Fz(\))23 b Fk(\025)g Fz(\(4)18 b(+)g(8)p Fq(y)j Fz(+)d Fq(y)1632 1684 y Fy(2)1687 1714 y Fz(+)g Fq(y)1814 1684 y Fy(3)1851 1714 y Fz(\))1883 1684 y Fy(2)1920 1714 y Fz(.)28 b(Th)n(us,)697 1889 y Fx(2)17 b(+)g(4)p Ft(y)j Fx(+)d(8)p Ft(y)1081 1858 y Fm(2)1132 1889 y Fx(+)g(2)p Ft(y)1288 1858 y Fm(3)1340 1889 y Fx(+)1417 1820 y Fj(p)p 1500 1820 141 4 v 69 x Ft(d)p Fx(\()p Ft(y)s Fx(\))p 697 1930 943 4 v 929 1998 a(3\(2)h(+)f(4)p Ft(y)j Fx(+)c Ft(y)1343 1976 y Fm(2)1377 1998 y Fx(\))1671 1947 y Fw(\025)1762 1898 y Fx(2)h(+)g(4)p Ft(y)j Fx(+)d(8)p Ft(y)2146 1866 y Fm(2)2197 1898 y Fx(+)g(2)p Ft(y)2353 1866 y Fm(3)2405 1898 y Fx(+)g(\(4)g(+)g(8)p Ft(y)j Fx(+)c Ft(y)2857 1866 y Fm(2)2909 1898 y Fx(+)g Ft(y)3026 1866 y Fm(3)3061 1898 y Fx(\))p 1762 1930 1329 4 v 2187 1998 a(3\(2)i(+)f(4)p Ft(y)i Fx(+)e Ft(y)2601 1976 y Fm(2)2635 1998 y Fx(\))3122 1947 y Fw(\025)k Fx(1)551 2207 y Fz(for)27 b Fq(y)f Fk(\025)c Fz(0.)27 b(Therefore,)g Fq(h)p Fz(\()p Fq(y)s(;)14 b(k)s Fz(\))27 b(is)h(minimized)g(when)g Fq(k)e Fz(=)2458 2158 y Fy(2+4)p Fl(y)r Fy(+8)p Fl(y)2731 2133 y Fh(2)2763 2158 y Fy(+2)p Fl(y)2883 2133 y Fh(3)2916 2158 y Fy(+)2967 2102 y Fk(p)p 3036 2102 123 4 v 56 x Fl(d)p Fy(\()p Fl(y)r Fy(\))p 2458 2188 700 4 v 2630 2236 a(3\(2+4)p Fl(y)r Fy(+)p Fl(y)2929 2219 y Fh(2)2961 2236 y Fy(\))3168 2207 y Fz(.)i(Let)876 2487 y Fq(s)p Fz(\()p Fq(y)s Fz(\))23 b(=)g Fq(h)1196 2345 y Fj( )1261 2487 y Fq(y)s(;)1352 2429 y Fz(2)18 b(+)g(4)p Fq(y)i Fz(+)e(8)p Fq(y)1767 2399 y Fy(2)1822 2429 y Fz(+)g(2)p Fq(y)1991 2399 y Fy(3)2046 2429 y Fz(+)2129 2358 y Fj(p)p 2212 2358 152 4 v 71 x Fq(d)p Fz(\()p Fq(y)s Fz(\))p 1352 2468 1012 4 v 1599 2544 a(3\(2)g(+)g(4)p Fq(y)j Fz(+)d Fq(y)2047 2520 y Fy(2)2084 2544 y Fz(\))2374 2345 y Fj(!)1046 2759 y Fz(=)1144 2703 y(2\(\(28)f(+)h(83)p Fq(y)i Fz(+)f(16)p Fq(y)1791 2673 y Fy(2)1845 2703 y Fz(+)f Fq(y)1972 2673 y Fy(3)2009 2703 y Fz(\))p Fq(d)p Fz(\()p Fq(y)s Fz(\))h Fk(\000)f Fq(d)p Fz(\()p Fq(y)s Fz(\))2455 2651 y Fh(3)p 2456 2660 29 4 v 2456 2693 a(2)2498 2703 y Fz(\))p 1144 2740 1387 4 v 1539 2816 a(27\(2)g(+)g(4)p Fq(y)i Fz(+)e Fq(y)2028 2792 y Fy(2)2065 2816 y Fz(\))2097 2792 y Fy(2)1134 2994 y Fk(\000)1209 2938 y Fz(12\(64)e(+)i(456)p Fq(y)i Fz(+)e(1260)p Fq(y)1990 2908 y Fy(2)2043 2938 y Fz(+)g(1655)p Fq(y)2338 2908 y Fy(3)2392 2938 y Fz(+)g(889)p Fq(y)2645 2908 y Fy(4)2699 2938 y Fz(+)g(147)p Fq(y)2952 2908 y Fy(5)2987 2938 y Fz(\))p 1209 2975 1812 4 v 1816 3051 a(27\(2)f(+)h(4)p Fq(y)j Fz(+)d Fq(y)2305 3027 y Fy(2)2342 3051 y Fz(\))2374 3027 y Fy(2)3029 2994 y Fq(:)523 3231 y Fz(It)26 b(su\016ces)g(to)g(pro)n(v)n(e)f Fq(s)p Fz(\()p Fq(y)s Fz(\))e Fk(\025)g Fz(0.)i(Let)i Fq(t)p Fz(\()p Fq(y)s Fz(\))c(=)g(27\(2)14 b(+)h(4)p Fq(y)j Fz(+)d Fq(y)2437 3201 y Fy(2)2473 3231 y Fz(\))2505 3201 y Fy(2)2543 3231 y Fq(s)p Fz(\()p Fq(y)s Fz(\).)26 b(It)h(su\016ces)f (to)g(pro)n(v)n(e)523 3322 y Fq(t)p Fz(\()p Fq(y)s Fz(\))34 b Fk(\025)e Fz(0.)i(Notice)g(that)g Fq(t)p Fz(\(0\))f(=)g(0.)g(Th)n (us,)h(it)g(su\016ces)f(to)h(pro)n(v)n(e)e Fq(t)2694 3292 y Ff(0)2717 3322 y Fz(\()p Fq(y)s Fz(\))i Fk(\025)f Fz(0)g(for)g Fq(y)j Fk(\025)d Fz(0.)523 3413 y(Ho)n(w)n(ev)n(er,)26 b Fq(t)911 3383 y Ff(0)934 3413 y Fz(\()p Fq(y)s Fz(\))d(=)1242 3381 y Fy(3)p 1163 3395 192 4 v 1163 3404 a Fk(p)p 1232 3404 123 4 v 57 x Fl(d)p Fy(\()p Fl(y)r Fy(\))1364 3413 y Fq(v)s Fz(\()p Fq(y)s Fz(\),)28 b(where)533 3619 y Ft(v)s Fx(\()p Ft(y)s Fx(\))556 3750 y(=)23 b(2\(128)c(+)e(688)p Ft(y)k Fx(+)16 b(1922)p Ft(y)1359 3714 y Fm(2)1412 3750 y Fx(+)h(3216)p Ft(y)1682 3714 y Fm(3)1735 3750 y Fx(+)f(3055)p Ft(y)2004 3714 y Fm(4)2057 3750 y Fx(+)h(1562)p Ft(y)2327 3714 y Fm(5)2380 3750 y Fx(+)g(420)p Ft(y)2612 3714 y Fm(6)2664 3750 y Fx(+)g(56)p Ft(y)2858 3714 y Fm(7)2910 3750 y Fx(+)f(3)p Ft(y)3065 3714 y Fm(8)3100 3750 y Fx(\))3130 3677 y Fj(p)p 3213 3677 141 4 v 73 x Ft(d)p Fx(\()p Ft(y)s Fx(\))639 3875 y Fw(\000)p Fx(\(64)i(+)f(216)p Ft(y)j Fx(+)d(198)p Ft(y)1304 3839 y Fm(2)1357 3875 y Fx(+)f(68)p Ft(y)1550 3839 y Fm(3)1602 3875 y Fx(+)h(25)p Ft(y)1796 3839 y Fm(4)1848 3875 y Fx(+)g(6)p Ft(y)2004 3839 y Fm(5)2038 3875 y Fx(\))p Ft(d)p Fx(\()p Ft(y)s Fx(\))556 3991 y Fw(\025)23 b Fx(2\(128)c(+)e(688)p Ft(y)k Fx(+)16 b(1922)p Ft(y)1359 3955 y Fm(2)1412 3991 y Fx(+)h(3216)p Ft(y)1682 3955 y Fm(3)1735 3991 y Fx(+)f(3055)p Ft(y)2004 3955 y Fm(4)2057 3991 y Fx(+)h(1562)p Ft(y)2327 3955 y Fm(5)2380 3991 y Fx(+)g(420)p Ft(y)2612 3955 y Fm(6)2664 3991 y Fx(+)g(56)p Ft(y)2858 3955 y Fm(7)2910 3991 y Fx(+)f(3)p Ft(y)3065 3955 y Fm(8)3100 3991 y Fx(\))h Fw(\001)639 4107 y Fx(\(4)h(+)f(8)p Ft(y)j Fx(+)c Ft(y)1015 4071 y Fm(2)1066 4107 y Fx(+)h Ft(y)1184 4071 y Fm(3)1218 4107 y Fx(\))g Fw(\000)g Fx(\(64)h(+)f(216)p Ft(y)j Fx(+)d(198)p Ft(y)1947 4071 y Fm(2)2000 4107 y Fx(+)f(68)p Ft(y)2193 4071 y Fm(3)2245 4107 y Fx(+)h(25)p Ft(y)2439 4071 y Fm(4)2491 4107 y Fx(+)g(6)p Ft(y)2647 4071 y Fm(5)2681 4107 y Fx(\))p Ft(d)p Fx(\()p Ft(y)s Fx(\))556 4224 y(=)23 b(3)p Ft(y)718 4188 y Fm(2)753 4224 y Fx(\(912)18 b(+)f(5600)i(+)d (13212)p Ft(y)1470 4188 y Fm(2)1523 4224 y Fx(+)h(15184)p Ft(y)1831 4188 y Fm(3)1884 4224 y Fx(+)g(9604)p Ft(y)2154 4188 y Fm(4)2207 4224 y Fx(+)g(3914)p Ft(y)2477 4188 y Fm(5)2529 4224 y Fx(+)g(1175)p Ft(y)2799 4188 y Fm(6)2852 4224 y Fx(+)g(235)p Ft(y)3084 4188 y Fm(7)3136 4224 y Fx(+)g(21)p Ft(y)3330 4188 y Fm(8)3365 4224 y Fx(\))556 4340 y Fw(\025)23 b Fx(0)p Ft(:)3350 4510 y Fk(u)-55 b(t)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF