(original) (raw)

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b(impro)m(v)m(emen)m(t)g(of)h(SRPT)e(o)m(v)m(er)i(PS)f(b)s (oth)f(in)g(terms)h(of)g(the)g(mean)h(resp)s(onse)e(time)h(and)347 2743 y(in)44 b(terms)h(of)g(mean)g(n)m(um)m(b)s(er)f(of)h(jobs)f(in)g (system)h(is)f(substan)m(tial)g(\(close)i(to)f(an)g(order)g(of)347 2856 y(magnitude\).)256 3029 y Fn(\017)h Ft(The)34 b(mean)h(impro)m(v)m (emen)m(ts)g(of)g(SRPT)e(o)m(v)m(er)j(PS)e(are)h(greatest)i(when)c(the) i(o)m(v)m(erload)h(is)e(close)347 3142 y(to)d(1.)256 3315 y Fn(\017)46 b Ft(Unfairness)34 b(to)i(big)f(jobs)f(under)g(SRPT)h (as)g(compared)h(with)e(PS)g(is)h(virtually)e(non-existen)m(t)347 3428 y(under)c(our)h(Bounded)g(P)m(areto)h(w)m(orkload.)256 3600 y Fn(\017)46 b Ft(The)27 b(higher)f(the)h(load)g(during)e(the)j (lo)m(w)f(load)g(p)s(erio)s(d,)e(the)j(b)s(etter)f(SRPT)f(lo)s(oks)h (with)f(resp)s(ect)347 3713 y(to)31 b(all)f(of)g(the)h(ab)s(o)m(v)m(e)g (metrics.)256 3886 y Fn(\017)46 b Ft(F)-8 b(or)39 b(distributions)c (with)i(a)h(ligh)m(ter)g(tail,)h(e.g.)65 b(the)39 b(exp)s(onen)m(tial)e (distribution,)g(the)h(results)347 3999 y(are)g(not)f(m)m(uc)m(h)g(w)m (orse.)61 b(In)37 b(fact)g(w)m(e)h(can)f(pro)m(v)m(e)h(that)g(for)f(an) g(exp)s(onen)m(tial)f(distribution,)f(if)347 4112 y(the)28 b(load)g(during)e(the)i(o)m(v)m(erload)h(p)s(erio)s(d)c(exceeds)k(2,)g (then)f(ev)m(ery)g(single)f(job)h(p)s(erforms)e(b)s(etter)347 4225 y(under)j(SRPT)g(than)h(under)f(PS,)h(in)f(exp)s(ectation.)261 4461 y(The)d(last)h(observ)-5 b(ation)26 b(ab)s(o)m(v)m(e)i(leads)e(us) f(to)j(sp)s(eculate)e(on)g(the)h(class)f(of)h(distributions)c(for)j (whic)m(h)120 4574 y(SRPT)d(impro)m(v)m(es)i(up)s(on)e(PS)g(under)g (transien)m(t)i(o)m(v)m(erload.)39 b(This)23 b(discussion)f(is)i(co)m (v)m(ered)i(in)d(Section)h(7,)120 4686 y(where)30 b(w)m(e)h(pro)m(vide) e(analyses,)i(observ)-5 b(ations,)30 b(and)f(op)s(en)h(problems)f(on)h (this)f(topic.)261 4867 y(The)23 b(ab)s(o)m(v)m(e)h(results)e(are)i (encouraging)f(with)f(resp)s(ect)h(to)h(the)f(p)s(oten)m(tial)g(real-w) m(orld)f(applicabilit)m(y)120 4980 y(of)31 b(SRPT)e(sc)m(heduling.)1867 5229 y(2)p eop %%Page: 3 5 3 4 bop 120 91 a Fp(2)135 b(Mo)t(del,)33 b(Relev)-7 b(an)l(t)31 b(Previous)g(W)-11 b(ork,)33 b(and)d(Simplifying)h(Assump-)322 241 y(tions)120 515 y Fm(2.1)112 b(Mo)s(del)120 754 y Ft(Throughout)26 b(this)g(pap)s(er)f(w)m(e)j(will)c(assume)i(an)h (M/G/1)i(queue.)39 b(The)27 b(job)f(sizes)h(will)d(b)s(e)i(assumed)g (to)120 867 y(b)s(e)32 b(indep)s(enden)m(t)f(and)h(iden)m(tically)f (distributed)f(with)i(c.d.f.)49 b Fl(F)13 b Ft(\()p Fl(x)p Ft(\))33 b(and)g(p.d.f.)48 b Fl(f)10 b Ft(\()p Fl(x)p Ft(\).)48 b(The)32 b(arriv)-5 b(al)120 980 y(pro)s(cess)37 b(will)e(consist)j(of)g(alternating)f(p)s(erio)s(ds)e(of)j(high)f(and)g (lo)m(w)g(loads.)63 b(During)36 b(the)i(high)e(load)120 1093 y(p)s(erio)s(d)29 b(\(also)i(called)f(the)h(\\HIGH")h(p)s(erio)s (d\),)d(jobs)i(arriv)m(e)f(with)g(mean)h(arriv)-5 b(al)29 b(rate)j Fl(\025)3170 1108 y Fk(h)3245 1093 y Ft(and)e(create)120 1206 y(a)h(load)f(of)g Fl(\032)543 1221 y Fk(h)613 1206 y Fl(>)25 b Ft(1.)41 b(The)30 b(high)f(load)h(p)s(erio)s(d)e(has)i (\014xed)g(duration)e Fl(t)2472 1221 y Fk(h)2517 1206 y Ft(.)41 b(During)29 b(the)h(lo)m(w)g(load)g(p)s(erio)s(d)120 1319 y(\(also)f(called)g(the)g(\\LO)m(W")h(p)s(erio)s(d\))e(jobs)g (arriv)m(e)h(with)f(mean)h(arriv)-5 b(al)28 b(rate)h Fl(\025)2819 1334 y Fk(l)2874 1319 y Ft(and)g(create)i(a)e(load)g(of) 120 1432 y Fl(\032)167 1447 y Fk(l)218 1432 y Fl(<)c Ft(1.)41 b(The)30 b(lo)m(w)g(load)g(p)s(erio)s(d)e(has)j(\014xed)e (duration)g Fl(t)2040 1447 y Fk(l)2066 1432 y Ft(.)261 1612 y(Let)i Fl(\032)f Ft(denote)h(the)g(a)m(v)m(erage)i(system)d (load.)41 b(Th)m(us)1400 1847 y Fl(\032)26 b Ft(=)1664 1785 y Fl(t)1697 1800 y Fk(h)p 1579 1826 248 4 v 1579 1909 a Fl(t)1612 1924 y Fk(h)1676 1909 y Ft(+)20 b Fl(t)1800 1924 y Fk(l)1836 1847 y Fl(\032)1883 1862 y Fk(h)1948 1847 y Ft(+)2143 1785 y Fl(t)2176 1800 y Fk(l)p 2049 1826 V 2049 1909 a Fl(t)2082 1924 y Fk(h)2147 1909 y Ft(+)g Fl(t)2271 1924 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y(LO)m(W)h(p)s(erio)s(d.)38 b(This)29 b(is)g(exp)s(ected)i(since)f (the)g(a)m(v)m(erage)j(load)d(is)g(less)f(than)h(1.)261 3013 y(Some)h(additional)d(notation:)41 b(The)30 b(n)m(um)m(b)s(er)f (of)h(jobs)g(in)f(the)i(system)f Fl(t)g Ft(time)g(after)h(the)g(onset)g (of)120 3126 y(the)h(HIGH)h(p)s(erio)s(d)d(will)g(b)s(e)h(denoted)i(b)m (y)f Fl(N)1682 3141 y Fk(h)1727 3126 y Ft(\()p Fl(t)p Ft(\).)47 b(Lik)m(ewise)31 b(the)h(n)m(um)m(b)s(er)f(of)i(jobs)e(in)g (the)i(system)f Fl(t)120 3239 y Ft(time)e(after)h(the)g(onset)f(of)h (the)g(LO)m(W)f(p)s(erio)s(d)e(will)g(b)s(e)i(denoted)g(b)m(y)g Fl(N)2566 3254 y Fk(l)2593 3239 y Ft(\()p Fl(t)p Ft(\).)120 3550 y Fm(2.2)112 b(Relev)-6 b(an)m(t)37 b(Previous)f(W)-9 b(ork)120 3789 y Ft(T)h(o)43 b(the)f(b)s(est)g(of)g(our)g(kno)m (wledge,)k(the)c(HIGH/LO)m(W)i(w)m(orkload)e(mo)s(del)f(with)g(general) h(job)g(size)120 3902 y(distribution)27 b(has)j(not)g(b)s(een)g (studied)f(under)g(either)g(PS)h(or)g(SRPT)g(sc)m(heduling.)261 4083 y(F)-8 b(or)33 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b(particular)f(constan)m(t)i(in)d(the)j(limit) c(as)k Fl(t)f Fn(!)g(1)p Ft(.)67 b(They)38 b(also)h(deriv)m(e)f(the)i (distribution)35 b(on)k(the)120 204 y(residual)28 b(job)i(sizes)g(in)f (the)i(limit)d(as)j Fl(t)25 b Fn(!)g(1)p Ft(.)261 385 y(Chen)47 b(et.)j(al.'s)e(w)m(ork)h([5])g(can)g(b)s(e)e(view)m(ed)h(as) h(analyzing)e(only)h(the)g(LO)m(W)h(p)s(erio)s(d)d(of)j(the)120 498 y(HIGH/LO)m(W)36 b(mo)s(del)d(under)g(PS)h(sc)m(heduling.)51 b(Sp)s(eci\014cally)-8 b(,)34 b(they)h(assume)f(a)h(PS)f(serv)m(er)h (running)120 611 y(under)28 b(load)i Fl(\032)621 626 y Fk(l)672 611 y Fl(<)25 b Ft(1)31 b(whic)m(h)d(b)s(egins)h(with)f(a)j (set)f(of)h(initial)c(jobs)i(distributed)e(according)j(to)h(an)f(arbi-) 120 724 y(trary)i(distribution.)41 b(They)31 b(obtain)g(a)h(\015uid)e (appro)m(ximation)g(on)i Fl(N)2501 739 y Fk(l)2527 724 y Ft(,)g(ho)m(w)m(ev)m(er)h(it)f(is)e(not)i(expressed)120 837 y(as)i(a)g(function)e(of)i Fl(t)p Ft(,)g(but)f(rather)h(as)g(a)g (function)e(of)i Fl(v)s Ft(\()p Fl(t)p Ft(\),)h(where)e Fl(v)s Ft(\()p Fl(t)p Ft(\))i(is)d(the)i(cum)m(ulativ)m(e)f(amoun)m(t) 120 949 y(of)e(pro)s(cessing)e(time)h(p)s(er)f(customer)i(allo)s(cated) f(b)m(y)g(the)h(serv)m(er)g(up)e(to)i(time)f Fl(t)p Ft(.)120 1257 y Fm(2.3)112 b(Simplifying)35 b(Assumptions)120 1496 y Ft(It)c(seems)h(natural)e(to)i(try)f(to)h(com)m(bine)f(the)g (results)f(in)g([14)q(])h(and)g([5])h(to)g(analyze)f(the)h(HIGH/LO)m(W) 120 1609 y(mo)s(del)k(under)f(PS.)i(Our)f(goal)i(is)e(to)i(understand)d (the)i Fo(me)-5 b(an)40 b(r)-5 b(esp)g(onse)41 b(time)e(for)g(a)g(job)g (of)g(size)g Fl(x)120 1722 y Ft(under)31 b(the)j(HIGH/LO)m(W)g(mo)s (del.)47 b(Unfortunately)-8 b(,)34 b(w)m(e)f(could)f(not)h(see)h(ho)m (w)f(to)h(directly)d(com)m(bine)120 1835 y(these)j(results)f(to)i (deriv)m(e)e(mean)i(resp)s(onse)e(time.)51 b(P)m(art)34 b(of)h(the)f(problem)e(is)h(that)i(the)f(Chen)f(result)120 1948 y(do)s(esn't)e(actually)f(compute)h Fl(N)1225 1963 y Fk(l)1252 1948 y Ft(\()p Fl(t)p Ft(\).)42 b(Another)31 b(part)g(of)g(the)g(problem)f(is)g(that)h(the)g(Jean-Marie)h(and)120 2061 y(Rob)s(ert)25 b(result)e(requires)h Fl(t)h Fn(!)g(1)p Ft(,)h(whereas)f(w)m(e)g(w)m(an)m(t)h(to)g(consider)e(\014nite)f Fl(t)2720 2076 y Fk(h)2790 2061 y Ft(and)h Fl(t)2994 2076 y Fk(l)3020 2061 y Ft(.)39 b(In)24 b(our)g(analysis)120 2174 y(it)32 b(w)m(as)g(therefore)h(necessary)f(for)g(us)f(to)i(resort) f(to)h(t)m(w)m(o)65 b Fo(simplifying)35 b(assumptions)i(under)d(PS)g (and)120 2287 y(under)f(SRPT)p Ft(:)231 2522 y(1.)357 2477 y Fk(N)413 2489 y Fj(h)454 2477 y Fi(\()p Fk(t)p Fi(\))p 357 2501 177 4 v 433 2553 a Fk(t)569 2522 y Ft(=)25 b(constan)m(t)92 b Fn(8)p Fl(t;)45 b Ft(for)31 b(all)e(sample)g(paths,) i(and)231 2701 y(2.)46 b Fl(N)420 2716 y Fk(l)446 2701 y Ft(\()p Fl(t)p Ft(\))31 b(is)f(the)g(same)h(for)f(all)f(sample)h (paths)p Fl(:)120 2936 y Ft(These)40 b(simplifying)c(assumptions)i (will)f(serv)m(e)k(sev)m(eral)f(purp)s(oses:)59 b(First,)42 b(they)e(will)d(allo)m(w)j(us)f(to)120 3049 y(obtain)29 b(an)g(appro)m(ximation)g(for)g(exp)s(ected)h(resp)s(onse)e(time)h(for) g(a)h(job)f(of)h(size)f Fl(x)h Ft(for)f(a)h(HIGH/LO)m(W)120 3162 y(pro)s(cess)e(under)f(PS)h(sc)m(heduling)e(\(see)k(Section)e (3\).)41 b(Second,)28 b(they)h(will)d(allo)m(w)i(us)g(to)h(obtain)f (appro)m(x-)120 3275 y(imations)35 b(for)g(all)g(p)s(erformance)g (metrics)h(of)g(in)m(terest)g(for)g(a)g(HIGH/LO)m(W)h(pro)s(cess)f (under)e(SRPT)120 3388 y(sc)m(heduling)20 b(\()h(exp)s(ected)h(resp)s (onse)f(time)g(for)g(a)h(job)f(of)h(size)f Fl(x)p Ft(,)i Fl(N)2310 3403 y Fk(h)2355 3388 y Ft(\()p Fl(t)p Ft(\),)h Fl(N)2580 3403 y Fk(l)2606 3388 y Ft(\()p Fl(t)p Ft(\)\))e(\(See)g (Section)g(4\).)38 b(Third,)120 3501 y(they)28 b(will)d(allo)m(w)i(us)h (to)g(compare)g(SRPT)f(and)g(PS)g(p)s(erformance)h(in)e(a)i(HIGH/LO)m (W)h(w)m(orkload)f(\(see)120 3614 y(Section)i(5\).)261 3794 y(Our)36 b(simplifying)d(assumptions)j(ma)m(y)i(seem)g(quite)f (strong.)61 b(Ho)m(w)m(ev)m(er,)42 b(in)36 b(Section)h(6)h(w)m(e)g (will)120 3907 y(compare)j(the)f(p)s(erformance)g(n)m(um)m(b)s(ers)f (deriv)m(ed)g(using)g(the)h(simplifying)c(assumptions)i(with)h(our)120 4020 y(sim)m(ulation)32 b(results)h(whic)m(h)f(mak)m(e)j(no)f (assumptions.)49 b(W)-8 b(e)36 b(will)31 b(\014nd)h(that)j(pro)m(vided) d Fl(t)3187 4035 y Fk(h)3266 4020 y Ft(and)h Fl(t)3479 4035 y Fk(l)3539 4020 y Ft(are)120 4133 y(not)h(to)s(o)g(small)e(\(at)i (least)g(100)h(times)e(the)g(mean)h(job)f(size\),)h(the)g(sim)m (ulation)e(n)m(um)m(b)s(ers)g(agree)i(with)120 4246 y(the)d (analytically-deriv)m(ed)d(n)m(um)m(b)s(ers.)120 4596 y Fp(3)135 b(Analytical)46 b(Results)g(for)f(PS)f(under)h(HIGH/LO)l(W)g (mo)t(del)120 4867 y Ft(The)32 b(goal)h(of)g(this)f(section)h(is)f(to)h (deriv)m(e)f(the)h(exp)s(ected)g(resp)s(onse)f(time)g(for)h(a)g(job)f (of)h(size)g Fl(x)f Ft(under)120 4980 y(the)38 b(HIGH/LO)m(W)h(mo)s (del)d(with)g(PS)h(sc)m(heduling.)60 b(Throughout)37 b(w)m(e)h(will)d(apply)h(our)h(simplifying)1867 5229 y(4)p eop %%Page: 5 7 5 6 bop 120 91 a Ft(assumptions)29 b(from)h(Section)g(2.)261 272 y(Before)f(w)m(e)f(b)s(egin)e(w)m(e)j(will)c(need)i(to)i(review)e (the)h(results)e(of)i(Jean-Marie)g(and)f(Rob)s(erts)g([14)r(])h(and)120 385 y(Chen)j(et.)47 b(al.)f([5)q(].)g(In)32 b(reviewing)e(these)j (results,)f(w)m(e)g(will)e(also)i(use)g(our)g(simplifying)c (assumptions)120 498 y(to)j(pro)m(vide)f(in)m(tuitiv)m(e)f(deriv)-5 b(ations)29 b(of)h(these)h(results)e(for)h(the)h(b)s(ene\014t)e(of)i (the)f(reader.)120 809 y Fm(3.1)112 b(Analysis)37 b(of)h(the)f(HIGH)f (load)i(p)s(erio)s(d)f(only)120 1048 y Fq(Lemma)c(0.1)46 b Fo(\(due)39 b(to)h([14)q(]\))f(Consider)h(an)g(M/G/1/PS)g(queue)e (with)i(lo)-5 b(ad)42 b Fl(\032)37 b(>)f Ft(1)k Fo(and)g(aver)-5 b(age)120 1161 y(arrival)39 b(r)-5 b(ate)38 b Fl(\025)662 1176 y Fk(h)707 1161 y Fo(.)55 b(L)-5 b(et)38 b Fl(N)1025 1176 y Fk(h)1069 1161 y Ft(\()p Fl(t)p Ft(\))g Fo(denote)g(the)g(numb) -5 b(er)37 b(of)h(jobs)f(in)g(the)h(system)g(at)f(time)h Fl(t)p Fo(.)55 b(Then)37 b(for)120 1274 y(almost)d(al)5 b(l)34 b(sample)g(p)-5 b(aths,)1614 1430 y Ft(lim)1594 1485 y Fk(t)p Fh(!1)1786 1368 y Fl(N)1859 1383 y Fk(h)1903 1368 y Ft(\()p Fl(t)p Ft(\))p 1786 1409 222 4 v 1880 1492 a Fl(t)2042 1430 y Ft(=)25 b Fl(a)1358 b Ft(\(1\))120 1628 y Fo(wher)-5 b(e)34 b Fl(a)e Fo(is)h(the)g(solution)h(to)f Fl(\025)1212 1643 y Fk(h)1257 1628 y Ft(\(1)21 b Fn(\000)1449 1559 y Fg(R)1504 1586 y Fh(1)1488 1656 y Fi(0)1594 1628 y Fl(f)10 b Ft(\()p Fl(x)p Ft(\))p Fl(e)1813 1595 y Fh(\000)p Fk(ax)1949 1628 y Fl(dx)p Ft(\))26 b(=)f Fl(a)p Fo(.)261 1908 y Fq(Pro)s(of)35 b(via)f(Simplifying)g(Assumptions)p Ft(:)41 b(The)29 b(follo)m(wing)f(is)h(a)h(rough)f(pro)s(of)g(based)h (on)f(our)120 2021 y(appro)m(ximations)34 b(made)h(in)f(Section)h(2.)56 b(Since)34 b(the)h(w)m(ork)g(in)f(the)i(system)f(b)m(y)g(time)g Fl(t)g Ft(increases)g(as)120 2134 y(\()p Fl(\032)202 2149 y Fk(h)259 2134 y Fn(\000)12 b Ft(1\))p Fl(t)g Ft(+)g Fl(o)p Ft(\()p Fl(t)p Ft(\),)28 b(it)e(is)f(not)h(di\016cult)f(to)i (see)f(that)h(for)f(almost)h(all)e(sample)g(paths,)i Fl(N)3042 2149 y Fk(h)3087 2134 y Ft(\()p Fl(t)p Ft(\))f(=)f Fl(at)12 b Ft(+)g Fl(o)p Ft(\()p Fl(t)p Ft(\),)120 2247 y(for)28 b(some)g Fl(a)p Ft(.)40 b(T)-8 b(o)28 b(determine)f Fl(a)p Ft(,)i(let)f(us)f(ignore)h(the)g Fl(o)p Ft(\()p Fl(t)p Ft(\))g(term)g(and)g(appro)m(ximate)g Fl(N)3079 2262 y Fk(h)3124 2247 y Ft(\()p Fl(t)p Ft(\))g(b)m(y)g Fl(at)p Ft(,)h(and)120 2360 y(assume)h(that)h(this)e(holds)g(for)h(all) g Fl(t)g Ft(and)g(for)g(all)f(sample)h(paths.)261 2541 y(No)m(w,)f(consider)e(a)g(job)h(of)f(size)h Fl(x)p Ft(,)g(arriving)d (at)k(time)e Fl(t)2126 2555 y Fk(a)2167 2541 y Ft(.)40 b(Then)26 b(the)i(service)f(receiv)m(ed)h(b)m(y)g(the)f(job)120 2654 y(b)m(y)j(time)g Fl(t)486 2668 y Fi(0)556 2654 y Ft(is)1468 2809 y Fl(S)5 b Ft(\()p Fl(t)1597 2823 y Fk(a)1639 2809 y Fl(;)15 b(t)1712 2823 y Fi(0)1752 2809 y Ft(\))25 b(=)1908 2694 y Fg(Z)1991 2721 y Fk(t)2016 2730 y Fs(0)1954 2883 y Fk(t)1979 2891 y Fj(a)2151 2748 y Fl(dt)p 2081 2788 V 2081 2872 a(N)2154 2887 y Fk(h)2198 2872 y Ft(\()p Fl(t)p Ft(\))3544 2809 y(\(2\))120 3025 y(Clearly)31 b(the)i(job)f(departs)g(at)h(time)g Fl(t)1435 3040 y Fk(d)1507 3025 y Ft(suc)m(h)f(that)i Fl(S)5 b Ft(\()p Fl(t)2043 3039 y Fk(a)2084 3025 y Fl(;)15 b(t)2157 3040 y Fk(d)2198 3025 y Ft(\))29 b(=)g Fl(x)p Ft(.)47 b(So,)33 b Fl(t)2673 3040 y Fk(d)2742 3025 y Ft(=)c Fl(t)2875 3039 y Fk(a)2916 3025 y Fl(e)2958 2992 y Fk(ax)3040 3025 y Ft(.)47 b(Th)m(us,)32 b(at)h(time)120 3138 y Fl(t)p Ft(,)d(if)g(a)g(job)h(of)f(size)g Fl(x)h Ft(arriving)d(at)j(time)f Fl(t)1572 3152 y Fk(a)1644 3138 y Ft(is)f(presen)m(t)i(in)e(the)h (system,)h(then)f Fl(t)2881 3152 y Fk(a)2948 3138 y Fn(\025)25 b Fl(te)3119 3105 y Fh(\000)p Fk(ax)3255 3138 y Ft(.)261 3318 y(No)m(w)f(the)g(exp)s(ected)g(n)m(um)m(b)s(er)e(of)i(jobs)f(of)h (size)f(b)s(et)m(w)m(een)h Fl(x)g Ft(and)f Fl(x)7 b Ft(+)g Fl(dx)22 b Ft(whic)m(h)h(arriv)m(e)g(during)e(time)120 3431 y([0)p Fl(;)15 b(t)p Ft(])28 b(is)d Fl(\025)456 3446 y Fk(h)501 3431 y Fl(tf)10 b Ft(\()p Fl(x)p Ft(\))p Fl(dx)p Ft(.)39 b(Since)26 b(the)g(arriv)-5 b(al)25 b(pro)s(cess)h(is)g (P)m(oisson)g(these)h(jobs)f(can)h(b)s(e)e(assumed)h(to)h(arriv)m(e)120 3544 y(uniformly)k(o)m(v)m(er)k([0)p Fl(;)15 b(t)p Ft(].)51 b(Th)m(us,)34 b(w)m(e)g(exp)s(ect)g(that)g(ab)s(out)g Fl(\025)2185 3559 y Fk(h)2230 3544 y Fl(t)p Ft(\(1)23 b Fn(\000)f Fl(e)2501 3511 y Fh(\000)p Fk(ax)2637 3544 y Ft(\))p Fl(f)10 b Ft(\()p Fl(x)p Ft(\))p Fl(dx)35 b Ft(of)e(these)h(jobs)g(will)120 3657 y(still)22 b(b)s(e)i(presen)m(t)g (in)f(the)h(system)h(at)g(time)f Fl(t)p Ft(.)38 b(Av)m(eraging)25 b(o)m(v)m(er)h(the)e(p)s(ossible)e(job)i(sizes)g Fl(x)p Ft(,)h(w)m(e)g(get)g(that)120 3770 y Fl(N)193 3785 y Fk(h)238 3770 y Ft(\()p Fl(t)p Ft(\))h(=)463 3701 y Fg(R)518 3727 y Fh(1)502 3798 y Fi(0)608 3770 y Fl(\025)661 3785 y Fk(h)706 3770 y Ft(\()p Fl(t)q Fn(\000)q Fl(te)922 3737 y Fh(\000)p Fk(ax)1057 3770 y Ft(\))p Fl(f)10 b Ft(\()p Fl(x)p Ft(\))p Fl(dx)p Ft(.)38 b(Equating)20 b Fl(N)1887 3785 y Fk(h)1932 3770 y Ft(\()p Fl(t)p Ft(\))h(to)h Fl(at)e Ft(giv)m(es)h(us)f Fl(a)26 b Ft(=)2748 3701 y Fg(R)2804 3727 y Fh(1)2788 3798 y Fi(0)2894 3770 y Fl(\025)2947 3785 y Fk(h)2992 3770 y Ft(\(1)q Fn(\000)q Fl(e)3187 3737 y Fh(\000)p Fk(ax)3323 3770 y Ft(\))p Fl(f)10 b Ft(\()p Fl(x)p Ft(\))p Fl(dx)p Ft(.)120 3883 y Ff(2)p Ft(.)261 4064 y(Since)21 b(w)m(e)i(will)c(b)s(e)j(in)m(terested)g(in)f (ev)-5 b(aluating)21 b(v)-5 b(arious)21 b(metrics)h(for)g(PS,)g(w)m(e)g (will)e(mak)m(e)j(the)f(results)120 4176 y(more)40 b(compact.)72 b(Let)41 b Fl(L)1021 4190 y Fk(g)1061 4176 y Ft(\()p Fl(s)p Ft(\))g(denote)g(the)f(Laplace)h(transform)e(of)i(a)g(function)e Fl(g)s Ft(,)k Fo(i.e.)70 b Fl(L)3394 4190 y Fk(g)3434 4176 y Ft(\()p Fl(s)p Ft(\))42 b(=)120 4220 y Fg(R)175 4247 y Fh(1)159 4317 y Fi(0)265 4289 y Fl(g)s Ft(\()p Fl(x)p Ft(\))p Fl(e)475 4256 y Fh(\000)p Fk(sx)608 4289 y Fl(dx)p Ft(.)k(Th)m(us)30 b(w)m(e)j(observ)m(e)f(that)h Fl(a)e Ft(satis\014es)h Fl(a)27 b Ft(=)h Fl(\025)2312 4304 y Fk(h)2357 4289 y Ft(\(1)22 b Fn(\000)f Fl(L)2613 4304 y Fk(f)2658 4289 y Ft(\()p Fl(a)p Ft(\)\).)46 b(Moreo)m(v)m(er)34 b(observing)120 4402 y(that)39 b Fl(L)p 387 4378 59 4 v 31 x Fk(F)445 4402 y Ft(\()p Fl(s)p Ft(\))g(=)e(\(1)26 b Fn(\000)g Fl(L)970 4417 y Fk(f)1015 4402 y Ft(\()p Fl(s)p Ft(\)\))p Fl(=s)p Ft(,)41 b(w)m(e)d(can)h(write)e Fl(a)h Ft(in)f(the)h(follo)m(wing)e(form)i(whic)m(h)f(will)e(b)s(e)j (useful)120 4515 y(later.)1651 4644 y Fl(L)p 1713 4619 V 30 x Fk(F)1771 4644 y Ft(\()p Fl(a)p Ft(\))26 b(=)2047 4582 y(1)p 2021 4622 98 4 v 2021 4706 a Fl(\025)2074 4721 y Fk(h)3544 4644 y Ft(\(3\))1867 5229 y(5)p eop %%Page: 6 8 6 7 bop 261 91 a Ft(W)-8 b(e)35 b(no)m(w)g(obtain)e(an)h(appro)m (ximation)f(for)h(the)g(n)m(um)m(b)s(er)f(of)h(jobs)g(with)f(remaining) f(size)i Fl(>)d(y)37 b Ft(at)120 204 y(the)31 b(end)e(of)i(the)f(HIGH)h (p)s(erio)s(d,)e(using)f(our)i(simplifying)c(assumptions)2669 171 y Fi(1)2709 204 y Ft(.)261 385 y(Consider)h(a)i(job)g(of)g (\(original\))e(size)i Fl(z)t Ft(.)40 b(Arguing)28 b(as)h(ab)s(o)m(v)m (e,)i(this)c(job)i(will)d(ha)m(v)m(e)k(remaining)d(size)120 498 y Fl(>)e(y)33 b Ft(at)e(time)f Fl(t)645 513 y Fk(h)720 498 y Ft(i\013)g(it's)g(arriv)-5 b(al)29 b(time,)h Fl(t)1532 512 y Fk(a)1573 498 y Ft(,)h(is)e(suc)m(h)h(that)h Fl(t)2155 512 y Fk(a)2222 498 y Fl(>)25 b(t)2351 513 y Fk(h)2396 498 y Fl(e)2438 465 y Fh(\000)p Fk(a)p Fi(\()p Fk(z)s Fh(\000)p Fk(y)r Fi(\))261 678 y Ft(Th)m(us)h(the)g(total)i(n)m(um)m(b) s(er)d(of)i(jobs)f(of)g(size)h(\()p Fl(z)t(;)15 b(z)j Ft(+)13 b Fl(dz)t Ft(\))26 b(whic)m(h)g(ha)m(v)m(e)i(remaining)c(size)j Fl(>)e(y)k Ft(at)e(time)120 791 y Fl(t)153 806 y Fk(h)228 791 y Ft(will)h(b)s(e)1395 904 y Fl(\025)1448 919 y Fk(h)1493 904 y Fl(t)1526 919 y Fk(h)1571 904 y Fl(f)10 b Ft(\()p Fl(z)t Ft(\)\(1)21 b Fn(\000)f Fl(e)1976 867 y Fh(\000)p Fk(a)p Fi(\()p Fk(z)s Fh(\000)p Fk(y)r Fi(\))2256 904 y Ft(\))p Fl(dz)1164 b Ft(\(4\))120 1071 y(In)m(tegrating)35 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2056 y Ft(\()p Fl(y)s Ft(\))26 b(=)1530 1912 y Fg(R)1585 1938 y Fh(1)1569 2009 y Fk(y)1675 1981 y Fl(\025)1728 1996 y Fk(h)1773 1981 y Fl(f)10 b Ft(\()p Fl(z)t Ft(\)\(1)21 b Fn(\000)f Fl(e)2178 1948 y Fh(\000)p Fk(a)p Fi(\()p Fk(z)s Fh(\000)p Fk(y)r Fi(\))2457 1981 y Ft(\))p Fl(dz)p 1530 2035 1057 4 v 2034 2118 a(a)3544 2056 y Ft(\(6\))120 2297 y(After)38 b(some)g(manipulation,)f(w)m(e)i (observ)m(e)f(that)h(the)f(Laplace)g(T)-8 b(ransform)36 b(of)p 2932 2224 97 4 v 38 w Fl(F)2990 2311 y Fk(r)3067 2297 y Ft(can)i(b)s(e)f(written)120 2409 y(simply)28 b(as)1446 2569 y Fl(L)p 1508 2544 84 4 v 30 x Fk(F)1553 2607 y Fj(r)1591 2569 y Ft(\()p Fl(s)p Ft(\))e(=)1836 2504 y(1)20 b Fn(\000)g Fl(\025)2045 2519 y Fk(h)2090 2504 y Fl(L)p 2152 2479 59 4 v 30 x Fk(F)2211 2504 y Ft(\()p Fl(s)p Ft(\))p 1836 2548 489 4 v 1979 2631 a Fl(s)g Fn(\000)g Fl(a)3544 2569 y Ft(\(7\))120 2911 y Fm(3.2)112 b(Analysis)37 b(of)h(the)f(LO)m(W)g(load)h(p)s(erio)s(d)f (only)120 3150 y Ft(W)-8 b(e)25 b(no)m(w)f(consider)f(just)g(the)h(lo)m (w)g(load)f(p)s(erio)s(d)f(\()p Fl(\032)1831 3165 y Fk(l)1882 3150 y Fl(<)j Ft(1\),)i(starting)c(with)g Fl(N)2716 3165 y Fk(l)2742 3150 y Ft(\(0\))i(jobs)e(with)g(remaining)120 3263 y(sizes)31 b(distributed)d(according)j(to)h(c.d.f.)43 b Fl(F)1612 3277 y Fk(r)1650 3263 y Ft(.)g(New)32 b(jobs)e(arriv)m(e)h (in)m(to)g(the)g(system)h(at)f(rate)h Fl(\025)3370 3278 y Fk(l)3427 3263 y Ft(where)120 3376 y(the)37 b(new)g(job)g(sizes)g(ha) m(v)m(e)i(c.d.f.)61 b Fl(F)13 b Ft(.)62 b(W)-8 b(e)38 b(no)m(w)g(use)f(our)f(simplifying)e(assumptions)h(to)j(deriv)m(e)f(an) 120 3489 y(expression)27 b(for)h(the)g(n)m(um)m(b)s(er)f(of)i(jobs)f (in)f(the)h(system.)40 b(Chen,)28 b(Kella)f(and)h(W)-8 b(eiss)28 b([5)q(])h(ha)m(v)m(e)g(analyzed)120 3602 y(the)k(lo)m (w-load)g(only)f(system)g(and)h(obtained)f(a)h(\015uid)e(limit)f (expression)i(for)g(the)h(n)m(um)m(b)s(er)f(of)h(jobs)f(in)120 3715 y(the)f(system.)40 b(The)30 b([5)q(])h(result)e(is)g(similar)f(to) 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b(job)h(will)120 4528 y(complete)g(at)g(time)f Fl(t)873 4543 y Fk(d)950 4528 y Ft(suc)m(h)g(that)i(the)e(service)h(receiv)m(ed)f(b)m(y)h(the)g (job)f(during)e(time)i Fl(t)3180 4542 y Fk(a)3258 4528 y Ft(to)h Fl(t)3408 4543 y Fk(d)3485 4528 y Ft(is)f Fl(x)p Ft(.)120 4641 y(No)m(w,)31 b(the)g(service)f(receiv)m(ed)h(b)m(y)f(the) h(job)f(during)e(time)i Fl(t)2119 4655 y Fk(a)2191 4641 y Ft(to)h Fl(t)2335 4656 y Fk(d)2405 4641 y Ft(under)e(PS,)h(will)e(b)s (e)3127 4572 y Fg(R)3182 4598 y Fk(t)3207 4610 y Fj(d)3166 4668 y Fk(t)3191 4676 y Fj(a)3322 4605 y Fk(dt)p 3273 4620 161 4 v 3273 4673 a(N)3329 4685 y Fj(l)3353 4673 y Fi(\()p Fk(t)p Fi(\))3443 4641 y Ft(.)p 120 4718 1416 4 v 224 4771 a Fs(1)258 4803 y Fr([14)q(])e(also)h(study)e(the)g (problem)g(of)i(residual)f(job)g(sizes.)1867 5229 y Ft(6)p eop %%Page: 7 9 7 8 bop 261 91 a Ft(Let)21 b(us)f(de\014ne)f Fl(v)s Ft(\()p Fl(t)p Ft(\))26 b(=)1045 22 y Fg(R)1101 49 y Fk(t)1084 119 y Fi(0)1205 51 y Fk(dy)p 1155 70 173 4 v 1155 123 a(N)1211 135 y Fj(l)1235 123 y Fi(\()p Fk(y)r Fi(\))1338 91 y Ft(,)c(where)e Fl(v)s Ft(\()p Fl(t)p Ft(\))h(is)e(the)i(cum)m (ulativ)m(e)f(service)g(p)s(er)f(customer)i(allo)s(cated)120 204 y(b)m(y)26 b(the)h(serv)m(er)g(up)e(to)j(time)e Fl(t)p Ft(.)39 b(Th)m(us)25 b(a)i(job)f(of)h(size)f Fl(x)h Ft(arriving)d(at)k (time)e Fl(t)2666 218 y Fk(a)2734 204 y Ft(is)f(presen)m(t)i(in)e(the)i (system)120 317 y(at)32 b(time)f Fl(t)g Ft(i\013)f Fl(v)s Ft(\()p Fl(t)p Ft(\))22 b Fn(\000)e Fl(v)s Ft(\()p Fl(t)991 331 y Fk(a)1033 317 y Ft(\))27 b Fl(<)f(x)p Ft(.)43 b(Th)m(us)30 b(the)i(n)m(um)m(b)s(er)e(of)h(T)m(yp)s(e)f(1)i(jobs)f(still)e(presen)m (t)i(at)h(time)f Fl(t)g Ft(will)120 430 y(b)s(e)36 b Fl(N)323 445 y Fk(l)349 430 y Ft(\(0\))p 464 358 97 4 v Fl(F)522 444 y Fk(r)561 430 y Ft(\()p Fl(v)s Ft(\()p Fl(t)p Ft(\)\).)60 b(T)-8 b(o)37 b(obtain)e(the)i(n)m(um)m(b)s(er)e(of) h(T)m(yp)s(e)g(2)g(jobs,)i(consider)d(the)i(jobs)e(whic)m(h)g(arriv)m (e)120 543 y(during)30 b(time)h Fl(y)k Ft(and)c Fl(y)24 b Ft(+)d Fl(dy)s Ft(.)45 b(There)31 b(will)e(b)s(e)i(appro)m(ximately)g Fl(\025)2417 558 y Fk(l)2443 543 y Fl(dy)k Ft(suc)m(h)d(jobs.)44 b(Out)32 b(of)f(these)i(the)120 656 y(n)m(um)m(b)s(er)f(still)f(presen) m(t)i(at)h(time)f Fl(t)g Ft(will)d(b)s(e)j Fl(\025)1697 671 y Fk(l)1723 656 y Fl(dy)p 1818 584 72 4 v 3 w(F)13 b Ft(\()p Fl(v)s Ft(\()p Fl(t)p Ft(\))24 b Fn(\000)d Fl(v)s Ft(\()p Fl(y)s Ft(\)\).)51 b(In)m(tegrating)33 b(o)m(v)m(er)i Fl(y)h Ft(from)d(0)g(to)h Fl(t)120 769 y Ft(will)26 b(giv)m(e)i(us)g(the)h(total)g(n)m(um)m(b)s(er)e(of)h(T)m (yp)s(e)g(2)g(jobs)g(still)f(presen)m(t)h(at)h(time)f Fl(t)p Ft(.)40 b(Adding)26 b(the)j(n)m(um)m(b)s(er)e(of)120 882 y(T)m(yp)s(e)j(1)h(and)e(T)m(yp)s(e)h(2)h(jobs)f(w)m(e)h(th)m(us)f (obtain,)970 1114 y Fl(N)1043 1129 y Fk(l)1069 1114 y Ft(\()p Fl(t)p Ft(\))c(=)f Fl(N)1367 1129 y Fk(l)1393 1114 y Ft(\(0\))p 1508 1042 97 4 v Fl(F)1566 1128 y Fk(r)1606 1114 y Ft(\()p Fl(v)s Ft(\()p Fl(t)p Ft(\)\))c(+)1938 999 y Fg(Z)2021 1025 y Fk(t)1984 1188 y Fi(0)2066 1114 y Fl(\025)2119 1129 y Fk(l)p 2145 1042 72 4 v 2145 1114 a Fl(F)13 b Ft(\()p Fl(v)s Ft(\()p Fl(t)p Ft(\))22 b Fn(\000)d Fl(v)s Ft(\()p Fl(y)s Ft(\)\))p Fl(dy)739 b Ft(\(8\))120 1336 y(W)-8 b(e)31 b(do)f(not)g(kno)m(w)g(of)g(a)g(w)m(a)m (y)h(to)f(solv)m(e)h(Equation)e(8)h(directly)f(to)h(obtain)g(the)g(n)m (um)m(b)s(er)e(in)h(system)h(as)120 1449 y(a)35 b(function)e(of)h (time.)53 b(Ho)m(w)m(ev)m(er,)37 b(w)m(e)e(can)g(obtain)e(this)h (indirectly)d(b)m(y)k(solving)e(for)h(the)g(n)m(um)m(b)s(er)f(in)120 1562 y(system)24 b(and)g(solving)f(for)h Fl(t)g Ft(as)h(a)g(function)e (of)h Fl(v)s Ft(.)39 b(W)-8 b(e)26 b(will)21 b(denote)k(these)g(b)m(y)f Fl(N)2843 1529 y Fk(v)2833 1588 y(l)2908 1562 y Ft(and)g Fl(t)3112 1529 y Fk(v)3177 1562 y Ft(resp)s(ectiv)m(ely)-8 b(.)261 1742 y(Observing)29 b(that)i Fl(v)s Ft(\(0\))26 b(=)f(0)31 b(and)1451 1707 y Fk(d)p 1438 1722 62 4 v 1438 1774 a(dt)1510 1742 y Fl(v)s Ft(\()p Fl(t)p Ft(\))26 b(=)1836 1707 y Fi(1)p 1792 1722 124 4 v 1792 1775 a Fk(n)p Fi(\()p Fk(t)p Fi(\))1925 1742 y Ft(,)k(Equation)g(8)h(can)f(b)s (e)g(written)g(as)948 2015 y Fl(N)1031 1977 y Fk(v)1021 2037 y(l)1072 2015 y Ft(\()p Fl(v)s Ft(\))c(=)f Fl(N)1384 2030 y Fk(l)1411 2015 y Ft(\(0\))p 1526 1943 97 4 v Fl(F)1584 2029 y Fk(r)1623 2015 y Ft(\()p Fl(v)s Ft(\))c(+)1852 1900 y Fg(Z)1935 1926 y Fk(v)r Fi(\()p Fk(t)p Fi(\))1898 2088 y(0)2072 2015 y Fl(\025)2125 2030 y Fk(l)p 2151 1943 72 4 v 2151 2015 a Fl(F)13 b Ft(\()p Fl(v)24 b Fn(\000)c Fl(z)t Ft(\))p Fl(N)2580 1977 y Fk(v)2570 2037 y(l)2621 2015 y Ft(\()p Fl(z)t Ft(\))p Fl(dz)718 b Ft(\(9\))120 2237 y(W)-8 b(e)32 b(can)e(no)m(w)h(solv)m(e)f(Equation)g(9.)41 b(Using)30 b(Laplace)g(T)-8 b(ransforms)30 b(this)f(giv)m(es)i(us)1114 2426 y Fl(L)1176 2440 y Fk(N)1239 2418 y Fj(v)1232 2465 y(l)1280 2426 y Ft(\()p Fl(s)p Ft(\))26 b(=)f Fl(N)1588 2441 y Fk(l)1614 2426 y Ft(\(0\))p Fl(L)p 1791 2402 84 4 v 31 x Fk(F)1836 2465 y Fj(r)1875 2426 y Ft(\()p Fl(s)p Ft(\))c(+)f Fl(\025L)p 2215 2402 59 4 v 31 x Fk(F)2273 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3802 y Ft(\()p Fl(s)p Ft(\)\))3498 3740 y(\(13\))120 4108 y Fm(3.3)112 b(Analysis)37 b(of)h(Resp)s(onse)g(times)e(for)h(the)g (HIGH/LO)m(W)g(mo)s(del)120 4348 y Ft(Using)g(the)h(results)f(ab)s(o)m (v)m(e)i(w)m(e)g(can)f(no)m(w)g(obtain)g(the)g(expressions)f(for)g(the) i(resp)s(onse)e(times)g(as)i(a)120 4461 y(function)29 b(of)i(job)f(size)g(under)f(transien)m(t)h(o)m(v)m(erload.)261 4641 y(W)-8 b(e)27 b(classify)d(the)i(jobs)f(in)m(to)h(3)f(t)m(yp)s (es.)40 b(Jobs)25 b(whic)m(h)f(arriv)m(e)h(during)e(the)j(HIGH)g(p)s (erio)s(d)d(and)i(\014nish)120 4754 y(during)h(the)i(HIGH)h(p)s(erio)s (d)d(itself)h(are)h(t)m(yp)s(e)h(1)f(jobs.)40 b(Jobs)27 b(whic)m(h)g(arriv)m(e)h(during)e(the)j(HIGH)f(p)s(erio)s(d)120 4867 y(but)i(\014nish)e(in)i(the)g(LO)m(W)h(p)s(erio)s(d)e(are)i(t)m (yp)s(e)g(2)g(jobs.)41 b(Jobs)30 b(whic)m(h)g(arriv)m(e)g(during)e(the) j(LO)m(W)g(p)s(erio)s(d)120 4980 y(and)f(\014nish)e(during)g(the)i (same)h(LO)m(W)g(p)s(erio)s(d)d(are)j(t)m(yp)s(e)f(3)h(jobs.)1867 5229 y(7)p eop %%Page: 8 10 8 9 bop 261 91 a Ft(Note,)32 b(that)f(since)e(the)h(a)m(v)m(erage)j (load)d(is)f(less)g(than)h(1,)h(v)m(ery)g(few)f(jobs)f(whic)m(h)g (arriv)m(e)h(during)e(the)120 204 y(LO)m(W)g(p)s(erio)s(d)c(will)h(con) m(tin)m(ue)i(during)e(the)j(next)f(HIGH)h(p)s(erio)s(d.)37 b(Th)m(us)26 b(w)m(e)i(consider)e(only)g(the)i(three)120 317 y(t)m(yp)s(es)i(of)h(jobs)f(men)m(tioned)g(ab)s(o)m(v)m(e.)261 498 y(Supp)s(ose)36 b Fl(J)46 b Ft(is)36 b(a)i(job)f(of)g(size)g Fl(x)p Ft(.)61 b(If)37 b(it)g(is)f(a)h(t)m(yp)s(e)h(1)f(job,)i(then)e (w)m(e)h(kno)m(w)f(that)h(it)f(m)m(ust)g(ha)m(v)m(e)120 611 y(arriv)m(ed)e(during)f(time)i(0)h(to)f Fl(t)1172 626 y Fk(h)1217 611 y Fl(e)1259 578 y Fh(\000)p Fk(ax)1396 611 y Ft(.)58 b(Assuming)34 b(it)i(arriv)m(es)f(uniformly)f(during)f (this)i(in)m(terv)-5 b(al,)37 b(w)m(e)120 724 y(obtain)1455 920 y Fl(E)5 b Ft([)p Fl(T)13 b Ft(\()p Fl(x)p Ft(\))p Fn(j)p Ft(Job)31 b(is)f(of)g(t)m(yp)s(e)h(1])1301 1120 y(=)1570 1058 y(1)p 1465 1099 257 4 v 1465 1182 a Fl(t)1498 1197 y Fk(h)1543 1182 y Fl(e)1585 1156 y Fh(\000)p Fk(ax)1746 1005 y Fg(Z)1829 1031 y Fk(t)1854 1043 y Fj(h)1895 1031 y Fk(e)1928 1007 y Fe(\000)p Fj(ax)1792 1193 y Fi(0)2068 1120 y Fl(y)s Ft(\()p Fl(e)2193 1082 y Fk(ax)2295 1120 y Fn(\000)20 b Ft(1\))p Fl(dy)1301 1350 y Ft(=)1465 1289 y(1)p 1465 1329 46 4 v 1465 1412 a(2)1520 1350 y Fl(t)1553 1365 y Fk(h)1598 1350 y Ft(\(1)h Fn(\000)f Fl(e)1832 1313 y Fh(\000)p Fk(ax)1969 1350 y Ft(\))1494 b(\(14\))261 1645 y(Supp)s(ose)25 b Fl(J)35 b Ft(is)25 b(of)i(t)m(yp)s(e)f(2,)i (then)e(it)f(m)m(ust)i(ha)m(v)m(e)g(arriv)m(ed)f(during)e(time)h Fl(t)2724 1660 y Fk(h)2769 1645 y Fl(e)2811 1612 y Fh(\000)p Fk(ax)2974 1645 y Ft(to)i Fl(t)3114 1660 y Fk(h)3159 1645 y Ft(.)39 b(If)26 b(it)f(arriv)m(es)120 1758 y(at)35 b(time)f Fl(y)s Ft(,)i(then)e(it)g(sp)s(ends)e Fl(t)1194 1773 y Fk(h)1262 1758 y Fn(\000)22 b Fl(y)37 b Ft(time)d(during)e(the)j (HIGH)g(p)s(erio)s(d)d(and)i(its)f(remaining)g(size)h(at)120 1871 y(the)27 b(b)s(eginning)e(of)i(the)g(LO)m(W)h(p)s(erio)s(d)d(is)h Fl(r)s Ft(\()p Fl(y)s Ft(\))f(=)g Fl(x)14 b Fn(\000)1998 1835 y Fi(1)p 1997 1850 38 4 v 1997 1902 a Fk(a)2060 1871 y Ft(log)2202 1831 y Fk(t)2227 1843 y Fj(h)p 2202 1850 66 4 v 2217 1902 a Fk(y)2278 1871 y Ft(.)40 b(Observ)m(e)27 b(that)h(b)m(y)f(Equation)f(13)i(this)120 1984 y(job)i(\014nishes)e(at) j(time)f Fl(t)944 1951 y Fk(v)985 1984 y Ft(\()p Fl(r)s Ft(\()p Fl(y)s Ft(\)\).)42 b(Th)m(us)29 b(w)m(e)i(obtain,)1177 2180 y Fl(E)5 b Ft([)p Fl(T)13 b Ft(\()p Fl(x)p Ft(\))p Fn(j)p Ft(Job)32 b(is)d(of)i(t)m(yp)s(e)f(2)q(])1024 2365 y(=)1251 2304 y(1)p 1187 2344 173 4 v 1187 2427 a Fl(k)s Ft(\()p Fl(x)p Ft(\))1385 2250 y Fg(Z)1468 2276 y Fk(t)1493 2288 y Fj(h)1431 2439 y Fk(t)1456 2451 y Fj(h)1497 2439 y Fk(e)1530 2420 y Fe(\000)p Fj(ax)1655 2365 y Ft([\()p Fl(t)1748 2380 y Fk(h)1813 2365 y Fn(\000)20 b Fl(y)s Ft(\))g(+)g Fl(t)2131 2328 y Fk(v)2172 2365 y Ft(\()p Fl(x)g Fn(\000)2382 2304 y Ft(1)p 2380 2344 49 4 v 2380 2427 a Fl(a)2454 2365 y Ft(log)2596 2304 y Fl(t)2629 2319 y Fk(h)p 2596 2344 78 4 v 2611 2427 a Fl(y)2684 2365 y Ft(\)])p Fl(dy)1024 2613 y Ft(=)1187 2552 y(1)p 1187 2592 46 4 v 1187 2676 a(2)1243 2613 y Fl(k)s Ft(\()p Fl(x)p Ft(\))h(+)1600 2552 y(1)p 1537 2592 173 4 v 1537 2676 a Fl(k)s Ft(\()p Fl(x)p Ft(\))1735 2498 y Fg(Z)1818 2525 y Fk(t)1843 2537 y Fj(h)1781 2687 y Fk(t)1806 2699 y Fj(h)1847 2687 y Fk(e)1880 2668 y Fe(\000)p Fj(ax)2019 2613 y Fl(t)2052 2576 y Fk(v)2093 2613 y Ft(\()p Fl(x)f Fn(\000)2302 2552 y Ft(1)p 2301 2592 49 4 v 2301 2676 a Fl(a)2374 2613 y Ft(log)2517 2552 y Fl(t)2550 2567 y Fk(h)p 2517 2592 78 4 v 2532 2676 a Fl(y)2604 2613 y Ft(\))p Fl(dy)1024 2852 y Ft(=)1187 2790 y(1)p 1187 2831 46 4 v 1187 2914 a(2)1243 2852 y Fl(k)s Ft(\()p Fl(x)p Ft(\))h(+)1681 2790 y(1)p 1537 2831 336 4 v 1537 2914 a(1)f Fn(\000)g Fl(e)1735 2888 y Fh(\000)p Fk(ax)1897 2737 y Fg(Z)1980 2763 y Fk(x)1943 2926 y Fi(0)2039 2852 y Fl(at)2120 2814 y Fk(v)2161 2852 y Ft(\()p Fl(x)g Fn(\000)g Fl(z)t Ft(\))p Fl(e)2482 2814 y Fh(\000)p Fk(az)2615 2852 y Fl(dz)794 b Ft(\(15\))120 3090 y(where)30 b Fl(k)s Ft(\()p Fl(x)p Ft(\))c(=)f Fl(t)710 3105 y Fk(h)755 3090 y Ft(\(1)c Fn(\000)f Fl(e)989 3057 y Fh(\000)p Fk(ax)1125 3090 y Ft(\).)261 3271 y(Finally)34 b(w)m(e)h(consider)g(the)h(case)g(when)f Fl(J)44 b Ft(is)35 b(a)g(job)h(of)f(t)m(yp)s(e)h(3.)56 b(T)-8 b(o)36 b(do)f(this,)h(w)m(e) g(need)f(to)i(\014rst)120 3384 y(digress)29 b(and)h(de\014ne)g(quan)m (tit)m(y)g Fl(t)1255 3398 y Fk(w)1312 3384 y Ft(.)261 3564 y(Consider)g(the)i(amoun)m(t)g(of)f(w)m(ork)h(accum)m(ulated)g (during)d(the)j(HIGH)g(p)s(erio)s(d.)42 b(This)30 b(is)g(appro)m(xi-) 120 3677 y(mately)i(\()p Fl(\032)504 3692 y Fk(h)571 3677 y Fn(\000)21 b Ft(1\))p Fl(t)776 3692 y Fk(h)821 3677 y Ft(.)46 b(Th)m(us)31 b(the)h(time)g(un)m(til)e(this)h(w)m(ork)h (is)f(remo)m(v)m(ed)i(from)f(the)g(system)g(during)e(the)120 3790 y(LO)m(W)h(p)s(erio)s(d)d(will)g(b)s(e)h(around)h Fl(t)1289 3804 y Fk(w)1370 3790 y Ft(=)25 b(\()p Fl(\032)1548 3805 y Fk(h)1614 3790 y Fn(\000)20 b Ft(1\))p Fl(t)1818 3805 y Fk(h)1863 3790 y Fl(=)p Ft(\(1)h Fn(\000)f Fl(\032)2147 3805 y Fk(l)2173 3790 y Ft(\).)261 3971 y(Returning)31 b(to)j(the)f(case)h(of)f(a)g(job)g(of)g(t)m(yp)s(e)g(3,)h(w)m(e)g(can)f (separate)h(jobs)e(of)h(t)m(yp)s(e)g(3)h(in)m(to)e(2)i(cases:)120 4084 y(those)d(whic)m(h)e(arriv)m(e)h(b)s(efore)g(time)g Fl(t)1389 4098 y Fk(w)1476 4084 y Ft(and)g(those)g(whic)m(h)g(arriv)m (e)g(after)h(time)f Fl(t)2864 4098 y Fk(w)2920 4084 y Ft(.)261 4265 y(Let)j(us)e(\014rst)h(consider)f(the)i(jobs)e(whic)m(h)g (arriv)m(e)h(during)e(time)i Fl(t)2494 4279 y Fk(w)2583 4265 y Ft(from)f(the)i(start)g(of)f(the)h(LO)m(W)120 4377 y(p)s(erio)s(d.)261 4558 y(Supp)s(ose)23 b(a)i(job)f(arriv)m(es)g (at)h(time)f Fl(t)1457 4572 y Fk(a)1523 4558 y Ft(from)g(the)h(start)g (of)g(the)g(LO)m(W)f(p)s(erio)s(d,)g(then)g(it)g(will)e(\014nish)g(at) 120 4671 y(time)j Fl(t)355 4638 y Fk(v)395 4671 y Ft(\()p Fl(x)9 b Ft(+)g Fl(v)s Ft(\()p Fl(t)686 4685 y Fk(a)729 4671 y Ft(\)\).)39 b(Giv)m(en)25 b(that)h(a)f(job)f(arriv)m(es)h (during)e(the)i(time)f(0)h(to)h Fl(t)2664 4685 y Fk(w)2721 4671 y Ft(,)g(it)e(will)e(arriv)m(e)j(uniformly)120 4784 y(during)j(this)h(in)m(terv)-5 b(al)30 b(\(since)g(the)h(arriv)-5 b(al)28 b(pro)s(cess)i(is)g(P)m(oisson\).)41 b(Th)m(us,)1092 4980 y Fl(E)5 b Ft([)p Fl(T)13 b Ft(\()p Fl(x)p Ft(\))p Fn(j)p Ft(Job)31 b(is)f(of)g(t)m(yp)s(e)h(3,)g(and)f(arriv)m(es)f(in)h (\(0)p Fl(;)15 b(t)2807 4994 y Fk(w)2864 4980 y Ft(\)])1867 5229 y(8)p eop %%Page: 9 11 9 10 bop 938 130 a Ft(=)1124 68 y(1)p 1102 108 90 4 v 1102 192 a Fl(t)1135 206 y Fk(w)1216 15 y Fg(Z)1300 41 y Fk(t)1325 49 y Fj(w)1263 203 y Fi(0)1379 130 y Ft(\()p Fl(t)1447 92 y Fk(v)1488 130 y Ft(\()p Fl(x)21 b Ft(+)f Fl(v)s Ft(\()p Fl(y)s Ft(\)\))h Fn(\000)f Fl(y)s Ft(\))p Fl(dy)938 359 y Ft(=)1124 297 y(1)p 1102 338 V 1102 421 a Fl(t)1135 435 y Fk(w)1216 244 y Fg(Z)1300 270 y Fh(1)1263 432 y Fi(0)1374 359 y Ft(\()p Fl(t)1442 321 y Fk(v)1483 359 y Ft(\()p Fl(z)25 b Ft(+)20 b Fl(x)p Ft(\))h Fn(\000)e Fl(t)1907 321 y Fk(v)1948 359 y Ft(\()p Fl(z)t Ft(\)\))p Fl(N)2182 321 y Fk(v)2172 381 y(l)2224 359 y Ft(\()p Fl(z)t Ft(\))p Fl(dz)1069 b Ft(\(16\))261 665 y(Observ)m(e)30 b(that)h(for)f(eac)m(h)h(of)f(the)h(T)m(yp)s(e)e(1,2)i(and)f(3)g(jobs)g (considered)f(ab)s(o)m(v)m(e,)j(the)e(resp)s(onse)f(time)120 778 y(of)i(a)f(job)g(is)g(prop)s(ortional)e(to)j Fl(t)1212 793 y Fk(h)1257 778 y Ft(.)261 959 y(F)-8 b(or)39 b(t)m(yp)s(e)f(3)g (jobs)g(whic)m(h)f(arriv)m(e)g(after)i(time)e Fl(t)1928 973 y Fk(w)1985 959 y Ft(,)j(their)d(resp)s(onse)g(time)h(will)d(b)s(e) i(indep)s(enden)m(t)120 1072 y(of)d Fl(t)260 1087 y Fk(h)305 1072 y Ft(.)51 b(Th)m(us)32 b(if)h Fl(t)734 1087 y Fk(h)812 1072 y Ft(is)g(quite)g(large)h(\(compared)h(to)f(the)g(mean)g(job)f (size\))h(w)m(e)h(can)f(assume)f(that)i(the)120 1185 y(resp)s(onse)e(time)h(for)g(these)g(jobs)g(is)f(negligible)f(compared) i(to)g(jobs)g(of)g(t)m(yp)s(e)g(1,)i(t)m(yp)s(e)e(2)h(and)e(jobs)h(of) 120 1298 y(t)m(yp)s(e)c(3)h(whic)m(h)e(arriv)m(e)h(during)e(time)i(0)h (to)g Fl(t)1633 1312 y Fk(w)1689 1298 y Ft(.)41 b(Hence)31 b(w)m(e)g(will)d(appro)m(ximate)i(the)g(resp)s(onse)g(time)g(of)120 1411 y(t)m(yp)s(e)h(3)f(jobs)g(arriving)f(during)f Fl(t)1253 1425 y Fk(w)1339 1411 y Ft(to)k Fl(t)1484 1426 y Fk(l)1540 1411 y Ft(b)m(y)e(0.)261 1591 y(So,)h(\014nally)d(w)m(e)j(obtain)1352 1795 y Fl(E)5 b Ft([)p Fl(T)13 b Ft(\()p Fl(x)p Ft(\))p Fn(j)p Ft(Job)31 b(is)f(of)g(t)m(yp)s(e)h(3])1198 1971 y(=)1369 1910 y(1)p 1362 1950 59 4 v 1362 2034 a Fl(t)1395 2049 y Fk(l)1446 1856 y Fg(Z)1529 1883 y Fh(1)1492 2045 y Fi(0)1604 1971 y Ft(\()p Fl(t)1672 1934 y Fk(v)1713 1971 y Ft(\()p Fl(z)25 b Ft(+)20 b Fl(x)p Ft(\))g Fn(\000)g Fl(t)2137 1934 y Fk(v)2178 1971 y Ft(\()p Fl(z)t Ft(\)\))p Fl(N)2412 1934 y Fk(v)2402 1994 y(l)2454 1971 y Ft(\()p Fl(z)t Ft(\))p Fl(dz)839 b Ft(\(17\))261 2279 y(Finally)30 b(to)j(obtain)e(the)h(appro)m(ximate)g(exp)s(ected)g(resp)s(onse)f (time)h(for)f(a)i(job)e(of)h(size)g Fl(x)p Ft(,)g(w)m(e)g(\014nd)120 2392 y(the)f(unconditional)c(resp)s(onse)j(time.)40 b(Th)m(us)29 b(w)m(e)i(obtain)f(that)845 2596 y Fl(E)5 b Ft([)p Fl(T)13 b Ft(\()p Fl(x)p Ft(\)])1155 2610 y Fk(P)d(S)1287 2596 y Ft(=)25 b Fl(p)1429 2610 y Fi(1)1468 2596 y Fl(E)5 b Ft(\()p Fl(T)1628 2610 y Fi(1)1668 2596 y Ft(\()p Fl(x)p Ft(\)\))21 b(+)f Fl(p)1983 2610 y Fi(2)2022 2596 y Fl(E)5 b Ft(\()p Fl(T)2182 2610 y Fi(2)2223 2596 y Ft(\()p Fl(x)p Ft(\)\))21 b(+)f Fl(p)2538 2610 y Fi(3)2577 2596 y Fl(E)5 b Ft(\()p Fl(T)2737 2610 y Fi(3)2777 2596 y Ft(\()p Fl(x)p Ft(\)\))564 b(\(18\))120 2800 y(where)45 b Fl(E)5 b Ft([)p Fl(T)548 2814 y Fk(i)577 2800 y Ft(\()p Fl(x)p Ft(\)])47 b(is)e(the)h(exp)s(ected)g(resp)s(onse)f(time)h(for)f(a)i(job)e(of)h(t) m(yp)s(e)g Fl(i)p Ft(,)k Fl(i)i Ft(=)f(1)p Fl(;)15 b Ft(2)p Fl(;)g Ft(3,)52 b(as)46 b(ob-)120 2913 y(tained)c(in)g (Equations)g(14,15)i(and)f(17.)78 b(The)43 b(probabilit)m(y)d Fl(p)2325 2927 y Fi(1)2407 2913 y Ft(that)j(a)g(job)g(of)g(size)f Fl(x)h Ft(is)f(of)h(t)m(yp)s(e)120 3026 y(1)48 b(is)e Fl(\025)374 3041 y Fk(h)419 3026 y Fl(t)452 3041 y Fk(h)496 3026 y Fl(e)538 2993 y Fh(\000)p Fk(ax)675 3026 y Fl(=)p Ft(\()p Fl(\025)808 3041 y Fk(h)853 3026 y Fl(t)886 3041 y Fk(h)962 3026 y Ft(+)32 b Fl(\025)1118 3041 y Fk(l)1144 3026 y Fl(t)1177 3041 y Fk(l)1202 3026 y Ft(\).)92 b(The)46 b(probabilit)m(y)f Fl(p)2085 3040 y Fi(2)2171 3026 y Ft(that)j(a)f(job)g(of)h(size)f Fl(x)g Ft(is)f(of)h(t)m(yp)s(e)g(2)h (is)120 3139 y(\()p Fl(\025)208 3154 y Fk(h)253 3139 y Fl(t)286 3154 y Fk(h)331 3139 y Ft(\(1)21 b Fn(\000)f Fl(e)565 3106 y Fh(\000)p Fk(ax)702 3139 y Ft(\)\))p Fl(=)p Ft(\()p Fl(\025)905 3154 y Fk(h)951 3139 y Fl(t)984 3154 y Fk(h)1049 3139 y Ft(+)g Fl(\025)1193 3154 y Fk(l)1219 3139 y Fl(t)1252 3154 y Fk(l)1278 3139 y Ft(\).)42 b(Finally)29 b(the)i(probabilit)m(y)d Fl(p)2358 3153 y Fi(3)2428 3139 y Ft(that)j(a)g(job)f(of)h(size)g Fl(x)f Ft(is)g(of)h(t)m(yp)s(e)g(3) 120 3252 y(is)e Fl(\025)264 3267 y Fk(l)290 3252 y Fl(t)323 3267 y Fk(l)349 3252 y Fl(=)p Ft(\()p Fl(\025)482 3267 y Fk(h)528 3252 y Fl(t)561 3267 y Fk(h)626 3252 y Ft(+)20 b Fl(\025)770 3267 y Fk(l)796 3252 y Fl(t)829 3267 y Fk(l)855 3252 y Ft(\).)261 3433 y(Substituting)28 b(the)i(v)-5 b(alues)30 b(in)f(Equation)h(18)h(and)f(doing)f(some)i(manipulation)d (giv)m(es)1009 3637 y Fl(E)5 b Ft([)p Fl(T)13 b Ft(\()p Fl(x)p Ft(\)])1319 3651 y Fk(P)d(S)856 3840 y Ft(=)1144 3779 y Fl(\025)1197 3794 y Fk(h)1242 3779 y Fl(t)1275 3794 y Fk(h)p 1019 3819 425 4 v 1019 3903 a Fl(\025)1072 3918 y Fk(h)1117 3903 y Fl(t)1150 3918 y Fk(h)1215 3903 y Ft(+)20 b Fl(\025)1359 3918 y Fk(l)1385 3903 y Fl(t)1418 3918 y Fk(l)1469 3696 y Fg(")1527 3779 y Ft(\(1)h Fn(\000)f Fl(e)1761 3746 y Fh(\000)p Fk(ax)1898 3779 y Ft(\))p Fl(t)1966 3794 y Fk(h)p 1527 3819 484 4 v 1746 3903 a Ft(2)2041 3840 y(+)2132 3725 y Fg(Z)2215 3752 y Fk(x)2178 3914 y Fi(0)2274 3840 y Fl(at)2355 3803 y Fk(v)2396 3840 y Ft(\()p Fl(x)g Fn(\000)g Fl(z)t Ft(\))p Fl(e)2717 3803 y Fh(\000)p Fk(az)2850 3840 y Fl(dz)2943 3696 y Fg(#)856 4100 y Ft(+)1192 4038 y Fl(\025)1245 4053 y Fk(l)p 1019 4079 425 4 v 1019 4162 a Fl(\025)1072 4177 y Fk(h)1117 4162 y Fl(t)1150 4177 y Fk(h)1215 4162 y Ft(+)g Fl(\025)1359 4177 y Fk(l)1385 4162 y Fl(t)1418 4177 y Fk(l)1469 3981 y Fg(\024)1513 3985 y(Z)1596 4011 y Fh(1)1559 4174 y Fi(0)1670 4100 y Ft(\()p Fl(t)1738 4062 y Fk(v)1780 4100 y Ft(\()p Fl(z)k Ft(+)c Fl(x)p Ft(\))h Fn(\000)f Fl(t)2204 4062 y Fk(v)2244 4100 y Ft(\()p Fl(z)t Ft(\)\))p Fl(N)2478 4062 y Fk(v)2468 4122 y(l)2521 4100 y Ft(\()p Fl(z)t Ft(\))p Fl(dz)2730 3981 y Fg(\025)3498 4100 y Ft(\(19\))261 4408 y(Observ)m(e)30 b(that)h(if)f Fl(\032)938 4423 y Fk(l)989 4408 y Ft(=)25 b(0,)825 4659 y Fl(E)5 b Ft([)p Fl(T)13 b Ft(\()p Fl(x)p Ft(\)])1135 4673 y Fk(P)d(S)1267 4659 y Ft(=)1373 4598 y(\(1)21 b Fn(\000)f Fl(e)1607 4565 y Fh(\000)p Fk(ax)1743 4598 y Ft(\))p Fl(t)1811 4613 y Fk(h)p 1373 4638 484 4 v 1592 4722 a Ft(2)1886 4659 y(+)1977 4544 y Fg(Z)2060 4571 y Fk(x)2023 4733 y Fi(0)2119 4659 y Fl(at)2200 4622 y Fk(v)2241 4659 y Ft(\()p Fl(x)h Fn(\000)e Fl(z)t Ft(\))p Fl(e)2562 4622 y Fh(\000)p Fk(az)2696 4659 y Fl(dz)713 b Ft(\(20\))1867 5229 y(9)p eop %%Page: 10 12 10 11 bop 120 91 a Fp(4)135 b(Analytical)46 b(Results)g(for)f(SRPT)f (under)g(HIGH/LO)l(W)i(mo)t(del)120 362 y Ft(In)37 b(the)g(previous)g (section)g(w)m(e)h(deriv)m(ed)f(an)g(appro)m(ximation)g(for)g(the)h (resp)s(onse)e(time)h(for)h(a)g(job)f(of)120 475 y(size)29 b Fl(x)f Ft(under)g(PS)g(sc)m(heduling.)38 b(The)28 b(goal)h(of)g(this) f(section)h(is)f(to)h(deriv)m(e)g(an)f(appro)m(ximation)g(for)h(the)120 588 y(resp)s(onse)g(time)h(for)h(a)f(job)g(of)h(size)f Fl(x)g Ft(under)f(the)i(HIGH/LO)m(W)h(mo)s(del)d(with)g(SRPT)g(sc)m (heduling.)120 899 y Fm(4.1)112 b(Num)m(b)s(er)37 b(of)h(jobs)g(during) f(HIGH)f(p)s(erio)s(d)120 1138 y Ft(Let)31 b Fl(x)335 1152 y Fk(o)403 1138 y Ft(b)s(e)f(de\014ned)f(suc)m(h)h(that)h Fl(\032)1289 1153 y Fk(h)1334 1138 y Ft(\()p Fl(x)1421 1152 y Fk(o)1460 1138 y Ft(\))25 b(=)g(1.)261 1319 y(Consider)k(a)i (job)f(of)g(size)h Fl(x)25 b(>)g(x)1372 1333 y Fk(o)1410 1319 y Ft(.)41 b(This)29 b(job)h(will)e(nev)m(er)j(run)e(as)h(long)g (as)h(there)g(are)f(jobs)g(of)h(size)120 1432 y(less)h(than)g Fl(x)h Ft(in)e(the)i(system.)47 b(Since)32 b Fl(\032)1486 1447 y Fk(h)1531 1432 y Ft(\()p Fl(x)p Ft(\))d Fl(>)g Ft(1,)34 b(it)e(follo)m(ws)g(that)h(for)f(almost)h(all)e(sample)h (paths)g Fl(!)120 1545 y Ft(there)i(will)c(b)s(e)j(a)h(time)f Fl(t)p Ft(\()p Fl(!)s Ft(\))h(suc)m(h)f(that)h(the)g(w)m(ork)f(made)h (up)e(b)m(y)i(jobs)f(of)g(size)g(less)g(than)g Fl(x)h Ft(will)d(b)s(e)120 1658 y(non-zero)i(for)f(all)f Fl(t)d(>)g(t)p Ft(\()p Fl(!)s Ft(\).)47 b(Hence,)34 b(if)d(the)i(job)f Fl(x)g Ft(arriv)m(es)g(after)h(time)f Fl(t)p Ft(\()p Fl(!)s Ft(\))g(it)g(will)e(remain)h(in)g(the)120 1771 y(system)37 b(at)g(least)f(un)m(til)f(the)h(end)g(of)h(the)f(HIGH)h(p)s (erio)s(d.)56 b(Assuming)35 b(that)i Fl(t)2869 1786 y Fk(h)2950 1771 y Ft(is)e(large)i(compared)120 1884 y(to)31 b Fl(t)p Ft(\()p Fl(!)s Ft(\),)h(w)m(e)f(will)c(use)k(the)g(appro)m (ximation)e(that)i(all)f(jobs)g(of)g(size)h Fl(>)25 b(x)2615 1898 y Fk(o)2684 1884 y Ft(are)31 b(held)e(bac)m(k)i(during)e(the)120 1996 y(HIGH)i(p)s(erio)s(d.)261 2177 y(Secondly)-8 b(,)28 b(for)f(jobs)h(of)g(size)f Fl(x)e(<)g(x)1477 2191 y Fk(o)1543 2177 y Ft(the)j(exp)s(ected)g(resp)s(onse)f(time)g(for)h(a)g(job)f(of)h (size)g Fl(x)g Ft(is)e Fl(o)p Ft(\()p Fl(t)3554 2192 y Fk(h)3599 2177 y Ft(\),)120 2290 y(as)32 b(sho)m(wn)f(in)g([2)q(].)45 b(The)32 b(reason)g(for)g(this)f(is)f(that)j(if)e(w)m(e)h(consider)f (the)h(busy)f(p)s(erio)s(ds)f(during)f(whic)m(h)120 2403 y(jobs)d(of)h(size)g Fn(\024)e Fl(x)h Ft(are)h(executed,)i(then)e(at)g (most)g(one)g(job)g(of)f(size)h Fl(>)e(x)i Ft(can)g(a\013ect)h(a)f (particular)e(busy)120 2516 y(p)s(erio)s(d.)38 b(Hence,)31 b(the)e(resp)s(onse)g(time)g(for)g(a)h(job)f(of)g(size)h Fl(x)f Ft(is)f(not)i(a\013ected)h(signi\014can)m(tly)d(b)m(y)h(jobs)g (of)120 2629 y(size)h Fl(>)25 b(x)p Ft(,)31 b(and)e(hence)i(is)e Fl(o)p Ft(\()p Fl(t)1128 2644 y Fk(h)1173 2629 y Ft(\),)i(if)f Fl(\032)p Ft(\()p Fl(x)p Ft(\))c Fl(<)f Ft(1.)261 2809 y(Using)30 b(the)g(observ)-5 b(ations)30 b(ab)s(o)m(v)m(e)i(w)m(e)f (appro)m(ximate)f(the)h(n)m(um)m(b)s(er)e(of)h(jobs)g(at)h(time)f Fl(t)g Ft(as)1537 3014 y Fl(N)1610 3029 y Fk(h)1655 3014 y Ft(\()p Fl(t)p Ft(\))c(=)f Fl(\025)1933 3029 y Fk(h)1978 3014 y Fl(t)p 2011 2942 72 4 v(F)13 b Ft(\()p Fl(x)2169 3028 y Fk(o)2207 3014 y Ft(\))1256 b(\(21\))120 3218 y(and)30 b(the)g(distribution)d(of)k(the)f(job)g(sizes)g(of)h(the)f (accum)m(ulated)h(jobs)f(as)1302 3492 y Fl(f)1347 3506 y Fk(r)1385 3492 y Ft(\()p Fl(x)p Ft(\))c(=)1629 3348 y Fg(\()1771 3385 y Fk(f)7 b Fi(\()p Fk(x)p Fi(\))p 1747 3408 184 4 v 1747 3418 55 4 v 1747 3474 a Fk(F)i Fi(\()p Fk(x)1868 3482 y Fj(o)1903 3474 y Fi(\))1941 3429 y Fl(;)197 b(x)25 b(>)g(x)2388 3443 y Fk(o)1769 3569 y Ft(0)182 b Fl(other)s(w)r(ise)3498 3492 y Ft(\(22\))120 3876 y Fm(4.2)150 b(Num)m(b)s(er)36 b(of)i(jobs)g(during)f(LO)m(W)h(p)s(erio)s (d)120 4115 y Ft(During)29 b(the)i(LO)m(W)g(p)s(erio)s(d,)e Fl(\032)1189 4130 y Fk(l)1241 4115 y Fl(<)c Ft(1,)31 b(and)f(th)m(us)h(the)g(jobs)f(accum)m(ulated)h(during)d(the)j(HIGH)g (p)s(erio)s(d)120 4228 y(will)j(start)j(receiving)f(some)h(share)g(of)f (the)h(pro)s(cessor.)59 b(Let)38 b(us)e(consider)f(what)i(the)g(accum)m (ulated)120 4341 y(jobs)c(lo)s(ok)h(lik)m(e)f(during)f(the)i(LO)m(W)g (p)s(erio)s(d.)49 b(A)m(t)35 b(time)e Fl(t)e Ft(=)g(0,)k(the)f(jobs)g (in)e(the)j(bag)f(are)g(of)g(size)g Fl(x)3622 4355 y Fk(o)120 4454 y Ft(and)d(greater.)44 b(As)31 b(time)g(progresses,)h (jobs)e(will)f(b)s(e)h(cleared)i(from)e(the)i(bag)f(starting)g(from)g (the)g(jobs)120 4567 y(of)e(size)g Fl(x)445 4581 y Fk(o)483 4567 y Ft(.)40 b(Let)29 b Fl(x)p Ft(\()p Fl(t)p Ft(\))g(b)s(e)f(the)h (size)g(of)g(the)g(smallest)f(job)g(remaining)f(in)h(the)h(bag)g(at)g (time)g Fl(t)p Ft(.)40 b(W)-8 b(e)30 b(will)120 4679 y(\014rst)g(appro)m(ximate)g Fl(x)p Ft(\()p Fl(t)p Ft(\))h(as)g(a)f (function)f(of)i Fl(t)p Ft(.)261 4860 y(Clearly)f Fl(x)p Ft(\(0\))f(=)d Fl(x)923 4874 y Fk(o)962 4860 y Ft(.)44 b(Consider)30 b(the)h(scenario)h(at)g(time)f Fl(t)p Ft(.)44 b(Consider)30 b(ho)m(w)i(m)m(uc)m(h)f(time)g(it)g(tak)m(es)120 4973 y(to)k(adv)-5 b(ance)35 b Fl(x)p Ft(\()p Fl(t)p Ft(\))g(b)m(y)f Fl(dx)h Ft(amoun)m(t.)53 b(This)32 b(will)g(simply)g(b) s(e)h(the)i(amoun)m(t)g(of)f(w)m(ork)h(of)f(size)g(b)s(et)m(w)m(een) 1845 5229 y(10)p eop %%Page: 11 13 11 12 bop 120 91 a Fl(x)p Ft(\()p Fl(t)p Ft(\))35 b(and)e Fl(x)p Ft(\()p Fl(t)p Ft(\))23 b(+)g Fl(dx)34 b Ft(presen)m(t)g(in)f (the)h(system)h(at)g(time)e Fl(t)p Ft(.)52 b(Supp)s(ose)33 b(it)g(tak)m(es)j Fl(dt)e Ft(time)g(to)h(do)f(this.)120 204 y(Let)h(us)e(estimate)i Fl(dt)f Ft(in)e(terms)i(of)g Fl(dx)p Ft(.)52 b(The)34 b(exp)s(ected)g(amoun)m(t)h(of)f(w)m(ork)g (that)g(needs)g(to)h(b)s(e)e(done)120 317 y(during)28 b(the)j Fl(dt)f Ft(time)g(can)h(b)s(e)f(divided)d(in)m(to)k(3)f(parts:) 231 573 y(1.)46 b(The)30 b(exp)s(ected)g(w)m(ork)h(made)f(up)f(of)h (jobs)g(in)f(the)h(original)e(bag)j(with)d(sizes)i(b)s(et)m(w)m(een)h Fl(x)p Ft(\()p Fl(t)p Ft(\))g(and)347 685 y Fl(x)p Ft(\()p Fl(t)p Ft(\))21 b(+)f Fl(dx)p Ft(.)41 b(This)28 b(w)m(ork)j(totals)g (to)g Fl(\025)1626 700 y Fk(h)1671 685 y Fl(t)1704 700 y Fk(h)1748 685 y Fl(x)p Ft(\()p Fl(t)p Ft(\))p Fl(f)10 b Ft(\()p Fl(x)p Ft(\()p Fl(t)p Ft(\)\))p Fl(dx)p Ft(.)231 873 y(2.)46 b(P)m(ending)30 b(w)m(ork)h(due)f(to)i(the)f(new)f(arriv)-5 b(als)30 b(\(during)f(the)i(LO)m(W)g(p)s(erio)s(d\))e(with)h(sizes)g(b) s(et)m(w)m(een)347 986 y Fl(x)p Ft(\()p Fl(t)p Ft(\))d(and)f Fl(x)p Ft(\()p Fl(t)p Ft(\))13 b(+)g Fl(dx)p Ft(.)40 b(Note)28 b(that)f(these)g(jobs)f(w)m(ere)h(not)g(w)m(ork)m(ed)g(up)s (on,)f(un)m(til)f(no)m(w.)40 b(This)24 b(new)347 1099 y(w)m(ork)31 b(will)c(b)s(e)j Fl(\025)918 1114 y Fk(l)944 1099 y Fl(tx)p Ft(\()p Fl(t)p Ft(\))p Fl(f)10 b Ft(\()p Fl(x)p Ft(\()p Fl(t)p Ft(\)\))p Fl(dx)p Ft(.)231 1287 y(3.)46 b(The)29 b(fresh)f(w)m(ork)h(whic)m(h)f(arriv)m(es)h(during)e (this)h Fl(dt)h Ft(amoun)m(t)h(of)f(time.)40 b(W)-8 b(e)31 b(need)e(only)f(consider)347 1399 y(w)m(ork)f(made)h(up)e(b)m(y)h(jobs) f(of)i(size)f(less)f(than)h Fl(x)p Ft(\()p Fl(t)p Ft(\),)h(since)f Fl(x)p Ft(\()p Fl(t)p Ft(\))g(will)e(b)s(e)h(a\013ected)j(b)m(y)e (other)g(jobs.)347 1512 y(This)i(w)m(ork)h(will)e(b)s(e)i Fl(\032)1120 1527 y Fk(l)1146 1512 y Ft(\()p Fl(x)p Ft(\()p Fl(t)p Ft(\)\))p Fl(dt)p Ft(.)120 1768 y(Using)g(these)g(observ)-5 b(ations)30 b(w)m(e)h(can)g(write,)810 1972 y Fl(dt)25 b Ft(=)g Fl(\025)1064 1987 y Fk(h)1109 1972 y Fl(t)1142 1987 y Fk(h)1186 1972 y Fl(x)p Ft(\()p Fl(t)p Ft(\))p Fl(f)10 b Ft(\()p Fl(x)p Ft(\()p Fl(t)p Ft(\)\))p Fl(dx)22 b Ft(+)e Fl(\025)1886 1987 y Fk(l)1912 1972 y Fl(tx)p Ft(\()p Fl(t)p Ft(\))p Fl(f)10 b Ft(\()p Fl(x)p Ft(\()p Fl(t)p Ft(\)\))p Fl(dx)21 b Ft(+)f Fl(\032)2638 1987 y Fk(l)2664 1972 y Ft(\()p Fl(x)p Ft(\()p Fl(t)p Ft(\)\))p Fl(dt)120 2176 y Ft(Hence,)1512 2270 y Fl(dt)p 1353 2311 399 4 v 1353 2394 a(\025)1406 2409 y Fk(h)1450 2394 y Fl(t)1483 2409 y Fk(h)1548 2394 y Ft(+)g Fl(\025)1692 2409 y Fk(l)1718 2394 y Fl(t)1786 2332 y Ft(=)1892 2270 y Fl(x)p Ft(\()p Fl(t)p Ft(\))p Fl(f)10 b Ft(\()p Fl(x)p Ft(\()p Fl(t)p Ft(\)\))p Fl(dx)p 1892 2311 536 4 v 1932 2394 a Ft(1)21 b Fn(\000)e Fl(\032)2135 2409 y Fk(l)2161 2394 y Ft(\()p Fl(x)p Ft(\()p Fl(t)p Ft(\)\))3498 2332 y(\(23\))120 2547 y(In)m(tegrating)43 b(the)g(l.h.s.)76 b(of)42 b(Equation)g(23)h(from)f(0)h(to)g Fl(t)f Ft(and)g(the)h(r.h.s.) 76 b(from)42 b Fl(x)3112 2561 y Fk(o)3193 2547 y Ft(to)h Fl(x)p Ft(\()p Fl(t)p Ft(\))g(and)120 2673 y(observing)29 b(that)734 2628 y Fk(d\032)806 2640 y Fj(l)831 2628 y Fi(\()p Fk(x)p Fi(\))p 734 2651 192 4 v 792 2704 a Fk(dx)961 2673 y Ft(=)c Fl(\025)1110 2688 y Fk(l)1136 2673 y Fl(xf)10 b Ft(\()p Fl(x)p Ft(\))30 b(and)g Fl(\032)1619 2688 y Fk(h)1663 2673 y Ft(\()p Fl(x)1750 2687 y Fk(o)1789 2673 y Ft(\))c(=)f(1,)31 b(w)m(e)f(get,)1162 2872 y(1)p 1146 2912 80 4 v 1146 2996 a Fl(\025)1199 3011 y Fk(l)1250 2933 y Ft(log)1392 2872 y Fl(\025)1445 2887 y Fk(h)1490 2872 y Fl(t)1523 2887 y Fk(h)1588 2872 y Ft(+)20 b Fl(\025)1732 2887 y Fk(l)1758 2872 y Fl(t)p 1392 2912 399 4 v 1504 2996 a(\025)1557 3011 y Fk(h)1602 2996 y Fl(t)1635 3011 y Fk(h)1826 2933 y Ft(=)1949 2872 y(1)p 1932 2912 80 4 v 1932 2996 a Fl(\025)1985 3011 y Fk(l)2036 2933 y Ft(log)2211 2872 y(1)h Fn(\000)f Fl(\032)2415 2887 y Fk(l)2441 2872 y Ft(\()p Fl(x)2528 2886 y Fk(o)2566 2872 y Ft(\))p 2179 2912 456 4 v 2179 2996 a(1)g Fn(\000)g Fl(\032)2382 3011 y Fk(l)2408 2996 y Ft(\()p Fl(x)p Ft(\()p Fl(t)p Ft(\)\))120 3194 y(Th)m(us)29 b(w)m(e)i(obtain)737 3161 y Fi(2)1471 3332 y Fl(\032)1518 3347 y Fk(l)1544 3332 y Ft(\()p Fl(x)p Ft(\()p Fl(t)p Ft(\)\))26 b(=)1915 3270 y Fl(\025)1968 3285 y Fk(l)1994 3270 y Ft(\()p Fl(t)20 b Ft(+)g Fl(t)2206 3285 y Fk(h)2251 3270 y Ft(\))p 1901 3311 399 4 v 1901 3394 a Fl(\025)1954 3409 y Fk(h)1999 3394 y Fl(t)2032 3409 y Fk(h)2097 3394 y Ft(+)g Fl(\025)2241 3409 y Fk(l)2267 3394 y Fl(t)3498 3332 y Ft(\(24\))120 3539 y(Ha)m(ving)30 b(obtained)e Fl(x)p Ft(\()p Fl(t)p Ft(\),)j Fl(N)1086 3554 y Fk(l)1112 3539 y Ft(\()p Fl(t)p Ft(\))f(can)f(b)s(e)g(readily)f(determined,)h(since)f(b)m(y)i(the)f(ab) s(o)m(v)m(e)i(argumen)m(ts)f(the)120 3652 y(n)m(um)m(b)s(er)f(of)i (jobs)e(at)j(time)e Fl(t)g Ft(will)e(b)s(e)h(appro)m(ximately)h(\()p Fl(\025)2108 3667 y Fk(h)2153 3652 y Fl(t)2186 3667 y Fk(h)2251 3652 y Ft(+)20 b Fl(\025)2395 3667 y Fk(l)2421 3652 y Fl(t)p Ft(\))p 2489 3580 72 4 v Fl(F)13 b Ft(\()p Fl(x)p Ft(\()p Fl(t)p Ft(\)\).)42 b(Th)m(us,)1345 3856 y Fl(N)1418 3871 y Fk(l)1444 3856 y Ft(\()p Fl(t)p Ft(\))26 b(=)f(\()p Fl(\025)1757 3871 y Fk(h)1802 3856 y Fl(t)1835 3871 y Fk(h)1900 3856 y Ft(+)20 b Fl(\025)2044 3871 y Fk(l)2070 3856 y Fl(t)p Ft(\))p 2138 3784 V Fl(F)13 b Ft(\()p Fl(x)p Ft(\()p Fl(t)p Ft(\)\))1064 b(\(25\))261 4128 y(Finally)-8 b(,)28 b(w)m(e)i(obtain)f(the)h(in)m(v)m(erse)f(of)g Fl(x)p Ft(\()p Fl(t)p Ft(\))h(whic)m(h)e(w)m(e)i(denote)g(b)m(y)2555 4112 y(~)2554 4128 y Fl(t)p Ft(\()p Fl(x)p Ft(\).)41 b(Th)m(us)3006 4112 y(~)3004 4128 y Fl(t)p Ft(\()p Fl(x)p Ft(\()p Fl(t)p Ft(\)\))27 b(=)e Fl(t)p Ft(.)40 b(This)120 4241 y(will)24 b(b)s(e)i(useful)f(in)h(obtaining)f(the)i(appro)m (ximation)f(for)g(the)h(exp)s(ected)g(resp)s(onse)f(time.)39 b(Clearly)3506 4225 y(~)3505 4241 y Fl(t)o Ft(\()p Fl(x)p Ft(\))120 4354 y(is)29 b(only)h(de\014ned)f(for)h Fl(x)25 b(>)g(x)1088 4368 y Fk(o)1127 4354 y Ft(.)40 b(Then)30 b(Equation)f(24)i(giv)m(es)2135 4321 y Fi(3)1519 4584 y Ft(~)1517 4601 y Fl(t)p Ft(\()p Fl(x)p Ft(\))26 b(=)f Fl(t)1827 4616 y Fk(h)1882 4540 y Fl(\032)1929 4555 y Fk(h)1973 4540 y Ft(\()p Fl(x)p Ft(\))c Fn(\000)f Ft(1)p 1882 4580 372 4 v 1891 4664 a(1)h Fn(\000)f Fl(\032)2095 4679 y Fk(l)2121 4664 y Ft(\()p Fl(x)p Ft(\))3498 4601 y(\(26\))p 120 4748 1416 4 v 224 4802 a Fs(2)258 4833 y Fr(F)-6 b(or)26 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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 eofill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap } bdef /FA { newpath tMatrix 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dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray 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 eofill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap } bdef /FA { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke } bdef /PA { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill } bdef /FAn { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke } bdef /PAn { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill } bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR { /vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath } bdef /FRR { MRR stroke } bdef /PRR { MRR fill } bdef /MlrRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath } bdef /FlrRR { MlrRR stroke } bdef /PlrRR { MlrRR fill } bdef /MtbRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath } bdef /FtbRR { MtbRR stroke } bdef /PtbRR { MtbRR fill } bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 34 189 547 611 MathWorks begin bpage %%EndPageSetup %%BeginObject: obj1 bplot /dpi2point 12 def portraitMode 0216 7344 csm 199 11 6158 5058 MR c np 92 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 6914 5188 PR 6 w 0 4226 5356 0 0 -4226 899 4615 4 MP PP -5356 0 0 4226 5356 0 0 -4226 899 4615 5 MP stroke 4 w DO SO 6 w 0 sg 899 4615 mt 6255 4615 L 899 389 mt 6255 389 L 899 4615 mt 899 389 L 6255 4615 mt 6255 389 L 899 4615 mt 6255 4615 L 899 4615 mt 899 389 L 899 4615 mt 899 4561 L 899 389 mt 899 443 L 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389 mt 6201 389 L 437 460 mt (1600) s 899 4615 mt 6255 4615 L 899 389 mt 6255 389 L 899 4615 mt 899 389 L 6255 4615 mt 6255 389 L gs 899 389 5357 4227 MR c np 24 w 0 -6 0 -16 0 -10 0 -11 0 -12 0 -15 0 -16 0 -19 1 -21 0 -24 0 -28 1 -31 1 -36 1 -41 3 -47 3 -53 5 -62 7 -70 10 -83 16 -99 23 -125 34 -165 50 -217 75 -268 111 -307 164 -325 165 -230 0 0 79 -87 361 -292 535 -253 793 -212 1175 -171 1743 -134 899 4184 35 MP stroke DA 0 -10 0 -24 0 -14 0 -17 0 -18 0 -22 0 -24 0 -28 1 -32 0 -36 0 -42 1 -47 1 -53 1 -62 3 -70 3 -79 5 -91 7 -104 10 -118 16 -134 23 -154 34 -175 50 -199 75 -228 111 -259 164 -296 165 -237 0 -1321 79 0 361 0 535 0 793 0 1175 0 1743 0 899 4615 35 MP stroke gr 24 w DA 2653 5024 mt (Percentile of Job Size) s 382 3650 mt -90 rotate ( Expected Response Time ) s 90 rotate SO 6 w 1 sg 0 477 1152 0 0 -477 958 967 4 MP PP -1152 0 0 477 1152 0 0 -477 958 967 5 MP stroke 4 w DO SO 6 w 0 sg 958 967 mt 2110 967 L 958 490 mt 2110 490 L 958 967 mt 958 490 L 2110 967 mt 2110 490 L 958 967 mt 2110 967 L 958 967 mt 958 490 L 958 967 mt 2110 967 L 958 490 mt 2110 490 L 958 967 mt 958 490 L 2110 967 mt 2110 490 L 1491 686 mt (PS ) s 1491 907 mt (SRPT) s gs 958 490 1153 478 MR c np 24 w 319 0 1065 621 2 MP stroke DA 319 0 1065 843 2 MP stroke gr 24 w DA SO 6 w end eplot %%EndObject epage end showpage %%Trailer %%EOF %%EndDocument endTexFig 278 1731 a Ft(\(a\))32 b Fl(\032)472 1746 y Fk(h)542 1731 y Ft(=)25 b(1)p Fl(:)p Ft(3)p Fl(;)15 b(\032)840 1746 y Fk(l)893 1731 y Ft(=)25 b(0)1861 b(\(b\))30 b Fl(\032)3093 1746 y Fk(h)3163 1731 y Ft(=)25 b(2)p Fl(;)15 b(\032)3391 1746 y Fk(l)3443 1731 y Ft(=)25 b(0)120 2015 y(Figure)g(2:)39 b Fo(A)n(nalytic)28 b(r)-5 b(esults:)40 b(R)-5 b(esp)g(onse)31 b(times)d(as)h(a)f(function)h(of)f(job)g(size)h (for)f(\(a\))h Fl(\032)3110 2030 y Fk(h)3180 2015 y Ft(=)c(1)p Fl(:)p Ft(3)p Fo(,)p Fl(\032)3466 2030 y Fk(l)3519 2015 y Ft(=)g(0)120 2128 y Fo(and)35 b(\(b\))f Fl(\032)495 2143 y Fk(h)568 2128 y Ft(=)28 b(2)p Fo(,)34 b Fl(\032)821 2143 y Fk(l)875 2128 y Ft(=)28 b(0)p Fo(.)47 b(Both)35 b(\014gur)-5 b(es)34 b(assume)h(the)f Fl(B)5 b Ft(\(1)p Fl(:)p Ft(5\))36 b Fo(distribution.)47 b Fl(t)2931 2143 y Fk(h)3010 2128 y Fo(is)34 b(assume)-5 b(d)35 b(to)g(b)-5 b(e)120 2241 y(1000,)34 b Fl(\032)413 2256 y Fk(l)465 2241 y Ft(=)24 b(0)33 b Fo(and)h Fl(t)848 2256 y Fk(l)906 2241 y Fo(is)f(chosen)g(such)g(that)h(the)f(aver)-5 b(age)33 b(lo)-5 b(ad)35 b(is)d(0.7.)-34 2821 y 15156103 10419816 2828615 12432752 35982622 40192655 startTexFig -34 2821 a %%BeginDocument: final_plots/resp_bp_1.3_.5.eps %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The Mathworks, Inc. %%Title: resp_bp_1.3_.5.eps %%CreationDate: 04/17/ 1 14:47:20 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Pages: 1 %%BoundingBox: 43 189 547 611 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray 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 eofill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap } bdef /FA { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke } bdef /PA { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill } bdef /FAn { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke } bdef /PAn { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill } bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR { /vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath } bdef /FRR { MRR stroke } bdef /PRR { MRR fill } bdef /MlrRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath } bdef /FlrRR { MlrRR stroke } bdef /PlrRR { MlrRR fill } bdef /MtbRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath } bdef /FtbRR { MtbRR stroke } bdef /PtbRR { MtbRR fill } bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 43 189 547 611 MathWorks begin bpage %%EndPageSetup %%BeginObject: obj1 bplot /dpi2point 12 def portraitMode 0216 7344 csm 306 11 6051 5058 MR c np 92 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 6914 5188 PR 6 w 0 4226 5356 0 0 -4226 899 4615 4 MP PP -5356 0 0 4226 5356 0 0 -4226 899 4615 5 MP stroke 4 w DO SO 6 w 0 sg 899 4615 mt 6255 4615 L 899 389 mt 6255 389 L 899 4615 mt 899 389 L 6255 4615 mt 6255 389 L 899 4615 mt 6255 4615 L 899 4615 mt 899 389 L 899 4615 mt 899 4561 L 899 389 mt 899 443 L %%IncludeResource: font Helvetica /Helvetica /ISOLatin1Encoding 192 FMSR 846 4828 mt (0) s 1970 4615 mt 1970 4561 L 1970 389 mt 1970 443 L 1837 4828 mt (20) s 3041 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-70 482 -53 462 -41 421 -32 369 -24 314 -19 261 -14 213 -11 172 -8 137 -7 108 -5 85 -3 66 -3 52 -3 40 -1 32 -2 24 -1 18 0 15 -1 947 4613 37 MP stroke DA 0 0 0 -1 0 -1 0 -4 0 -5 0 -9 0 -13 0 -20 1 -26 0 -35 0 -44 1 -53 1 -61 0 -68 2 -75 1 -79 2 -82 1 -85 2 -86 3 -88 3 -91 3 -94 4 -98 4 -103 5 -109 7 -115 8 -119 9 -124 10 -125 12 -127 14 -125 16 -123 0 -1635 5247 0 899 4615 35 MP stroke gr 24 w DA 2653 5024 mt (Percentile of Job Size) s 489 3650 mt -90 rotate ( Expected Response Time ) s 90 rotate SO 6 w 1 sg 0 476 1152 0 0 -476 5043 925 4 MP PP -1152 0 0 476 1152 0 0 -476 5043 925 5 MP stroke 4 w DO SO 6 w 0 sg 1081 1065 mt 2233 1065 L 1081 589 mt 2233 589 L 1081 1065 mt 1081 589 L 2233 1065 mt 2233 589 L 1081 1065 mt 2233 1065 L 1081 1065 mt 1081 589 L 1081 1065 mt 2233 1065 L 1081 589 mt 2233 589 L 1081 1065 mt 1081 589 L 2233 1065 mt 2233 589 L 1614 784 mt (PS ) s 1614 1005 mt (SRPT) s gs 1081 589 1153 477 MR c np 24 w 319 0 1188 720 2 MP stroke DA 319 0 1188 941 2 MP stroke gr 24 w DA SO 6 w end eplot %%EndObject epage end showpage %%Trailer %%EOF %%EndDocument endTexFig 2007 2821 a 15156103 10419816 2236579 12432752 35982622 40192655 startTexFig 2007 2821 a %%BeginDocument: final_plots/resp_bp_2_.5.eps %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The Mathworks, Inc. %%Title: resp_bp_2_.5.eps %%CreationDate: 04/17/ 1 17:36:23 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Pages: 1 %%BoundingBox: 34 189 547 611 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray 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 eofill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap } bdef /FA { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke } bdef /PA { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill } bdef /FAn { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke } bdef /PAn { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill } bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR { /vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath } bdef /FRR { MRR stroke } bdef /PRR { MRR fill } bdef /MlrRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath } bdef /FlrRR { MlrRR stroke } bdef /PlrRR { MlrRR fill } bdef /MtbRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath } bdef /FtbRR { MtbRR stroke } bdef /PtbRR { MtbRR fill } bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 34 189 547 611 MathWorks begin bpage %%EndPageSetup %%BeginObject: obj1 bplot /dpi2point 12 def portraitMode 0216 7344 csm 199 11 6158 5058 MR c np 92 dict begin 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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 eofill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap } bdef /FA { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke } bdef /PA { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill } bdef /FAn { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke } bdef /PAn { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill } bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR { /vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath } bdef /FRR { MRR stroke } bdef /PRR { MRR fill } bdef /MlrRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath } bdef /FlrRR { MlrRR stroke } bdef /PlrRR { MlrRR fill } bdef /MtbRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath } bdef /FtbRR { MtbRR stroke } bdef /PtbRR { MtbRR fill } bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 34 189 556 611 MathWorks begin bpage %%EndPageSetup %%BeginObject: obj1 bplot /dpi2point 12 def portraitMode 0216 7344 csm 199 11 6265 5058 MR c np 92 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 6914 5188 PR 6 w 0 4226 5356 0 0 -4226 899 4615 4 MP PP -5356 0 0 4226 5356 0 0 -4226 899 4615 5 MP stroke 4 w DO SO 6 w 0 sg 899 4615 mt 6255 4615 L 899 389 mt 6255 389 L 899 4615 mt 899 389 L 6255 4615 mt 6255 389 L 899 4615 mt 6255 4615 L 899 4615 mt 899 389 L 899 4615 mt 899 4561 L 899 389 mt 899 443 L %%IncludeResource: font Helvetica /Helvetica /ISOLatin1Encoding 192 FMSR 846 4828 mt (0) s 1970 4615 mt 1970 4561 L 1970 389 mt 1970 443 L 1864 4828 mt (20) s 3041 4615 mt 3041 4561 L 3041 389 mt 3041 443 L 2935 4828 mt (40) s 4113 4615 mt 4113 4561 L 4113 389 mt 4113 443 L 4007 4828 mt (60) s 5184 4615 mt 5184 4561 L 5184 389 mt 5184 443 L 5078 4828 mt (80) s 6255 4615 mt 6255 4561 L 6255 389 mt 6255 443 L 6095 4828 mt (100) s 899 4615 mt 953 4615 L 6255 4615 mt 6201 4615 L 758 4686 mt (0) s 899 3911 mt 953 3911 L 6255 3911 mt 6201 3911 L 544 3982 mt (200) s 899 3206 mt 953 3206 L 6255 3206 mt 6201 3206 L 544 3277 mt (400) s 899 2502 mt 953 2502 L 6255 2502 mt 6201 2502 L 544 2573 mt (600) s 899 1798 mt 953 1798 L 6255 1798 mt 6201 1798 L 544 1869 mt (800) s 899 1093 mt 953 1093 L 6255 1093 mt 6201 1093 L 437 1164 mt (1000) s 899 389 mt 953 389 L 6255 389 mt 6201 389 L 437 460 mt (1200) s 899 4615 mt 6255 4615 L 899 389 mt 6255 389 L 899 4615 mt 899 389 L 6255 4615 mt 6255 389 L gs 899 389 5357 4227 MR c np DA 10 74 11 -321 11 -124 11 104 10 -326 11 94 11 -211 10 81 11 -402 11 96 11 -16 10 21 11 -225 11 78 10 -93 11 24 11 -265 11 92 10 -26 11 -113 11 -76 10 8 11 90 11 27 11 -48 10 -234 11 78 11 -21 10 -152 11 104 11 -78 10 98 11 -111 11 10 11 -42 10 -29 11 -38 11 -38 10 -77 11 105 11 -59 11 5 10 -66 11 46 11 -50 10 33 11 -163 11 120 11 -8 10 -143 11 -21 11 59 10 -43 11 -11 11 25 11 70 10 -157 11 57 11 -25 10 -7 11 38 11 17 11 -10 10 -81 11 -2 11 -24 10 32 11 -111 11 152 11 -141 10 38 11 0 11 -17 10 -17 11 1 11 -70 11 52 10 -31 11 52 11 -37 10 -65 11 14 11 -27 11 -25 10 87 11 -103 11 58 10 -75 11 -33 11 106 11 -31 10 -18 11 30 11 -41 10 1 11 16 11 3 10 -22 11 2 5184 3673 100 MP stroke 11 -23 11 35 10 -72 11 37 11 -31 10 -24 11 20 11 -34 11 45 10 -58 11 -2 11 22 10 -44 11 37 11 -28 11 -19 10 14 11 -40 11 88 10 -27 11 -44 11 26 11 -60 10 -5 11 11 11 -35 10 63 11 -72 11 22 11 0 10 20 11 -32 11 -14 10 -27 11 55 11 -19 11 -14 10 -65 11 23 11 -17 10 100 11 -50 11 -28 11 58 10 -58 11 -10 11 14 10 8 11 7 11 -24 11 -27 10 55 11 -88 11 15 10 45 11 1 11 -30 10 19 11 -57 11 -4 11 28 10 -7 11 0 11 -16 10 9 11 -38 11 53 11 -11 10 -63 11 44 11 -28 10 -6 11 31 11 -46 11 -4 10 -3 11 -3 11 36 10 29 11 -35 11 -17 11 -18 10 62 11 -70 11 8 10 -4 11 -26 11 -4 11 27 10 1 11 -19 11 8 10 0 11 -4 11 -54 11 36 10 12 11 -23 11 -3 4123 4133 100 MP stroke 10 11 11 5 11 1 11 -14 10 -12 11 29 11 -43 10 33 11 -2 11 -39 11 33 10 -2 11 -25 11 -10 10 15 11 -5 11 -6 11 -27 10 62 11 -16 11 -56 10 3 11 14 11 7 10 -25 11 25 11 0 11 -10 10 -63 11 32 11 9 10 24 11 -44 11 17 11 39 10 -24 11 -7 11 -40 10 32 11 -32 11 26 11 -33 10 23 11 14 11 -8 10 -36 11 4 11 8 11 -5 10 14 11 -24 11 -2 10 16 11 0 11 -6 11 9 10 -5 11 -8 11 -16 10 -2 11 13 11 -32 11 5 10 27 11 -31 11 -8 10 17 11 -11 11 -12 11 11 10 11 11 0 11 -41 10 42 11 -11 11 -14 11 -17 10 37 11 -6 11 -34 10 26 11 -20 11 6 10 17 11 -22 11 -1 11 -6 10 0 11 -3 11 12 10 -5 11 -17 11 -5 11 11 10 -4 11 -19 11 18 10 12 11 -34 3063 4363 100 MP stroke 11 12 11 18 10 -8 11 -26 11 -3 10 23 11 -18 11 10 11 -8 10 -10 11 -11 11 9 10 -13 11 16 11 -4 11 0 10 -2 11 -15 11 7 10 -8 11 12 11 -2 11 -22 10 32 11 -25 11 -3 10 2 11 5 11 3 11 5 10 -18 11 15 11 -7 10 -1 11 -10 11 -1 11 -12 10 6 11 0 11 -2 10 2 11 -4 11 -2 11 -9 10 -2 11 6 11 -13 10 12 11 6 11 -14 10 7 11 -7 11 3 11 0 10 3 11 -21 11 19 10 -11 11 -14 11 15 11 -7 10 -10 11 2 11 2 10 7 11 -14 11 -2 11 6 10 -8 11 16 11 -6 10 -4 11 -4 11 -5 11 4 10 -1 11 2 11 -7 10 -9 11 5 11 -5 11 -3 10 6 11 -6 11 6 10 -9 11 1 11 5 11 -1 10 -8 11 -3 11 2 10 2 11 -3 11 -10 11 -3 10 -1 11 9 11 -2 2002 4497 100 MP stroke 10 -3 11 -1 11 -9 11 -3 10 8 11 1 11 -2 10 -2 11 -8 11 2 10 -2 11 3 11 -9 11 2 10 -7 11 6 11 -3 10 1 11 3 11 -2 11 -2 10 -5 11 0 11 -4 10 3 11 -1 11 -9 11 10 10 -11 11 2 11 -2 10 -1 11 1 11 3 11 -7 10 2 11 -5 11 -1 10 -4 11 4 11 -5 11 1 10 1 11 -2 11 -1 10 -1 11 0 11 -4 11 -3 10 -1 11 4 11 -6 10 4 11 -3 11 -3 11 0 10 0 11 0 11 -1 10 -2 11 -4 11 2 11 1 10 -7 11 2 11 -1 10 1 11 -3 11 -2 11 -2 10 3 11 -3 11 -2 10 -1 11 -1 11 0 10 -2 11 0 11 -1 11 -2 10 -1 11 -1 11 -2 10 1 11 -1 11 -2 11 -1 10 0 11 -1 11 -1 10 -1 11 -2 11 0 11 -2 10 -1 11 -1 11 -1 10 -1 11 -1 942 4612 100 MP stroke 11 0 11 -2 10 0 11 -1 899 4615 5 MP stroke SO 10 -19 11 -340 11 -179 11 44 10 -280 11 17 11 -209 10 84 11 -357 11 45 11 5 10 19 11 -196 11 31 10 -72 11 10 11 -213 11 35 10 41 11 -107 11 -88 10 36 11 68 11 -27 11 -1 10 -204 11 43 11 -41 10 -102 11 51 11 -27 10 47 11 -91 11 -19 11 -30 10 -34 11 -63 11 10 10 -74 11 60 11 -34 11 -15 10 -134 11 55 11 -39 10 28 11 -69 11 -38 11 30 10 -158 11 22 11 -74 10 -25 11 -19 11 -59 11 -51 10 -75 11 -38 11 -18 10 -5 11 -27 11 -23 11 -20 10 -22 11 -22 11 5 10 -14 11 -10 11 3 11 -10 10 -7 11 -5 11 2 10 -1 11 -3 11 -2 11 -8 10 -1 11 -1 11 -4 10 1 11 -2 11 0 11 -2 10 2 11 -3 11 2 10 -4 11 -1 11 1 11 -1 10 -1 11 0 11 1 10 -2 11 0 11 0 10 0 11 -1 5184 4601 100 MP stroke 11 0 11 -1 10 -1 11 0 11 -1 10 0 11 0 11 0 11 0 10 0 11 -1 11 0 10 0 11 0 11 0 11 -1 10 0 11 0 11 0 10 -1 11 0 11 1 11 -1 10 0 11 0 11 0 10 0 11 -1 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 -1 11 1 11 -1 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 -1 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 -1 11 1 11 0 10 -1 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 4123 4611 100 MP stroke 10 -1 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 -1 11 0 11 0 10 0 11 1 11 -1 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 3063 4613 100 MP stroke 11 0 11 0 10 0 11 0 11 0 10 0 11 -1 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 2002 4614 100 MP stroke 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 -1 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 942 4615 100 MP stroke 11 0 11 0 10 0 11 0 899 4615 5 MP stroke DA 0 -1 0 -1 0 -7 1 -20 1 -45 3 -77 4 -112 6 -146 10 -180 15 -214 25 -252 36 -288 51 -316 72 -334 111 -332 177 -314 264 -280 356 -238 431 -195 474 -156 482 -121 462 -94 421 -72 369 -54 314 -42 261 -31 213 -25 172 -18 137 -14 108 -11 85 -8 66 -6 52 -5 40 -4 32 -3 24 -2 18 -1 15 -2 48 -4 899 4615 40 MP stroke gr DA gs 826 517 5503 4172 MR c np SO 24 w 899 4615 PD 947 4611 PD 962 4609 PD 980 4608 PD 1004 4606 PD 1036 4603 PD 1076 4599 PD 1128 4594 PD 1194 4588 PD 1279 4580 PD 1387 4569 PD 1524 4555 PD 1696 4537 PD 1909 4512 PD 2170 4481 PD 2484 4439 PD 2853 4385 PD 3274 4313 PD 3736 4219 PD 4218 4098 PD 4692 3942 PD 5123 3747 PD 5479 3509 PD 5743 3229 PD 5920 2915 PD 6031 2583 PD 6103 2249 PD 6154 1933 PD 6190 1645 PD 6215 1393 PD 6230 1179 PD 6240 999 PD 6246 853 PD 6250 741 PD 6253 664 PD 6254 619 PD 6255 599 PD 6255 592 PD 6255 591 PD 6255 590 PD DA gr 24 w gs 899 389 5357 4227 MR c np DD 6 w 0 -2 0 -4 0 -7 0 -10 0 -16 0 -22 1 -29 0 -37 1 -45 0 -54 1 -61 1 -67 1 -72 2 -76 1 -79 2 -80 3 -82 2 -84 3 -86 4 -90 4 -95 5 -99 6 -105 7 -110 8 -115 10 -117 12 -118 13 -118 14 -116 17 -114 19 -112 21 -109 24 -108 27 -109 32 -109 38 -111 44 -114 53 -117 63 -118 74 -118 85 -117 2 -671 4756 0 899 4615 44 MP stroke gr 6 w DD gs 826 519 5503 4170 MR c np SO 24 w 899 4615 PD 5655 4615 PD 5657 3944 PD 5742 3827 PD 5816 3709 PD 5879 3591 PD 5932 3474 PD 5976 3360 PD 6014 3249 PD 6046 3140 PD 6073 3031 PD 6097 2923 PD 6118 2814 PD 6137 2702 PD 6154 2588 PD 6168 2472 PD 6181 2354 PD 6193 2236 PD 6203 2119 PD 6211 2004 PD 6218 1894 PD 6224 1789 PD 6229 1690 PD 6233 1595 PD 6237 1505 PD 6240 1419 PD 6242 1335 PD 6245 1253 PD 6247 1173 PD 6248 1094 PD 6250 1018 PD 6251 946 PD 6252 879 PD 6253 818 PD 6253 764 PD 6254 719 PD 6254 682 PD 6255 653 PD 6255 631 PD 6255 615 PD 6255 605 PD 6255 598 PD 6255 594 PD 6255 592 PD DD gr 24 w gs 899 389 5357 4227 MR c np gr 2653 5024 mt (Percentile of Job Size) s 382 3597 mt -90 rotate (Expected Response Time) s 90 rotate 2304 237 mt (Simulation comparison results) s SO 6 w 1 sg 0 918 2063 0 0 -918 4132 1367 4 MP PP -2063 0 0 918 2063 0 0 -918 4132 1367 5 MP stroke 4 w DO SO 6 w 0 sg 4132 1367 mt 6195 1367 L 4132 449 mt 6195 449 L 4132 1367 mt 4132 449 L 6195 1367 mt 6195 449 L 4132 1367 mt 6195 1367 L 4132 1367 mt 4132 449 L 4132 1367 mt 6195 1367 L 4132 449 mt 6195 449 L 4132 1367 mt 4132 449 L 6195 1367 mt 6195 449 L 4665 644 mt (PS simulation ) s 4665 865 mt (SRPT simulation) s 4665 1086 mt (PS theory ) s 4665 1307 mt (SRPT theory ) s gs 4132 449 2064 919 MR c np DA 319 0 4239 579 2 MP stroke SO 319 0 4239 801 2 MP stroke DA 319 0 4239 1022 2 MP stroke SO gs 4326 949 147 147 MR c np 24 w 4399 1022 PD gr 24 w DD 6 w 319 0 4239 1243 2 MP stroke SO gs 4326 1170 147 147 MR c np 24 w 4399 1243 PD gr 24 w gr 24 w 6 w end eplot %%EndObject epage end showpage %%Trailer %%EOF %%EndDocument endTexFig 120 3956 a Ft(Figure)41 b(4:)62 b(Sim)m(ulation)39 b(results)g(as)j (compared)f(with)e(analytical)i(appro)m(ximation)f(results,)i Fl(\032)3501 3971 y Fk(h)3589 3956 y Ft(=)120 4069 y(2)p Fl(;)15 b(\032)252 4084 y Fk(l)304 4069 y Ft(=)25 b(0)p Fl(:)p Ft(5.)261 4275 y(In)45 b(general)g(for)h(all)e(w)m(orkload)h (distributions)d(w)m(e)k(tested)2351 4242 y Fi(5)2391 4275 y Ft(,)k(w)m(e)45 b(found)g(that)h(our)f(analytical)120 4388 y(appro)m(ximation)32 b(results)g(matc)m(h)i(almost)f(exactly)h (with)d(sim)m(ulation)g(when)h Fl(t)2838 4403 y Fk(h)2916 4388 y Ft(is)g(large.)49 b(W)-8 b(e)34 b(found)120 4501 y(that)26 b(sim)m(ulation)e(matc)m(hed)j(analysis)d(ev)m(en)j(for)f Fl(t)1820 4516 y Fk(h)1890 4501 y Ft(only)f(ab)s(out)h(100-1000)j (times)c(the)h(mean)g(job)g(size,)120 4614 y(pro)m(vided)35 b Fl(\032)547 4629 y Fk(h)628 4614 y Ft(w)m(as)i(not)g(to)s(o)g(close)g (to)s(o)g(1.)59 b(F)-8 b(or)38 b(lo)m(w)e(v)-5 b(ariabilit)m(y)34 b(distributions)f Fl(t)3009 4629 y Fk(h)3090 4614 y Ft(and)j Fl(n)f Ft(ma)m(y)j(b)s(e)120 4727 y(lo)m(w)m(er.)g(The)22 b(comparison)f(of)i(sim)m(ulation)d(and)h(analysis)g(is)g(an)i(ev)m(en) f(closer)h(matc)m(h)g(when)e(considering)120 4840 y(the)31 b(metric)f(n)m(um)m(b)s(er)f(in)g(system)h(as)h(a)g(function)e(of)h (time.)p 120 4894 1416 4 v 224 4948 a Fs(5)258 4980 y Fr(More)d(plots)f(not)f(sho)n(wn)h(here)g(for)h(lac)n(k)e(of)i(space.) 1845 5229 y Ft(18)p eop %%Page: 19 21 19 20 bop 120 91 a Fp(7)135 b(SRPT)31 b(vs.)55 b(PS)32 b(under)f(more)h(general)h(w)l(orkloads)g({)e(Op)t(en)h(prob-)322 241 y(lems)120 511 y Ft(In)d(Section)i(5)f(w)m(e)h(compared)f(the)h(p)s (erformance)e(of)h(SRPT)g(vs.)40 b(PS)30 b(on)g(a)g Fl(B)5 b Ft(\()p Fl(\013)26 b Ft(=)f(1)p Fl(:)p Ft(5\))32 b(distribution)120 624 y(under)h(transien)m(t)i(o)m(v)m(erload.)55 b(W)-8 b(e)36 b(found)d(that)j(SRPT)d(had)i(t)m(w)m(o)h Fo(desir)-5 b(able)38 b(pr)-5 b(op)g(erties)p Ft(:)52 b(\(1\))36 b(It)f(im-)120 737 y(pro)m(v)m(ed)c(signi\014can)m(tly)e(up)s(on)h(PS)g (with)g(resp)s(ect)h(to)g(mean)g(resp)s(onse)f(time,)h(and)g(\(2\))h (SRPT)e(did)f(not)120 850 y(treat)i(large)g(jobs)f(w)m(orse)h(than)f (they)g(w)m(ould)f(b)s(e)h(treated)i(under)c(PS.)261 1031 y(The)g(p)s(oin)m(t)f(of)i(this)e(section)h(is)f(to)i(explore)f (whether)g(these)h Fo(desir)-5 b(able)32 b(pr)-5 b(op)g(erties)30 b Ft(of)f(SRPT)e(also)120 1144 y(translate)k(to)g(other)f (distributions.)261 1324 y(W)-8 b(e)27 b(\014rst)e(sho)m(w)h(that)g (the)g(desirable)e(prop)s(erties)h(for)g(lo)m(w)h(v)-5 b(ariabilit)m(y)23 b(distributions)f(as)27 b(w)m(ell.)38 b(W)-8 b(e)120 1437 y(pro)m(v)m(e)34 b(that)f(for)f(the)h(exp)s(onen)m (tial)f(distribution)d(\()p Fl(C)1962 1404 y Fi(2)2030 1437 y Ft(=)g(1\),)35 b(SRPT)c(impro)m(v)m(es)i(up)s(on)e(PS)h(for)g 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b(The)27 b(reason)h(is)f(that)h(while)d(it)i(seems)h (ob)m(vious)f(that)h(large)g(jobs)f(w)m(on't)h(receiv)m(e)g(service)f (under)120 3810 y(SRPT)k(during)g(the)i(o)m(v)m(erload)g(p)s(erio)s(d,) e(it)i(turns)e(out)i(that)g(they)g(also)f(don't)h(receiv)m(e)g(m)m(uc)m (h)g(service)120 3923 y(under)c(PS)i(during)e(the)i(o)m(v)m(erload)h(p) s(erio)s(d.)40 b(This)29 b(is)h(particularly)f(true)i(when)f(the)h (distribution)c(has)120 4036 y(a)k(hea)m(vy-tailed)f(prop)s(ert)m(y)-8 b(.)261 4216 y(Our)37 b(results)f(ha)m(v)m(e)j(implications)c(for)j (real)f(w)m(orld)g(systems,)j(where)d(\015uctuations)g(in)f(load)i(are) 120 4329 y(common.)64 b(Under)37 b(PS)h(sc)m(heduling,)g(these)h (\015uctuations)e(can)h(results)f(in)g(large)h(buildups)c(of)k(jobs)120 4442 y(whic)m(h)30 b(con)m(tin)m(ue)h(to)h(e\013ect)h(the)e(system)h (for)f(a)g(long)g(time.)43 b(In)30 b(SRPT,)h(as)g(w)m(e'v)m(e)i(sho)m (wn,)e(the)g(e\013ect)120 4555 y(of)g(load)f(\015uctuations)f(is)g(m)m (uc)m(h)i(less)e(sev)m(ere.)1845 5229 y(22)p eop %%Page: 23 25 23 24 bop 120 91 a Fp(9)135 b(Ac)l(kno)l(wledgemen)l(ts)120 362 y Ft(W)-8 b(e)31 b(w)m(ould)d(lik)m(e)h(to)i(thank)e(John)g(Leho)s (czky)h(for)f(p)s(oin)m(ting)f(us)h(to)h(the)g(appropriate)f(prior)f (literature)120 475 y(and)i(for)g(reading)f(early)h(drafts)g(of)h(this) e(w)m(ork.)120 792 y Fp(References)155 973 y Fi([1])45 b(G.)32 b(Banga,)i(P)-6 b(.)32 b(Drusc)n(hel,)h(and)g(J.)e(Mogul.)56 b(Better)33 b(op)r(erating)g(system)e(features)h(for)f(faster)h(net)n (w)n(ork)h(serv)n(ers.)55 b Fa(A)n(CM)275 1052 y(SIGMETRICS)27 b(Performanc)l(e)g(Evaluation)g(R)l(eview)p Fi(,)c(26\(3\):23{30,)j (1998.)155 1204 y([2])45 b(Nikhil)21 b(Bansal)h(and)h(Mor)e(Harc)n (hol-Balter.)28 b(Analysis)21 b(of)h(SRPT)g(sc)n(heduling:)31 b(In)n(v)n(estigating)24 b(unfairness.)k(In)23 b Fa(Pr)l(o)l(c)l(e)l(e) l(dings)275 1283 y(of)j(Performanc)l(e)h(and)f(A)n(CM)g(Sigmetrics)p Fi(,)c(2001.)155 1435 y([3])45 b(M.)30 b(Bender,)j(S.)d(Chakrabarti,)j (and)e(S.)f(Muth)n(ukrishnan.)53 b(Flo)n(w)31 b(and)g(stretc)n(h)h (metrics)d(for)h(sc)n(heduling)i(con)n(tinous)g(job)275 1514 y(streams.)e(In)24 b Fa(Pr)l(o)l(c)l(e)l(e)l(dings)k(of)e(the)f (9th)h(A)n(nnual)h(A)n(CM-SIAM)e(Symp)l(osium)j(on)e(Discr)l(ete)e(A)n (lgorithms)p Fi(,)h(1998.)155 1666 y([4])45 b(H.)22 b(Braun)h(and)g(K.) e(Cla\013y)-6 b(.)30 b(W)-6 b(eb)23 b(tra\016c)g(c)n(haracterization:) 32 b(an)23 b(assessmen)n(t)f(of)h(the)g(impact)f(of)g(cac)n(hing)i(do)r (cumen)n(ts)f(from)275 1745 y(NCSA's)g(Web)i(serv)n(er.)30 b(In)25 b Fa(Pr)l(o)l(c)l(e)l(e)l(dings)i(of)f(the)g(Se)l(c)l(ond)g (International)h(WWW)f(Confer)l(enc)l(e)p Fi(,)e(1994.)155 1897 y([5])45 b(H.)28 b(Chen,)j(O.)d(Kella,)h(and)g(G.)g(W)-6 b(eiss.)46 b(Fluid)28 b(appro)n(ximations)h(for)f(a)h(pro)r(cessor)g (sharing)f(queue.)48 b Fa(Queueing)29 b(Systems:)275 1976 y(The)l(ory)d(and)h(Applic)l(ations)p Fi(,)e(27:99{125,)g(1997.) 155 2128 y([6])45 b(M.)19 b(Cola)t(janni,)h(P)-6 b(.)19 b(S.)h(Y)-6 b(u,)20 b(and)g(D.)f(M.)g(Dias.)24 b(Analysis)19 b(of)g(task)i(assignmen)n(t)e(p)r(olicies)h(in)f(scalable)h (distributed)g(W)-6 b(eb-serv)n(er)275 2207 y(systems.)30 b Fa(IEEE)d(T)-5 b(r)l(ansactions)27 b(on)f(Par)l(al)t(lel)h(and)g (Distribute)l(d)e(Systems)p Fi(,)e(9\(6\),)h(1998.)155 2359 y([7])45 b(M.)33 b(Cro)n(v)n(ella)g(and)i(A.)e(Besta)n(vros.)61 b(Self-similarit)n(y)30 b(in)k(World)f(Wide)h(Web)h(tra\016c:)51 b(Evidence)35 b(and)f(p)r(ossible)g(causes.)275 2438 y(In)27 b Fa(Pr)l(o)l(c)l(e)l(e)l(dings)j(of)e(the)f(1996)j(A)n(CM)d (SIGMETRICS)i(International)g(Confer)l(enc)l(e)f(on)g(Me)l(asur)l (ement)h(and)f(Mo)l(deling)h(of)275 2517 y(Computer)e(Systems)p Fi(,)c(pages)h(160{169,)h(Ma)n(y)f(1996.)155 2669 y([8])45 b(Daniel)21 b(M.)f(Dias,)g(William)f(Kish,)h(Ra)t(jat)h(Mukherjee,)g (and)g(Ren)n(u)h(T)-6 b(ew)n(ari.)26 b(A)20 b(scalable)h(and)g(highly)g (a)n(v)l(ailable)g(w)n(eb)h(serv)n(er.)275 2748 y(In)i Fa(COMPCON)p Fi(,)f(pages)h(85{92,)h(1996.)155 2900 y([9])45 b(P)n(eter)31 b(Drusc)n(hel)f(and)h(Gaura)n(v)g(Banga.)52 b(Lazy)31 b(receiv)n(er)f(pro)r(cessing)h(\(LRP\):)g(A)f(net)n(w)n(ork) h(subsystem)f(arc)n(hitecture)i(for)275 2979 y(serv)n(er)24 b(systems.)30 b(In)24 b Fa(Pr)l(o)l(c)l(e)l(e)l(dings)k(of)e(OSDI)f ('96)p Fi(,)f(Octob)r(er)g(1996.)120 3131 y([10])45 b(James)24 b(Gw)n(ertzman)h(and)g(Margo)f(Seltzer.)35 b(The)25 b(case)g(for)f (geographical)h(push-cac)n(hing.)35 b(In)25 b Fa(Pr)l(o)l(c)l(e)l(e)l (dgins)k(of)e(HotOS)f('94)p Fi(,)275 3210 y(Ma)n(y)e(1994.)120 3363 y([11])45 b(Mor)17 b(Harc)n(hol-Balter,)h(Mark)g(Cro)n(v)n(ella,)g (and)g(Cristina)f(Murta.)22 b(On)17 b(c)n(ho)r(osing)i(a)f(task)g (assignmen)n(t)g(p)r(olicy)g(for)f(a)g(distributed)275 3441 y(serv)n(er)24 b(system.)30 b Fa(IEEE)d(Journal)g(of)f(Par)l(al)t (lel)h(and)g(Distribute)l(d)e(Computing)p Fi(,)f(59:204)h({)e(228,)h (1999.)120 3594 y([12])45 b(Mor)25 b(Harc)n(hol-Balter)h(and)g(Allen)f (Do)n(wney)-6 b(.)38 b(Exploiting)26 b(pro)r(cess)g(lifetime)e (distributions)h(for)g(dynamic)g(load)h(balancing.)275 3672 y(In)e Fa(Pr)l(o)l(c)l(e)l(e)l(dings)k(of)e(SIGMETRICS)h('96)p Fi(,)d(pages)g(13{24,)h(1996.)120 3825 y([13])45 b(A.)16 b(Iy)n(engar,)j(E.)e(MacNair,)g(and)g(T.)f(Nguy)n(en.)21 b(An)c(analysis)f(of)h(w)n(eb)g(serv)n(er)f(p)r(erformance.)k(In)d Fa(Pr)l(o)l(c)l(e)l(e)l(dings)k(of)f(GLOBECOM)275 3904 y('97)p Fi(,)k(1997.)120 4056 y([14])45 b(A.)25 b(Jean-Marie)h(and)g(P) -6 b(.)25 b(Rob)r(ert.)38 b(On)25 b(the)i(transien)n(t)f(b)r(eha)n (vior)g(of)g(the)g(pro)r(cessor-sharing)f(queue.)39 b Fa(Queueing)26 b(Systems:)275 4135 y(The)l(ory)g(and)h(Applic)l(ations) p Fi(,)e(17:129{136,)g(1994.)120 4287 y([15])45 b(M.)34 b(Kaasho)r(ek,)39 b(D.)34 b(Engler,)j(D.)e(W)-6 b(allac)n(h,)38 b(and)d(G.)g(Ganger.)66 b(Serv)n(er)35 b(op)r(erating)h(systems.)64 b(In)35 b Fa(SIGOPS)i(Eur)l(op)l(e)l(an)275 4366 y(Workshop)28 b('96)p Fi(,)23 b(pages)i(141{148,)g(1996.)120 4518 y([16])45 b(Dongeun)24 b(Kim,)c(Cheol)i(Ho)g(P)n(ark,)g(and)h(Daey)n(eon)g(P)n (ark.)29 b(Request)23 b(rate)g(adaptiv)n(e)g(dispatc)n(hing)h(arc)n (hitecture)f(for)e(scalable)275 4597 y(in)n(ternet)k(serv)n(er.)120 4749 y([17])45 b(Leonard)25 b(Kleinro)r(c)n(k.)31 b Fa(Queueing)25 b(Systems)p Fi(,)d(v)n(olume)i(I)r(I.)f(Computer)h(Applications.)31 b(John)25 b(Wiley)e(&)h(Sons,)f(1976.)120 4901 y([18])45 b(W.)25 b(E.)f(Leland)h(and)g(T.)f(J.)g(Ott.)35 b(Load-balancing)26 b(heuristics)e(and)h(pro)r(cess)g(b)r(eha)n(vior.)34 b(In)25 b Fa(Pr)l(o)l(c)l(e)l(e)l(dings)k(of)e(Performanc)l(e)275 4980 y(and)g(A)n(CM)e(Sigmetrics)p Fi(,)e(pages)h(54{69,)h(1986.)1845 5229 y Ft(23)p eop %%Page: 24 26 24 25 bop 120 91 a Fi([19])45 b(Je\013rey)20 b(C.)f(Mogul.)24 b(Net)n(w)n(ork)c(b)r(eha)n(vior)g(of)f(a)h(busy)g(w)n(eb)f(serv)n(er)g (and)h(its)g(clien)n(ts.)k(T)-6 b(ec)n(hnical)20 b(rep)r(ort,)g(DEC)f (WRL)h(Researc)n(h)275 170 y(Rep)r(ort)k(95/5,)h(Octob)r(er)f(1995.)120 324 y([20])45 b(Je\013rey)22 b(C.)f(Mogul)g(and)h(K.)f(K.)g (Ramakrishnan.)26 b(Eliminating)20 b(receiv)n(e)i(liv)n(elo)r(c)n(k)g (in)f(an)h(in)n(terrupt-driv)n(en)f(k)n(ernel.)28 b(In)22 b Fa(Pr)l(o)l(c.)275 403 y(USENIX)k(1996)h(T)-5 b(e)l(chnic)l(al)27 b(Confer)l(enc)l(e)p Fi(,)c(pages)i(99{111,)g(1996.)120 556 y([21])45 b(Da)n(vid)19 b(L.)g(P)n(eterson)h(and)f(Da)n(vid)g(B.)f (Adams.)k(F)-6 b(ractal)20 b(patterns)g(in)f(D)n(ASD)f(I/O)i(tra\016c.) j(In)c Fa(CMG)j(Pr)l(o)l(c)l(e)l(e)l(dings)p Fi(,)f(Decem)n(b)r(er)275 635 y(1996.)120 789 y([22])45 b(A.)23 b(Silb)r(ersc)n(hatz)i(and)f(P)-6 b(.)23 b(Galvin.)31 b Fa(Op)l(er)l(ating)26 b(System)g(Conc)l(epts,)g (5th)g(Edition)p Fi(.)31 b(John)25 b(Wiley)f(&)f(Sons,)h(1998.)120 942 y([23])45 b(W.)24 b(Stallings.)31 b Fa(Op)l(er)l(ating)26 b(Systems,)f(2nd)h(Edition)p Fi(.)32 b(Pren)n(tice)24 b(Hall,)f(1995.)120 1096 y([24])45 b(A.S.)23 b(T)-6 b(anen)n(baum.)32 b Fa(Mo)l(dern)26 b(Op)l(er)l(ating)g(Systems)p Fi(.)31 b(Pren)n(tice)24 b(Hall,)f(1992.)120 1249 y([25])45 b(Ronald)24 b(W.)g(W)-6 b(ol\013.)31 b Fa(Sto)l(chastic)c(Mo)l(deling)f(and)g(the)g (The)l(ory)g(of)g(Queues)p Fi(.)31 b(Pren)n(tice)24 b(Hall,)e(1989.) 1845 5229 y Ft(24)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF