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y(researc)o(h)j(will)f(serv)o(e)g(to)i(impro)o(v) o(e)d(presen)o(t)h(metho)q(ds)h(of)h(mo)q(delling)d(computer)h(net)o(w) o(orks)h([2].)112 1775 y(A.)48 b(Queueing)16 b(Theory)h(Bac)o(kground) 112 1924 y(Cen)o(tral)g(to)h(in)o(terpreting)d(results)i(from)f(an)o(y) h(queueing)g(mo)q(del)f(is)h(the)f(understanding)i(of)g(the)112 2024 y(underlying)g(queueing)g(pro)q(cess.)27 b(If)18 b(y)o(ou)g(no)o(w)h(refer)e(to)i(Fig.)f(1,)g(y)o(ou)g(will)g(see)f(an)i (example)d(of)112 2124 y(the)21 b(L)213 2118 y Fh(a)237 2124 y Fn(T)264 2139 y(E)291 2124 y(X)f(picture)g(en)o(vironmen)o(t.)32 b(Of)20 b(particular)g(imp)q(ortance)g(are)g(\014v)o(e)g(basic)h(c)o (haracter-)112 2223 y(istics)f([3]:)30 b(arriv)m(al)20 b(pattern)h(of)g(customers,)f(service)f(pattern)i(of)g(serv)o(ers,)f (queue)g(discipline,)112 2323 y(system)15 b(capacit)o(y)l(,)g(and)i (the)f(n)o(um)o(b)q(er)e(of)j(service)e(c)o(hannels.)p 112 2413 225 2 v 200 2485 a(Journal)i(mo)q(del)e(is)h 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535 V 1070 535 V 1188 535 V 1306 535 V 588 676 a Fn(Fig.)16 b(1.)22 b(A)15 b(sample)g Fl(picture)f Fn(en)o(vironmen)o(t.)112 819 y(B.)48 b(Net)o(w)o(ork)15 b(Applications)112 968 y(The)j(study)g(of)g(a)g(net)o(w)o(ork)g(of)g (queues)f(can)h(b)q(e)g(used)g(to)g(pro)o(vide)f(useful)g(information)g (for)h(the)112 1067 y(design)23 b(and)f(main)o(tenance)f(of)h(computer) f(net)o(w)o(orks,)i(where)e(sev)o(eral)g(computers)g(are)i(com-)112 1167 y(m)o(unicating)f(with)h(eac)o(h)f(other.)42 b(On)23 b(the)g(design)g(side,)h(the)f(mo)q(deling)f(of)h(a)h(net)o(w)o(ork)e (can)112 1267 y(pro)o(vide)d(statistics)h(suc)o(h)f(the)h(a)o(v)o (erage)f(n)o(um)o(b)q(er)f(of)i(pac)o(k)o(ets)f(w)o(aiting)g(to)h(b)q (e)g(transmitted)e(at)112 1366 y(eac)o(h)e(computer)f([4].)112 1541 y(C.)49 b(Solution)16 b(Metho)q(ds)112 1690 y(The)27 b(most)e(common)f(n)o(umerical)g(solution)i(to)h(the)f(transien)o(t)f (queueing)h(mo)q(del)f(is)h(found)112 1790 y(through)20 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1866 V 1476 1874 V 1476 1882 V 1476 1890 V 1476 1899 V 1476 1907 V 1476 1915 V 1476 1924 V 1476 1932 V 1476 1940 V 1476 1949 V 1476 1957 V 1476 1965 V 1476 1974 V 1476 1982 V 1476 1990 V 1476 1998 V 1476 2007 V 1476 2015 V 1476 2023 V 1476 2032 V 1476 2040 V 1476 2048 V 1476 2057 V 1476 2065 V 1476 2073 V 1476 2081 V 1476 2090 V 1476 2098 V 1476 2106 V 1476 2115 V 1476 2123 V 1476 2131 V 1476 2140 V 1476 2148 V 1476 2156 V 1476 2164 V 1476 2173 V 1476 2181 V 1476 2189 V 1476 2198 V 1476 2206 V 1476 2214 V 1476 2223 V 1476 2231 V 1476 2239 V 1476 2247 V 344 2436 a Fn(Fig.)h(4.)21 b(Sto)q(c)o(hastic)c(balance)f(for)g (tandem)f(queue)h(without)g(feedbac)o(k.)p eop %%Page: 7 16 7 15 bop 1888 -118 a Fn(7)542 123 y(+)108 b Fm(\015)713 130 y Fj(2)733 123 y Fm(P)764 130 y Fj(0)p Fk(;i)p Fd(\000)p Fj(1)862 123 y Fn(+)11 b Fm(\026)940 130 y Fj(1)961 123 y Fm(P)992 130 y Fj(1)p Fk(;i)p Fd(\000)p Fj(1)1474 123 y Fm(i)i Fn(=)h(1)p Fm(;)8 b Fn(2)p Fm(;)g Fn(3)p Fm(:::)98 b Fn(\(2.2\))304 201 y Fm(dP)360 208 y Fk(j;)p Fj(0)p 304 223 101 2 v 333 269 a Fm(dt)451 235 y Fn(=)42 b Fc(\000)8 b Fn(\()p Fm(\015)622 242 y Fj(1)653 235 y Fn(+)j Fm(\015)727 242 y Fj(2)758 235 y Fn(+)g Fm(\026)836 242 y Fj(1)856 235 y Fn(\))d Fm(P)914 242 y Fk(j;)p Fj(0)969 235 y Fn(+)j Fm(\015)1043 242 y Fj(1)1063 235 y Fm(P)1094 242 y Fk(j)r Fd(\000)p Fj(1)p Fk(;)p Fj(0)1197 235 y Fn(+)g Fm(\026)1275 242 y Fj(2)1295 235 y Fm(P)1326 242 y Fk(j;)p Fj(1)1468 235 y Fm(j)16 b Fn(=)e(1)p Fm(;)8 b Fn(2)p Fm(;)g Fn(3)p Fm(:::)98 b Fn(\(2.3\))310 320 y Fm(dP)366 327 y Fk(j;i)p 310 342 95 2 v 336 388 a Fm(dt)451 354 y Fn(=)42 b Fc(\000)8 b Fn(\()p Fm(\015)622 361 y Fj(1)653 354 y Fn(+)j Fm(\015)727 361 y Fj(2)758 354 y Fn(+)g Fm(\026)836 361 y Fj(1)867 354 y Fn(+)g Fm(\026)945 361 y Fj(2)965 354 y Fn(\))d Fm(P)1023 361 y Fk(j;i)1073 354 y Fn(+)j Fm(\015)1147 361 y Fj(1)1167 354 y Fm(P)1198 361 y Fk(j)r Fd(\000)p Fj(1)p Fk(;i)1295 354 y Fn(+)g Fm(\026)1373 361 y Fj(2)1393 354 y Fm(P)1424 361 y Fk(j;i)p Fj(+1)542 466 y Fn(+)108 b Fm(\015)713 473 y Fj(2)733 466 y Fm(P)764 473 y Fk(j;i)p Fd(\000)p Fj(1)859 466 y Fn(+)11 b Fm(\026)937 473 y Fj(1)957 466 y Fm(P)988 473 y Fk(j)r Fj(+1)p Fk(;i)p Fd(\000)p Fj(1)1447 466 y Fm(j;)d(i)14 b Fn(=)g(1)p Fm(;)8 b Fn(2)p Fm(;)g Fn(3)p Fm(:::)82 b Fn(\(2.4\))210 616 y(The)18 b(mean)f(and)i(v)m(ariance)f(statistics)g(for)g(the)g(second)g (queue)f(are)i(obtained)f(b)o(y)g(the)f(fol-)112 715 y(lo)o(wing)g(equations:)816 815 y Fm(M)863 822 y Fj(2)897 815 y Fn(=)961 761 y Fd(1)949 773 y Fi(X)950 864 y Fk(i)p Fj(=1)1017 815 y Fm(i)11 b Fc(\001)1083 761 y Fd(1)1070 773 y Fi(X)1070 864 y Fk(j)r Fj(=0)1139 815 y Fm(P)1170 822 y Fk(j;i)129 941 y Fn(and)742 1041 y Fm(V)770 1048 y Fj(2)804 1041 y Fn(=)869 987 y Fd(1)856 1000 y Fi(X)858 1091 y Fk(i)p Fj(=1)925 1041 y Fm(i)942 1021 y Fj(2)972 1041 y Fc(\001)1010 987 y Fd(1)998 1000 y Fi(X)997 1091 y Fk(j)r Fj(=0)1066 1041 y Fm(P)1097 1048 y Fk(j;i)1147 1041 y Fc(\000)g Fm(M)1249 1021 y Fj(2)1244 1053 y(2)1269 1041 y Fm(:)112 1168 y Fn(Calculation)20 b(of)g(the)g(mean)f(and)h(v)m (ariance)g(requires)e(the)i(truncation)g(of)g(the)g(M/M/1)p Fm(=)p Fc(1)g Fn(to)112 1267 y(some)c(maxim)o(um)d(n)o(um)o(b)q(er)i (of)i(states.)24 b(Stated)17 b(di\013eren)o(tly)l(,)e(the)i(M/M/1/)p Fc(1)h Fn(queue)e(mo)q(del)g(is)112 1367 y(appro)o(ximated)f(b)o(y)g (an)i(M/M/1/k)g(queue.)j(While)15 b(it)g(is)h(imp)q(ossible)e(to)j(ev)m (aluate)e(the)h(error)g(in)112 1467 y(this)d(appro)o(ximation,)f(an)h (indication)f(of)h(the)f(truncation)h(error)f(can)h(b)q(e)g(obtained)g (b)o(y)f(summing)112 1566 y(all)h(the)f(probabilit)o(y)g(states)h(up)f (to)h(state)g Fm(k)i Fn(and)e(subtracting)g(this)f(total)h(from)f(one.) 20 b(This)13 b(yields)112 1666 y(the)k(probabilit)o(y)e(of)i(b)q(eing)g (in)f(a)h(state)g(greater)f(than)h Fm(k)r Fn(.)22 b(If)16 b(this)h(v)m(alue)f(is)g(v)o(ery)g(small)e(then)j(its)112 1765 y(pro)q(duct)g(with)g Fm(i)e Fn(and)i Fm(i)551 1747 y Fj(2)587 1765 y Fn(will)e(also)i(b)q(e)f(small.)210 1865 y(It)21 b(is)f(easy)i(to)f(see)g(ho)o(w)g(large)g(and)h (complicated)c(the)j(Kolmogoro)o(v)f(equation)h(set)g(can)112 1965 y(b)q(ecome)e(for)h(just)h(a)f(small)f(net)o(w)o(ork,)h(and)g(the) g(usefulness)g(of)g(an)h(accurate,)f(state-reducing)112 2064 y(appro)o(ximation)c([11].)112 2239 y(B.)48 b(Appro)o(ximations)15 b(for)h(the)g(T)l(andem)g(Queue)626 2388 y(1.)49 b(Indep)q(enden)o(t)15 b(Queue)h(Assumption)112 2520 y(Jac)o(kson)f([12)q(])f(sho)o(w)o(ed)h (that)g(a)g(net)o(w)o(ork)f(of)h(queues)f(can)h(b)q(e)f(analyzed)h(as)g (a)g(group)h(of)f(indep)q(en-)112 2619 y(den)o(t)21 b(M/M/1)g(queues)g (when)g(the)g(net)o(w)o(ork)f(is)h(op)q(erating)g(under)g(steady-state) h(conditions.)p eop %%Page: 8 17 8 16 bop 1888 -118 a Fn(8)112 123 y(One)17 b(metho)q(d)f(to)h(appro)o (ximate)e(the)h(tandem)g(queue)f(state)i(space)g(is)f(to)h(assume)f (that)h(the)g(in-)112 223 y(dep)q(endence)c(holds)h(under)f(transien)o (t)h(conditions)f(as)h(w)o(ell.)19 b(By)13 b(assuming)g(the)g(t)o(w)o (o)g(queues)g(are)112 322 y(indep)q(enden)o(t,)19 b(the)g(join)o(t)g (probabilit)o(y)f Fm(P)893 329 y Fk(j;i)951 322 y Fn(simply)f(b)q (ecomes)h(the)h(pro)q(duct)h(of)g(the)f(marginal)112 422 y(probabilities,)h Fm(P)441 429 y Fk(j)480 422 y Fn(and)h Fm(P)610 429 y Fk(i)625 422 y Fn(.)33 b(Th)o(us,)21 b(the)f(n)o(um)o(b)q(er)e(of)j(states)g(needed)e(to)i(mo)q(del)e(the)h (tandem)112 521 y(M/M/1/50)e(queue)e(b)o(y)g(the)g(Kolmogoro)o(v)f (equations)i(decreases)f(from)f(2500)j(to)e(100.)210 621 y(Since)k(the)h(primary)f(motiv)m(ation)g(b)q(ehind)h(the)g(appro)o (ximation)f(metho)q(ds)h(is)g(to)g(obtain)112 721 y(accurate)f(mean)e (and)h(v)m(ariance)g(statistics)g(for)h(the)f(queues,)g(it)g(is)g(of)g (in)o(terest)f(to)i(in)o(v)o(estigate)112 820 y(errors)g(induced)e(b)o (y)g(assuming)h(the)f(queues)g(to)i(b)q(e)f(indep)q(enden)o(t.)28 b(The)19 b(mean)e(and)j(v)m(ariance)112 920 y(statistics)d(for)f(the)g (\014rst)h(and)f(second)h(queues)f(are)g(de\014ned)g(as)746 1169 y Fm(M)793 1176 y Fj(1)855 1169 y Fn(=)947 1115 y Fd(1)935 1128 y Fi(X)934 1219 y Fk(j)r Fj(=1)1004 1169 y Fm(j)e Fc(\001)d Fm(P)1094 1176 y Fk(j)765 1318 y Fm(V)793 1325 y Fj(1)855 1318 y Fn(=)947 1264 y Fd(1)935 1277 y Fi(X)934 1368 y Fk(j)r Fj(=1)1004 1318 y Fm(j)1027 1298 y Fj(2)1058 1318 y Fc(\001)f Fm(P)1113 1325 y Fk(j)1143 1318 y Fc(\000)h Fm(M)1245 1298 y Fj(2)1240 1331 y(1)1265 1318 y Fm(;)112 1468 y Fn(and)752 1617 y Fm(M)799 1624 y Fj(2)861 1617 y Fn(=)952 1563 y Fd(1)940 1576 y Fi(X)942 1667 y Fk(i)p Fj(=1)1008 1617 y Fm(i)g Fc(\001)g Fm(P)1092 1624 y Fk(i)1812 1617 y Fn(\(2.5\))771 1760 y Fm(V)799 1767 y Fj(2)861 1760 y Fn(=)952 1706 y Fd(1)940 1718 y Fi(X)942 1810 y Fk(i)p Fj(=1)1008 1760 y Fm(i)1025 1739 y Fj(2)1056 1760 y Fc(\001)g Fm(P)1112 1767 y Fk(i)1137 1760 y Fc(\000)g Fm(M)1239 1739 y Fj(2)1234 1772 y(2)1259 1760 y Fm(:)539 b Fn(\(2.6\))210 1909 y(The)14 b(accuracy)f(of)h Fm(P)591 1916 y Fk(j)624 1909 y Fn(for)g Fm(j)j(>)c Fn(0)i(will)d 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