An Efficient Algorithm for the Exact Analysis of Multiclass Queueing Networks with Large Population Sizes

Transcription

1 An Efficient Algoithm fo the Exact Analyi of Multicla Queueing Netwok with Lage Population Size iuliano Caale Neptuny R&D via Duando 0-, I-08 Milan, Italy and Politecnico di Milano - DEI Via Ponzio /, I-0 Milan,Italy giuliano.caale@polimi.it ABSTRACT We intoduce an efficient algoithm fo the exact analyi of cloed multicla poduct-fom ueueing netwok model with lage population ize. We adopt a novel appoach, baed on linea ytem of euation, which ignificantly educe the cot of computing nomalizing contant. With the popoed algoithm, the analyi of a model with N ciculating job of multiple clae euie eentially the olution of N linea ytem with ode independent of population ize. A ditinguihing featue of ou appoach i that we can immediately apply theoem, olution techniue, and decompoition fo linea ytem to ueueing netwok analyi. Following thi idea, we popoe a block tiangula fom of the linea ytem that futhe educe the euiement, in tem of both time and toage, of an exact analyi. An example illutate the efficiency of the eulting algoithm in peence of lage population. Categoie and Subject Decipto C. [Pefomance of Sytem]: Modeling techniue eneal Tem Algoithm, Pefomance, Theoy Keywod Poduct-fom ueueing netwok, computational algoithm, exact analyi, nomalizing contant, multicla model. INTRODUCTION Analytical pefomance modeling of compute and communication ytem i often caied out uing ueueing netwok model. Fo hitoical eaon, cloed poduct-fom netwok [] have been the focu of the field fo ove twenty Pemiion to make digital o had copie of all o pat of thi wok fo peonal o claoom ue i ganted without fee povided that copie ae not made o ditibuted fo pofit o commecial advantage and that copie bea thi notice and the full citation on the fit page. To copy othewie, to epublih, to pot on eve o to editibute to lit, euie pio pecific pemiion and/o a fee. SIMetic/Pefomance 0, June 0, 00, Saint Malo, Fance. Copyight 00 ACM /0/000...$.00. yea [8]. Uing thee model, claic pefomance poblem a the capacity planning of compute ytem can be eaily tackled, typically with much le computational effot than imulation (ee, e.g., [] fo an oveview of application). Moeove, computational algoithm fo model with poduct-fom olution have lagely inpied futhe eeach. Fo intance, ueueing netwok with extended featue that ae uccefully applied in eveal aea including oftwae pefomance evaluation, paallel ytem analyi, and modeling of netwok with pioitie o blocking (e.g., [,,,, ]), can be olved by mean of modified veion of the algoithm popoed fo poduct-fom netwok. The ditinguihing featue of poduct-fom model i that imple cloed-fom expeion ae known fo the euilibium ditibution of netwok tate pobabilitie. The challenge i to efficiently compute the nomalizing contant that aue that tate pobabilitie um to one. Excellent development in the field [,, 9, 0,,,,,,, ] have hown that exact olution method ae computationally feaible in eveal cae of inteet. Howeve, the computational euiement uually become pohibitive when a lage multicla population i peent in the netwok. Thi ha led to the well-known concluion that typically multicla ueueing netwok ae intactable fo lage population ize. Thi i a evee difficulty of the theoy that ha been addeed fo pactical pupoe only by appoximation techniue a local iteative method (e.g., [8,, ]), bound and aymptotic expanion (e.g., [, 9]), o with the tuncation of Eule ummation when numeically inveting the geneating function of the nomalizing contant [0]. The eult peented in thi pape indicate that efficient exact algoithm fo multicla model with lage population ae poible. In the following ection, we intoduce a novel appoach fo computing nomalizing contant that i efficient even in thi difficult cae. The algoithm i baed on the olution of ytem of linea euation involving nomalizing contant, and follow fom a imple conideation. Seveal ecuence euation fo nomalizing contant have been popoed in peviou wok [,9,,,,]. Clealy, each of them give an altenative deciption of the tuctue of the nomalizing contant. We popoe to ue the imultaneou infomation of a et of ecuence euation to educe the cot of computing nomalizing contant. In paticula, the popoed algoithm ha the following tuctue. We conide a et of netwok deived fom the

2 model to be olved, and defined in uch a way that all nomalizing contant can be elated by a ytem of linea euation. Then, we inceae in a linea fahion, and imultaneouly fo all netwok, the total population fom to N, being N the um of cla population in the oiginal model. Afte each inceae, we compute nomalizing contant uing the linea ytem. Finally, we detemine pefomance meaue of inteet fo the oiginal model. In thi way, an exact analyi euie eentially the olution of N linea ytem. Moeove, depite the compoition of the et of netwok, and hence linea ytem tuctue, may change while inceaing population, linea ytem ode i alway independent of population ize. A a coneuence, we how that the computational cot i motly detemined by the numbe of ueue and clae, athe than by the total population, which i uually the laget of the thee paamete. Thee ae benefit and dawback connected to the intoduction of linea ytem of euation into ueueing netwok analyi. Beide the ignificant eduction of computational complexity, that i anyway a fundamental motivation, thee ae othe ignificant advantage. Fo intance, the new appoach pomote the application of linea algeba techniue to the analyi of poduct-fom netwok. We how an application of thi idea by defining a block tiangula fom of the linea ytem coefficient matix that ignificantly educe time and toage euiement. Moeove, the diffuion of pecialized oftwae fo linea algeba, like compute algeba ytem (e.g., [8]), may educe implementation effot. We found intead two dawback egading uniuene and numeical accuacy of olution achievable with tandad (inexact) linea ytem olve. Concening the latte, we believe that the new algoithm, beide the elated theoetical eult, i alo of pactical inteet due to it efficiency. Thu, ganting eult coectne i a fundamental objective of the peent analyi. A imple and definitive olution to numeical accuacy poblem follow by adopting an exact linea ytem olve in implementation (e.g., []). Thi ha a limited impact on the aymptotic complexity of the algoithm that emain, alo in the wot cae, the exact techniue of choice fo multicla model with lage population ize. Futhe, an exact olution let u avoid floating-point ange exception that feuently aie in nomalizing contant algoithm. Concening uniuene, we dicu ome cae whee the linea ytem doe not have a uniue olution due to a ingula coefficient matix. We popoe tategie to adde the poblem. The pape i oganized a follow. In Section we give peliminay definition. Related wok i citically analyzed in Section, whee we alo intoduce the olution appoach baed on linea ytem. Section define the algoithm that i futhe examined in Section uing linea algeba techniue. Computational euiement ae dicued in Section. A numeical example i peented in Section. Finally, Section 8 give concluion and outline futue wok. Theoem poof ae epoted in the final appendix.. PRELIMINARIES AND NOTATION. Multicla Cloed Queueing Netwok Let u conide a cloed poduct-fom model compoed by M load-independent ueue, R job clae and an abitay numbe of delay. The population ize fo cla i N, and the population vecto i N (N,..., N,..., N R ). We de- P R note by N = = N the total population. The loading ρ k (alo called in the liteatue evice demand) i the poduct between the mean evice time pe viit and the mean numbe of viit of cla at ueue k. We denote by ρ 0 the um of delay fo cla. Exact algoithm conideed in the following ection pefom ecuion on the numbe of job and ueue in the model. Without lo of geneality, we indicate the population cuently poceed by the ecuion with the R- dimenional vecto n (n,..., n,..., n, 0,..., 0), whee n > 0,. Hence, R, i the numbe of non-empty P clae of the cuent population. We denote by n = = n the total population of n. Let 0 be the zeo vecto, and let e be a vecto of all zeo except fo the -th element that i one. In eveal algoithm, e.g., Convolution [,], LBANC [9], and MVA [], n pan the population et pod(n) = {n 0 n N}, () with cadinality cad(pod(n)) = Q R = (N + ). A we how late in the pape, a cla ecuion algoithm a RE- CAL [] may be intead efomulated a a ecuion on lin(n) = {0, e, e,..., N e, N e + e, N e + e,..., N e + N e,..., N}, () whee cad(lin(n)) = N +. A gaphical compaion of the two et i given in Figue. Note that fo lage multicla population cad(lin(n)) << cad(pod(n)). While ecuing, eveal intemediate model ae olved fo each population n. We denote by j j(n) the population of an intemediate model olved when the cuent population i n. The elated netwok tuctue i pecified by a multiplicity vecto m (m,..., m k,..., m ) uch that, fo each ueue type k, k, the intemediate model contain m k ueue of that type [, 0,, ]. Hee, M, i the numbe of ueue type fo the cuent population. The total numbe of ueue i m = P m k. All m k ueue of type k have identical loading fo the nonempty clae. With a light abue of notation, we denote by ρ k the loading of cla job at ueue of type k. If not othewie tated, we aume that fo two ueue type k and k uch that k k thee exit at leat one cla,, uch that ρ k ρ k. Hence, each ueue type define a balanced ubnetwok of maximum ize. Note that by thi definition (), and i non-deceaing with. We call input netwok a netwok that ha the ame M ueue of the model that we want to olve ecuively, but with population not neceaily eual to N. The tuctue of the input netwok fo the cuent population n i denoted by M (M,..., M k,..., M ), whee P M k = M. Vey often, intemediate model tuctue will be pecified uing M. Fo intance, an intemediate model with multiplicity vecto m = M + e can be obtained by adding to the input netwok with population n a ueue of type. Fo the et of the pape, if not othewie tated, k,, c, k,, c, will index ueue type and non-empty clae. Summaizing Example A a ummaizing example, let u aume that we want to olve ecuively a model with M =, R =, N = (8, ), delay ρ 0 =, ρ 0 =, and loading ρ = 0, ρ =,

3 (,) (,) (,) (,) (0,) (,0) (,) (0,) (,0) (0,) (0,0) (,) a) pod(n) (,0) (,) (,0) (,) (0,0) b) lin(n) (,) Figue : Population et fo N = (, ) ρ = 0, ρ = 9, ρ = 0, ρ = 9. Fo illutation pupoe, we ecuively poce, fo each n lin(n), a et of ( + ) intemediate model (m, j) with multiplicity and population m {M, M + e,..., M + e }, () j {n, n e }. () By the definition, we have the following cae: if n 8 and n = 0 then =. In thi cae all ueue have identical loading, thu =, ρ 0 =, ρ = 0, M = e, (m, j) {(e, n e ), (e, n e e ), (e, n e ), (e, n e e )}. If n 8 and n then =, =, ρ 0 =, ρ 0 =, ρ = 0, ρ =, ρ = 0, ρ = 9, M = e + e, (m, j) {(e + e, n e + n e ), (e + e, n e + n e e ), (e + e, n e + n e ), (e + e, n e + n e e ), (e + e, n e + n e ), (e + e, n e + n e e )}. Model fo n = 0 ae olved uing temination condition pecified late. Thu, we do not define a pecific notation.. Nomalizing Contant Euation We denote by (m, j) the nomalizing contant of a model with multiplicity m and j = (j,..., j,..., j, 0,..., 0) job. Fom the BCMP theoem [], it follow that X Y m = f i (j i ), () i=0 whee the ummation i taken on the tate-pace ( (j 0,..., j i,..., j m) mx i=0 j i = j, j i 0, whee j i = (j i,..., j i,..., j i ), i the population at tation i, and the poduct-fom facto f i fo ueue and delay (the latte i aociated to the index i = 0) ae defined a f i (j i ) = (Q P ρj 0 0 /j0, if i = 0, ( ji)q ρj i i /ji, if i m. () Note that Q the numbe of opeation euied by () i of the j ode of +m = j that i infeaible in mot cae. Howeve, can be computed uing ecuence euation involving intemediate model. Fo compactne, if (m, j), we denote by +k the nomalizing contant of a ), model with population j e and multiplicity m + e k. Similaly, k efe to a model with j e and m e k, efe to j e and m, +k efe to j and m + e k, and k efe to j and m e k. The emoval of multiple ueue o job i denoted, e.g., by k,i,c efeing to j e e c and m e k e i. Thoughout the pape, we conide the following nomalizing contant ecuence euation. The convolution expeion [, ] fo, i.e., = k + and fo +k, i.e., +k = + X = X = ρ k, k, () ρ k +k, k. (8) Note that, without lo of geneality, we conide only convolution expeion among the m poible, efeing to the emoval of any of the identical ueue of type k. We alo intoduce the netwok population containt fo the model with j = n, i.e., n = ρ 0 + X m k ρ k +k,. (9) which ae an application of Little Law [0], and genealize to the cae m k > a imila et of euation peented in []. A taightfowad poof of (9) i given at the end of thi ection. Temination condition fo ecuion baed on ()-(9) ae = f 0(j), fo nomalizing contant of model with m = 0, and =, fo nomalizing contant of model with j = 0. The condition ae conitent auming 0 0 =. Pefomance meaue can be computed uing the following fomula [] fo the mean thoughput of cla X (m, j) = /, (0) and fo the mean cla ueue length fo ueue of type k Q k (m, j) = ρ k +k /. () Related meaue, a utilization o epone time, can be eaily obtained fom the above meaue (e.g., [9]). Note that a taightfowad poof of (9) follow by ineting (0) and () fo j = n into the population containt fo a cloed model [], given by n = ρ 0X (m, n) + P k m kq k (m, n). Thi motivate the choice of the name fo (9).. ANALYSIS OF RELATED WORK Thi ection intoduce the new olution appoach baed on linea ytem of nomalizing contant ecuence euation. In ode to pepae backgound fo the new algoithm, we efomulate popula computational algoithm fo cloed netwok a linea ytem of nomalizing contant ecuence euation.. Recuive Solution Uing Linea Sytem Mot exact algoithm fo poduct-fom netwok compute nomalizing contant uing the linea ecuence euation ()-(9). In geneal, a olution can be obtained by diffeent techniue. We popoe a novel appoach whee, fo each population n conideed by the ecuion, a vecto g g(n) of nomalizing contant of intemediate model i computed

4 uing a linea ytem of euation ()-(9). The element of g change with the conideed algoithm, and will be decibed late. Due to the lineaity of ()-(9), the techniue take the fom of a euence of linea ytem Ag = X = B g, () whee the matix A A(n) decibe the linea dependencie between the vaiable in g, the g g(n e ) vecto ae computed ecuively, and the B B (n) matice detemine the linea ytem known tem fom the g vecto. In geneal, if n lin(n), then B i the zeo matix fo. Theefoe, fom () it follow that the ecuive olution of ueueing netwok model uing a euence of linea ytem euie to elect:. the et of poceed population n, typically eithe lin(n) o pod(n),. the element of the g vecto fo all poceed n,. the mix of euation ()-(9) uch that A i an invetible matix and hence () ha a uniue olution. In the next ection, we how how to efomulate popula computational algoithm in the fom ().. Analyi of the Convolution Algoithm We begin with an analyi of the Convolution algoithm. In the convolution appoach, () i conideed fo an abitay ueue k, and ecuively applied to the contant k and in the ight hand ide until the netwok contain ueue o job. Hence, a linea ytem efomulation may conide, fo all n pod(n), a vecto g compoed by the nomalizing contant of intemediate model (m, j) with population j = n and multiplicity m {M, M e, M e,..., M M e, M M e e, M M e e,..., e, 0}. The numbe of intemediate model conideed fo each n i thu cad(g) = M +. Fo each n pod(n), the mix of euation fo () i P k = ρ k, () = that i the et of all poible euation () uch that. the unknown nomalizing contant in the left hand ide ae all element of the g vecto,. the nomalizing contant in the ight hand ide ae known tem belonging to the g vecto, which ae ecuively computed. Thu, looking at (), the A matix i defined by the coefficient of the nomalizing contant in the left hand ide, while the B matice depend on the ight hand ide only. Thi notation will be ued thoughout the et of the pape. A an example, a model with M = ueue, = ueue type, and no delay can be olved by mean of the following linea ytem ,,, = X = ρ ρ ρ, 0, whee we ued the temination condition fo m = 0 in the lat euation. It i clea fom the example that the Convolution algoithm ha a coefficient matix independent of both population and loading, and that i alway invetible... Application Example We now how a fit application of linea ytem of nomalizing contant ecuence euation. We modify the Convolution algoithm by intoducing new euation. We coneuently modify the linea ytem and the ecuive tuctue of the algoithm to account fo the new unknown. In geneal, it i uite difficult to integate, at leat in an efficient manne, (8) and (9) within (), ince they intoduce eveal new +k unknown. Intead, it may be helpful to include additional convolution expeion () fo diffeent choice of k. We conide a ytem of euation () fo all k. In thi cae, the nomalizing contant in g ae of intemediate model (m, j) with population j = n and multiplicity {m 0 m M}. () Q Thu cad(g) = (m k + ). Conideing the ame example een befoe, and omitting,, = 0, we have now ,,, = X = ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ, ρ, ρ, which i an ovedetemined linea ytem with euation and jut unknown. Theefoe, uing the additional euation, and povided that the eulting matix i invetible, we may avoid to ecuively evaluate ome tem of the ight hand ide by moving them into the vecto of unknown. Fo intance, let u aume = and obeve that the contant,,, and, ae the nomalizing contant of netwok with a ingle ueue and no delay, thu eual to the poduct-fom facto f, f and f, epectively. Conideing a new unknown the nomalizing contant and k, k, and educing the matix to uae by dopping unneceay vaiable and euation, we get ρ ρ ρ ρ ρ ρ = ρ ρ f + ρ f + ρ f + ρ f + ρ which ha a uniue olution if all loading of cla ae ditinct. Thu, with the incluion of additional convolution expeion we found a way to avoid the ecuion on cla when =, o to pefom a ecuion on lin(n) intead than on pod(n). In thi way, the numbe of ecuively evaluated population dop fom the (N + )(N + ) of,,

5 the oiginal algoithm to the jut N + N + of the new fomulation. Thi clealy lead to ignificant computational aving fo model with lage population ize. Nevethele, new ovehead ae intoduced fo linea ytem olution (e.g., the cot of LU factoization [0]). Howeve, the main poblem i that, even if the eult genealize eaily to model with a lage numbe of ueue, fo > the inceaed numbe of unknown typically make impoible to poce a lin(n) population et, and thi i intinically elated to the ue of () in the linea ytem. Since the cope of the peent pape i to develop a geneal lin(n) ecuion, we hall not conide the popoed vaiant of the Convolution algoithm in the following ection, a well a the incluion of () in linea ytem.. Analyi of the RECAL Algoithm In thi ection, we dicu the elation between RECAL [] and a lin(n) ecuion baed on (9) fo =. We begin by fit eviewing RECAL. In thi algoithm, each ueue i univocally aociated to a pecial cla, called elflooping cla, compoed by job looping indefinitely though the tation. Queue have to be changed into load-dependent tation in ode to eve elf-looping job. The advantage i that the nomalizing contant can then be computed uing the following efficient cla ecuion. Let D n k = {(d,..., d n ) X t d t = k, d t 0, t n} () be the et of combination with epetition of k element among n, which ha cadinality cad(dk n ) = n+k k. Then, the nomalizing contant h(v, j) of a netwok with M ueue, a delay eve, population vecto v fo the M elf-looping clae, and cuent population j = n fo the emaining clae, atifie the following ecuence euation h(v, n) = X d D M+ n whee d (d 0, d,..., d k,..., d M ), c(v, d) = ρd 0 0 d 0 c(v, d)h(v + d, n n e ), () MY v k + d k v k ρ d k k, () and with temination condition h(v, 0) = fo all v. Note that the nomalizing contant of the input netwok with population N and no elf-looping clae i immediately obtained by computing h(0, N) with (). Futhemoe, if ρ 0 = 0, then the ummation i on (d,..., d M ) Dn M. We emak that we did not mention the m vecto ince RECAL aume that the numbe of ueue in a model i alway eual to M, and that intemediate model diffe only fo the numbe of elf-looping job and non-empty clae. Hence, RECAL ha no ueue ecuion. Howeve, we how that a diffeent intepetation of () i poible, a poved by the following theoem. Theoem. Let M fo all population, and let each type k, k M, be compoed by m k identical ueue with loading eual to thoe of ueue k, k M, of the input netwok. Then, the ecuive inetion n time of (9) fo cla into all it ight hand ide nomalizing contant give () by etting v k = m k, k M. Infomally, Theoem tate that () i euivalent to the cla- population containt, povided that the M ueue in the input netwok ae neve aggegated into the ame ueue type. Futhemoe, the theoem ha the impotant coneuence of howing that, fom the elation v k = m k, the addition of elf-looping job at ueue k i euivalent to the inetion of new ueue into the balanced ubnetwok k. Theefoe, two vey diffeent phyical intepetation of RECAL may be conideed. Moeove, Theoem how that the definition of ueue type implicitly aumed in RECAL diffe fom that adopted in Section.. But ince ou ueue type definition, which aggegate maximal balanced ubnetwok into a ingle type, i indeed the mot efficient fo model with a lage numbe of identical ueue, we conclude that RECAL can be lagely impoved in thi cae, a ecently hown by []. A poible appoach to adde the poblem conit in ecuively olving (9) fo cla uing the ueue type definition that aggegate maximal balanced ubnetwok. Note that the atifice of conideing ingle-job clae in ode to educe computational cot may alo be applied to (9). Fom the above obevation, we have the following linea ytem efomulation of RECAL n n = ρ 0 + P m kρ k +k, (8) fo a et of population n lin(n), and whee the left hand ide unknown efe to intemediate model (m, j) having population j = n and multiplicity m M + (d,..., d ) (d 0, d,..., d ) D + N n, with cad(g) = +N n N n. Thu, the cadinality of g inceae while ecuing on population. The extenion of (8) by including (8) into the linea ytem lead u to conide the LBANC algoithm.. Analyi of the LBANC Algoithm The LBANC algoithm [9] ha been hown in [] to be the unnomalized veion of the MVA algoithm []. The computational euiement of the two method ae vey imila, and the main diffeence i that LBANC ue nomalizing contant, while MVA wok diectly with mean pefomance indice. Concening linea ytem of nomalizing contant ecuence euation, LBANC illutate a imultaneou ue of (8) and (9). The population ecuion i n pod(n), and the fomulation of LBANC dicued in [] can be efomulated a the following linea ytem ( +k = P ρ k +k, k, n = ρ 0 + P k m kρ k +k, (9) whee the left hand ide unknown have population j = n and multiplicity m {M, M + e,..., M + e }. (0) Theefoe, LBANC compute cad(g) = + contant at each population, and it i ubtantially moe efficient than convolution when << M, an advantage that ha not been peviouly pointed out in the liteatue. Anothe advantage of LBANC i the diect availability of the tem +k fo (), which ae intead computed in the Convolution algoithm with additional ecuion. Nevethele, the main ouce of complexity in both LBANC and Convolution emain the pod(n) population ecuion. The incluion of new euation into the linea ytem (9) in ode to educe computational cot i addeed in the next ection.

6 . ALORITHM DESCRIPTION. Peliminaie We initially aume that ρ 0 = 0,. The definition of the new algoithm tat by obeving that the incluion of the population containt fo the non-empty clae into (9) doe not intoduce new unknown and may inceae the numbe of independent euation in the linea ytem. Thu, thee additional euation have the potential to ignificantly educe the cot of computing nomalizing contant. In paticula, a we pove late, if a cetain et of euation (8)-(9) i conideed, and povided that the eulting linea ytem i non-ingula, then we can olve model uing an efficient lin(n) ecuion with vey diffeent computational tadeoff with epect to exiting algoithm. We can how that the above obevation hold tue alo fo the geneal cae whee we dop the ρ 0 = 0 condition. Depite now the additional population containt intoduce new unknown into the linea ytem, i.e., the c contant fo c, we can alway exploit the fact that thee contant efe to model with j = n job of cla. Thi allow u to include into the ytem alo the cla population containt fo the model with j = n e c, i.e., n c = ρ 0 c, + X m k ρ k +k c,, c, () which allow to compute the new unknown diectly fom g. Hence, the incluion of the new c unknown i compenated by the availability of additional cla population containt. Finally, we point out that in geneal the conideed linea ytem may not have neceaily full ank, but fo the ake of implicity in peenting, we initially take thi aumption. Singulaity condition will be dicued in Section.., a well a tategie to adde the poblem... Peliminay Analyi We now illutate ome difficultie aiing when including (8) and (9) into a linea ytem with the aim of defining a lin(n) ecuion. We conide a linea ytem elating nomalizing contant of model with multiplicity m {M, M+e, M+e } and population j {n, n e }. Aume that the cuent population n i uch that = and =. We have that the linea ytem defined by (8), k, and (9),, fo a lin(n) ecuion i whee and Ag B g Ag = B g, () ρ ρ 0 0 M ρ 0 M ρ n ρ n n ρ ρ M ρ 0 M ρ 0 ρ M ρ 0 M ρ 0 ρ , + +, + +,,, Note that () i undedetemined, having moe unknown than euation. Thi indicate that the conideed ytem of euation (8)-(9) may be inufficient fo a population ecuion in lin(n). Thu, a diffeent appoach i euied to take full advantage of the additional population containt.. The Linea Sytem We popoe to olve cloed multicla poduct-fom netwok uing a population ecuion in lin(n). Fo each ecuively poceed n lin(n), we compute the following vecto of unknown + g =, () whee i a vecto of nomalizing contant of intemediate model (m, j) having population j {n, n e,..., n e } and multiplicity m {M + (d,..., d ) (d,..., d ) D }. () Intead, + include all nomalizing contant that can be obtained by adding a ueue k, k, to the nomalizing contant in, i.e.: + { +k, +k,..., +k k, {,,..., } }. () Thu, + include contant of intemediate model (m +, j) with population j {n, n e,..., n e } and multiplicity m + {M + (d,..., d ) (d,..., d ) D }. () We denote by + and the component of g. Note that by the definition, g ha cadinality cad(g) = cad( + ) + cad() = + + +, () which i independent of population ize. Thu, fo fixed and, g and g have identical cadinality, and the linea ytem ode doe not inceae with n. Fo the cuent population n, g i computed fom g by the following linea ytem: 8 >< >: P +k c= n c ρ 0c c n = ρ 0 + ρ kc +k c = ρ k +k, k P P n c = ρ 0 c, + P m k ρ kc +k c = 0, c m k ρ k +k, m k ρ k +k c,, c (8) whee { +k, +k c } +, {, c}, { +k, +k c,} +, and {, c, }. We denote the fou goup of euation, fom the top, by (8a), (8b), (8c) and (8d). We motivate the linea ytem by the following two obevation: fo each n, we can diectly compute all nomalizing contant in fom g by (8c)-(8d). Once that ha been computed, we can detemine + by olving the ubytem (8a)-(8b).

7 Thu, the olution of (8) i alway decompoable into two imple poblem, a we dicu in Section. Futhemoe, we jutify ()-() by the following theoem. Theoem. The linea ytem (8) ha alway a uae matix of coefficient. A a coneuence of thi eult, if the coefficient matix A i non-ingula, the ytem i neve undedetemined and (8) ha a uniue olution. It i impotant to point out that ()- () define the mallet cadinality vecto fo which the above theoem hold, but othe definition of and +k may alo be conideed. Numeical conditioning, ingulaity, and block tiangula fom of the coefficient matix of (8) ae dicued in Section... Illutative Example We now how a mall example of (8). We conide the ame model of Section... We have = and =, and o g depend on D = D = {(, 0), (0, )} and D = D = {(, 0), (, ), (0, )}. In paticula, we have g = + T with + = +, +, +, +, +, +, T, (9) = T, (0) and imilaly g = g = + T whee Let + = +, +,, +, +,, +, then (8) i whee Ag and B g +,, T, () = + +, +, + T. () t zk (M k + z)ρ k, Ag = B g, () ρ ρ ρ ρ t 0 t n ρ t 0 0 t 0 0 n ρ n n n n ρ ρ ρ ρ t 0 t ρ t 0 t ρ t 0 0 t ρ t 0 0 t ρ 0 +, +, +, +, +, +, , +,, +, +,, +, +,, + +, + +, Compaed to the linea ytem of Section.., the coefficient matix A i now uae, and povided that it i noningula, we have a uniue olution to the linea ytem. Hence, a lin(n) ecuion i now poible.,.. The Computational Algoithm The new algoithm ha the following tuctue: : Compute g(n, 0,..., 0) : fo = to R do : Initialize g(n,..., N, 0,..., 0) : fo n = to N do : Setup and olve (8) fo n = (N,..., N, n, 0,..., 0) : end fo : end fo 8: Compute pefomance meaue Computation of g(n, 0,..., 0). The ecuion on the fit cla can be pefomed uing ingle-cla algoithm popoed in peviou wok (e.g., [, 9]). Futhemoe, unde cetain aumption, ingle cla nomalizing contant can be computed uing cloed-fom fomula (ee, e.g., [, ]), but exact aithmetic may be euied to avoid numeical intabilitie. Initialization of g(n,..., N, 0,..., 0). At each change of the value of, alo ()-(8) change. In paticula, the ode of the coefficient matix gow with, and thi ha the following implication:. the olution of the linea ytem become moe expenive a inceae. Hence, it i bette to poce the clae with the laget population fit, ince they euie the laget numbe of iteation.. At the fit population poceed afte a change, the vecto + (N,..., N, 0,..., 0) i not available. Fo intance, when = the fit linea ytem olved euie + (N, N, 0,..., 0) with multiplicity depending on the et D, but duing ecuion fo = only intemediate model defined on D and D wee olved. The econd iue can be addeed a follow. Let be the new cla to be poceed, and aume n =. Note that the contant in (n e ) = (N,..., N, 0,..., 0) ae known fom the ecuion on cla. We compute the euied + (n e ) = + (N,..., N, 0,..., 0) fom (n e ) uing the following linea ytem: 8 >< >: P +k P c= ρ kc +k c =, k, m k ρ kc +k c = n c + ρ 0c c, c () whee {, c} (n e ), { +k, +k c } + (n e ). We point out that the matix of coefficient of () i the coefficient matix C fo cla that we will intoduce in the next ection. Pefomance meaue. In ode to compute pefomance meaue by (0) and (), we need to compute,, R, and +k, k, R, fo the input netwok with population N. Note that the intemediate model olved by the linea ytem fo n = N have multiplicity m M, thu computing pefomance meaue

8 euie to emove the ueue added by ()-(). Thi tak ha to be pefomed in diffeent way depending on netwok tuctue. We have the following two cae: all delay ρ 0, R, ae tictly poitive; o thee exit one o moe clae with zeo delay. In the fit cae, we can ecuively emove the ueue added by ()-() fom the linea ytem olution fo n = N uing 8 >< >: = +k P R N = ρ 0 X = ρ k +k, k, m k ρ k ρ 0 +k, R. () Intead, in the econd cae we pemute cla indice o to poce lat all clae with ρ 0 = 0. Then if N R R, the euied nomalizing contant can be computed fom g(n,..., N R, N R R) by ecuive application of (9) fo cla R. Thi alo let u kip the olution of the lat R linea ytem (8). Othewie, the ecuive olution of (9) come to a point whee the poblem i ecuively educed to emoving ueue in the nomalizing contant computed fo cla R. Thu, a ecuive pocedue can be ued to compute pefomance meaue in thi cae.. LINEAR ALEBRA RESULTS. Block Tiangula Fom We now apply linea algeba techniue to implify (8). We begin by expeing the linea ytem in the fom () A(n )g = B g, () whee A(n ) indicate that only the n coefficient change in A while ecuing on cla population. The main limitation of thi dependence i that we cannot eue the ame LU factoization fo linea ytem depending on the ame. Nevethele, we how that () implicitly define A(n ) into an uppe block tiangula fom that ugget a taightfowad olution to the poblem. Note that in (8) the population n multiplie all and only the contant in. Theefoe, () can be expeed uing only population-independent matice a + B C A = g 0 n I B, () + whee I i the identity matix of ode, and C + C(, ) i a uae diagonal block of ode. Uing block auian elimination we get = n B g, (8) and we only need to olve the linea ytem C + = B g A. (9) The lat two fomula pove that we can ecuively olve (8) uing matice that in no way depend on n. In paticula, only C euie a LU factoization, which can anyway be eued fo all cla population. Futhemoe, by ineting (8) into (9) we get and defining F C + = B n A B g, (0) C B, F 0 C A B B, whee the zeo matix in F ha ize cad() x cad(g ), we may compactly efomulate (8) a g = F + n F g, n =,..., N. () which i a et of ecuence euation in the n vaiable. Depite that a block auian elimination i typically moe efficient than a ecuive olution of (), becaue F and F may not peeve the paity of C, thi compact fomulation may be of inteet fo theoetical eaon due to it implicity. We alo point out that an advantage deiving fom the above eult i that, if loading and delay ae all tictly poitive, then the numbe and the placement of the non-zeo coefficient in linea ytem matice ae fixed fo all model with ame and. Thu, it i poible to optimize the elated data tuctue and algoithm by pecomputation. Fo intance, the poition of the non-zeo in C may be eaily pecomputed. Then, the matix can be intantiated in an aay of minimum ize that can be acceed efficiently uing hahing. We alo emak that linea ytem olve pefom the ame opeation and memoy accee fo fixed and, and thi may alo be ued to optimize implementation... Fine-ain Decompoition Diect olution method fo linea ytem, e.g., auian elimination, euie a computational effot that i typically cubic with the ode of the coefficient matix [0]. Stating fom thi fact, we popoe a decompoition of the coefficient matix C into block tiangula fom that allow ignificant computational aving by educing the ode of linea ytem. Depite thi i not euied to gant the coectne of the eult, it i ueful to limit gowth of time and toage euiement. The decompoition depend on a patitioning of the et D, which immediately implie by () a patitioning of + and thu a decompoition of C. We patition D o that combination (d,..., d ) with ame poition of nonzeo element ae included into the ame patition. We denote by P ht the t-th patition compoed by combination with h non-zeo in the ame poition. Fo example, D can be patitioned into the et P = {(, 0, 0)}, P = {(0,, 0)}, P = {(0, 0, )}, P = {(,, 0), (,, 0)}, P = {(, 0, ), (, 0, )}, P = {(0,, ), (0,, )}, P = {(,, )}. So + become + = + (P ) + (P )... + (P ). () whee each ubet + (P ht ) contain nomalizing contant of intemediate model (m ht, j) with population j {n, n e,..., n e } and multiplicity m ht {M + (d,..., d ) (d,..., d ) P ht }. () Note that, by definition of D, the numbe of non-zeo element h mut ange in h H whee H min{, }. () Futhe, note that the numbe of poible choice fo the poition of the h non-zeo element i h. The emaining h element ae fee to be aigned to any of the h poition, and thi can be modeled by a et of combination with epetition D h. h Fom thi conideation, we have immediately cad (P ht ) = h + ( h) =, () h h

9 which give cad + = H X h= and the lat expeion implie + = h= h h h. () h, () which i an application of Vandemonde convolution [], and ummaize the effect of the patitioning. We alo emak that in (8) two nomalizing contant in + appea in the ame euation only if they efe to model diffeing by at mot one ueue aigned to diffeent type. Thi mean that non-zeo coefficient in C can elate eithe nomalizing contant of the ame + (P ht ), o contant depending on P ht and P h t with h h =. Thu, pemuting the column of C to eflect the patitioning of +, and in paticula oting column accoding to the numbe of non-zeo, we get the block tiangula fom whee C C C C C C C H C H,H C H, (8) C h diag(c h, c h,..., c ht,...), h H, (9) i a diagonal matix of h block cht that contain the coefficient fo the nomalizing contant in + (P ht ). We now pove the following theoem. Theoem. The h diagonal block cht C h ae uae matice of ode h, whee h h ow ae convolution expeion (8a), and h ( h) ow ae population containt (8b). Theefoe, thi fom geatly implifie the olution of (9), ince LU factoization i now euied only fo the diagonal block that have ode <<. h. Numeical Popetie +.. Eo Popagation Peliminay implementation of the popoed algoithm ugget that oundoff eo tend to accumulate while iteating on population [9]. Even on model whee C i numeically well conditioned (i.e., ha a low condition numbe [0]), the computation typically fail if the netwok contain moe than few ten of job. In ode to adde the poblem in a imple and effective way, we popoe to exactly olve (e.g., []) all linea ytem (8). Thi mean that exact aithmetic ha to be adopted thoughout all computation. If thi i implemented uing ational multipeciion aithmetic libaie (e.g., [8, ]) all digit of opeand ae kept, and no oundoff i pefomed. Depite the additional digit intoduce ome ovehead, exact ational aithmetic doe not uffe any kind of numeical intability. So we alo avoid undeflow o oveflow poblem that typically affect nomalizing contant evaluation. Moeove, nowaday multipeciion libaie (e.g., MP []) offe vey good pefomance fo opeand with thouand of digit, a we expect in nomalizing contant of lage model. Othe techniue fo exact linea algeba may alo be adopted, e.g. modula aithmetic [] that alo povide a natual way fo paallelizing linea ytem olution. We point to Section fo a dicuion on the cot of adopting exact multipeciion aithmetic... Singula Linea Sytem We now conide condition that poduce a ingula C matix, and dicu poible olution. Noting that the loading fo cla R ae neve included in C, we identify the following citical cae:. thee exit one o moe ueue type k with ρ kc = 0 fo ome c R. Thi condition yield column of all zeo in C. Thu, ome unknown of g ae not included into any euation.. Thee exit linea elation between loading that let ome C h block become ingula. The fit cae can be eaily addeed by emoving the unknown in + aociated with the zeo column. The eulting ytem i ovedetemined and till admit a uniue olution. The econd cae i moe difficult, becaue dependencie between loading may eult in a ingula C matix. Fo intance, thee exit ingula diagonal block if two ueue type k and k have ρ k c = ρ k c fo ome c. Singulaity condition can be enumeated by a ymbolic computation of the deteminant of the C h block. Fo intance, the C diagonal block fo =, = i C = ρ ρ ρ ρ, (0) ρ ρ which ha deteminant det(c ) = ρ ρ +ρ ρ +ρ ρ ρ ρ ρ ρ ρ ρ. Conideing ρ, we ee that fo ρ ρ the value ρ = ρ ρ ρ ρ + ρ ρ ρ ρ ρ ρ () give a zeo deteminant. Note that the complexity of ingulaity condition gow uickly with block ode (e.g., the deteminant of the C block fo = and = include ove 0 poduct tem). So, a numeical diagnoi of ingulaity i ecommended in implementation. We popoe to adde ingulaity by moving back into the vecto of known tem the unknown eponible fo the ingulaity of (8). Thi mean that an hybid population ecuion ha to be pefomed, whee ome nomalizing contant ae computed fom linea ytem, while the emaining one ae ecuively evaluated. Fo intance, if the nomalizing contant +k c and +k c of the lat t clae ae moved back to the ight hand ide of (8), then we can olve the model fo all population in pod(n R t+,..., N R ) uing lin(n,..., N R t ) ecuion. In thi way, the new nomalizing contant in the ight hand ide ae diectly available fom peviou ecuion. Finally, the ecuion fo the lat population (N R t+,..., N R ) olve the input netwok with population N. The computational cot in peence of a ingula coefficient matix i thu an intemediate cae between that fo non-ingula ytem and that of exiting pod(n) ecuion.

10 . COMPUTATIONAL REQUIREMENTS Finally, we dicu the computational euiement of the popoed algoithm. In paticula, we conide the cot of olving the euence of ytem (9). Thi tak i typically eveal ode of magnitude lowe than all othe opeation, e.g., (8) and the computation of pefomance meaue. Thoughout thi ection, we aume to ue LU factoization, fowad and back ubtitution [0] to olve (9). Since the additional complexity due to numeical tabilization i not a theoetical limit to the pefomance of the popoed algoithm, but athe chaacteize the bet cuently available table implementation, we fit uantify the cot of olving the euence of linea ytem egadle of accuacy of olution. Then, we dicu the aymptotic cot of adopting an exact linea ytem olve in implementation.. Solving the Seuence of Linea Sytem Let LU t(θ) (/)θ and LU (θ) θ be epectively the numbe of opeation and the toage euied by a LU factoization of a full-ank uae matix of ode θ. Similaly, let BS t (θ) θ be the total numbe of opeation of fowad and back ubtitution. Aume that C i in the fom (8), and denote by θ h h the ode of the diagonal block c ht when thee ae non-empty clae. Then, fom the conideation in Section.., the time euiement i of the ode of RX = h= RX h = h= < RX " = h= h LU t (θ h ) + " h h h h (h ) XN n =0 # " BS t (θ h ) # h + N + h (h ) + N + #, () whee the cae n = 0 account fo the initialization of g afte a change, and H min{, }. Note that the fomula i typically an uppe bound, becaue it doe not account fo the computational aving due to the paity of C. Moeove, auming to keep in memoy only the LU factoization fo the cuent value of, toage euiement i maximum fo = R, and it i of the ode of h= h LS (θ Rh ) h= h " < h= # R R R h h h which i independent of population ize. R h (h ), (). Aymptotic Cot of Stabilization If we adopt in implementation an exact olve baed on ational multipeciion aithmetic, then we have to account alo fo the inceaing cot of aithmetic opeation. Thu, we have in geneal that the time and toage euiement have to be pecified a LU t (θ) LU t (θ, n), LU (θ) LU (θ, n), and BS t BS t(θ, n). The exact numbe of opeation euied by multipeciion aithmetic tongly depend with implementation, thu we limit to tudy aymptotic behavio. Becaue the numbe of digit of the coefficient in C i contant, opeation euied to olve (9) have computational euiement that gow with the numbe of digit of the unknown in g and g. Let u aume all loading to be caled to intege [], and let u aume to keep with mall ovehead a common denominato between the vaiable, e.g., n n fo the nomalizing contant in g, and n n (n ) fo thoe in g. Then, the time cot of ational aithmetic opeation gow linealy with the numbe of digit of the numeato, while toage depend by both the ize of numeato and of the common denominato. Taking the logaithm of () in the wot-cae of a model with balanced m ueue, intege loading all eual to ρ max k, {ρ k }, 0 k, and population n, we get the following expeion log = log n m +n n n n ρ n < log (m + n ) n ρ n, () n n whee both numeato and denominato of the logaithm agument ae intege. Thu, we ee that the gowth of the numbe of digit of the numeato of i O(n log(m + n)). Thu, fo the contant in g, which have m M + + accounting alo the poduct-fom facto of the delay, the time and toage ovehead of exact aithmetic gow at mot a O(n log(m + + n)), which may be egaded a a cale facto fo the computational cot in the cae without tabilization.. Model with Lage Population Size Time and toage complexity fomula pove that the peented algoithm i vey efficient fo lage population ize, even accounting fo tabilization. In paticula, auming contant, M and R, and conideing N = κ fo all clae, we have that the time euiement gow fo κ + a O(κ), o O(κ log κ) accounting alo fo numeical tabilization, while unde the ame condition the Convolution, LBANC, and the MVA algoithm ae appoximately O(κ R ). Similaly, RECAL and othe efficient cla ecuion (e.g, [,, ]) euie fo lage population ize appoximately O(κ M ) o O(κ ) time. Similaly, accounting fo tabilization, the toage euiement of the algoithm, gow a O(κ log κ), that i typically eveal ode of magnitude malle than exiting algoithm. Theefoe, the popoed appoach may povide ubtantial computational aving fo model with lage population.. NUMERICAL EXAMPLE Finally, we compae the efficiency of the popoed algoithm with epect to exiting techniue by a numeical example. We olve a model with N = 000 job, M = ueue, = ueue type fo all population, and R = clae. The loading ρ k fo the conideed model ae given in Table. In the example, the multiplicitie M k neve change duing ecuion. Since complexity fomula fo RE- CAL ae available only fo the cae with no delay [], fo the pupoe of compaion we et ρ 0 = 0 fo all clae. We ued a peliminay implementation of the algoithm witten in C, and baed on the exact ational aithmetic outine of the MP libay [] veion... The algoithm wa un on a AMD Athlon 000+ poceo uing the -bit mode. The olution of the linea ytem in block tiangula fom wa pefomed by euing the ame LU factoization fo all population of cla. The computed nomalizing contant wa (N) = Since we ued exact aithmetic, all digit of wee available at the

11 Table : Example multicla model with N = 000 job belonging to R = clae. Queue Cla of euet Type k M k ρ k ρ k ρ k ρ k ρ k ρ k Population N end of the computation. We alo detemined thoughput and ueue-length fo all clae and ueue. Fo intance, the thoughput of cla i X (N) =. 0 job pe econd, while the aveage cla ueue-length at ueue of type i Q (N) =. job. Accounting multipeciion ovehead and all opeation not modeled by (), the olution wa obtained in. On the conideed achitectue thi i appoximately 0 opeation. Duing execution, the amount of allocated memoy wa alway le than 0 MByte, alo thank to the ue of pecialized data tuctue fo pae matice. Fo the ame model, the theoetical numbe of opeation of LBANC i 0 9, 0 0 fo Convolution, and 0 fo RECAL. Theefoe, compaed to the othe method, the popoed algoithm i the only computationally feaible on the conideed example. 8. CONCLUSIONS We peented a new efficient exact algoithm fo computing nomalizing contant of cloed multicla poduct-fom netwok. The method i baed on the new poweful concept of linea ytem of nomalizing contant ecuence euation, which may indicate futhe eeach development in the aea. An example peented in the pape how that the new algoithm can be eveal ode of magnitude fate than exiting techniue on model with lage population. A compaion on non-aymptotic population, a well a on model with lage numbe of ueue type and clae, i left a futue wok. It i cuently not clea whethe the eult peented in the pape may be extended to the load-dependent cae. Lack of eult concening the genealization of (8)-(9) to the loaddependent cae make it difficult to etablih whethe uch extenion i poible. In paticula, the lack of a uitable genealization of () to the load-dependent cae eem to be the main iue. 9. ACKNOWLEDEMENTS Additional mateial of inteet fo thi wok can be found in []. I wih to thank my advio, pof.. Seazzi, fo hi help and continuou uppot of my eeach. I thank J. Anelmi and the anonymou SIMetic eviewe fo uggetion that ubtantially helped in inceaing the uality of the pape. I am alo in debt with pof. P. Cemonei fo ueful dicuion. I thank D. Adagna, M. Rovei, C. Spelta and S. Zaneo fo contuctive comment. 0. REFERENCES [] S. Balamo, V. De Nitto Peoné, and R. Onvual. Analyi of ueueing netwok with blocking, 00. [] F. Bakett, K.M. Chandy, R.R. Muntz, and F.. Palacio. Open, cloed, and mixed netwok of ueue with diffeent clae of cutome. J.ACM, ():8 0, 9. [] A. Betozzi and J. McKenna. Multidimenional eidue, geneating function, and thei application to ueueing netwok. SIAM Rev., ():9 8, 99. [] R.M. Byant, A.E.Kzeinki, M.S.Lakhmi, and K.M. Chandy. The mva pioity appoximation. ACM Tan. Comp. Sy., (): 9, 98. [] J.P. Buzen. Computational algoithm fo cloed ueueing netwok with exponential eve. Comm. ACM, (9):, 9. [] S. Cabay. Exact olution of linea euation. In Poc. of the nd ACM SYMSAC Sympoium, page ACM Pe, 9. []. Caale. The Thoughput Analyi of Poduct-Fom Queueing Netwok. PhD thei, Politecnico di Milano, Milan, Italy, 00. [8] K.M. Chandy and D. Neue. Lineaize: A heuitic algoithm fo ueuing netwok model of computing ytem. Comm. ACM, ():, 98. [9] K.M. Chandy and C.H. Saue. Computational algoithm fo poduct-fom ueueing netwok model of computing ytem. Comm. ACM, (0): 8, 980. [0].L. Choudhuy, K.K. Leung, and W. Whitt. Calculating nomalization contant of cloed ueuing netwok by numeically inveting thei geneating function. J.ACM, ():9 90, 99. [] D.J.A. Cohen. Baic Techniue of Combinatoial Theoy. John Wiley and Son, 98. [] A.E. Conway, E. de Soua e Silva, and S.S. Lavenbeg. Mean value analyi by chain of poduct fom ueueing netwok. IEEE Tan. Comp., 8():, 989. [] A.E. Conway and N.D. eogana. RECAL - a new efficient algoithm fo the exact analyi of multiple-chain cloed ueueing netwok. J.ACM, ():8 9, 98. [] P. Cemonei and C. ennao. Integated pefomance model fo pmd application and mimd achitectue. IEEE Tan. Pa. Dit. Sy., ():0, 00. [] P. Cemonei, P.J. Schweitze, and. Seazzi. A unifying famewok fo the appoximate olution of cloed multicla ueuing netwok. IEEE Tan. Comp., :, 00. [] E. de Soua e Silva and S.S. Lavenbeg. Calculating joint ueue-length ditibution in poduct-fom ueueing netwok. J.ACM, ():9 0, 989. [] E. de Soua e Silva and R.R. Muntz. Queueing netwok: Solution and application. Technical Repot CSD-8900, CS Dep. - UCLA, Sep 989. [8] J.. Duma et al. Linbox: A geneic libay fo exact linea algeba. In Poc. ICMS 0, page 0 0, Wold Scientific, Singapoe, 00.

12 [9] David oldbeg. What evey compute cientit hould know about floating-point aithmetic. ACM Comp. Suv., (): 8, 99. [0]. H. olub and C. F. Van Loan. Matix computation. John Hopkin Univ. Pe, 99. [] T. anlund. NU MP: The NU Multiple Peciion Aithmetic Libay, Aug [] P.. Haion. On nomalizing contant in ueueing netwok. Opeation eeach, : 8, 98. [] P.. Haion and S. Couy. On the aymptotic behaviou of cloed multicla ueueing netwok. Pef. Eval., (): 8, 00. [] P.. Haion and T.T.Lee. A new ecuive algoithm fo computing geneating function in cloed ueueing netwok. In Poc. of the 00 IEEE Int l MASCOTS Sympoium, page 0. IEEE Pe, 00. [] J. Kiz. Thoughput bound fo cloed ueueing netwok. Pef. Eval., (): 0, 98. [] S. Lam. Dynamic caling and gowth behavio of ueueing netwok nomalization contant. J.ACM, 9():9, 98. [] S. Lam. A imple deivation of the mva and lbanc algoithm fom the convolution algoithm. IEEE Tan. Comp., :0 0, 98. [8] S.S. Lavenbeg. A pepective on ueueing model of compute pefomance. Pef. Eval., 0():, 989. [9] E.D. Lazowka, J. Zahojan,.S. aham, and K.C. Sevcik. Quantitative Sytem Pefomance. Pentice-Hall, 98. [0] J. D. C. Little. A poof of the ueueing fomula L = λw. Opeation Reeach, page 9:8 8, 9. [] Michael T. McClellan. The exact olution of ytem of linea euation with polynomial coefficient. J.ACM, 0(): 88, 9. [] J. McKenna and D. Mita. Aymptotic expanion and integal epeentation of moment of ueue length in cloed makovian netwok. J.ACM, (): 0, Ap 98. [] M. Reie and H. Kobayahi. Queueing netwok with multiple cloed chain: Theoy and computational algoithm. IBM J. Re. Dev., 9():8 9, 9. [] M. Reie and S.S. Lavenbeg. Mean-value analyi of cloed multichain ueueing netwok. J.ACM, ():, 980. [] J.A. Rolia and K.C. Sevcik. The method of laye. IEEE Tan. Soft. Eng., (8):89 00, 99. [] P.J. Schweitze. Appoximate analyi of multicla cloed netwok of ueue. In Int l Conf. on Stoch. Contol and Optim., Amtedam, page 9, 99. [] M. Seeno. Mean value analyi of poduct fom olution ueueing netwok with epetitive evice blocking. Pef. Eval., :9, 999. [8] The PARI oup, Bodeaux. PARI/P, veion.., [9] J. Zahojan, K.C. Sevcik, D.L. Eage, and B. alle. Balanced job bound analyi of ueueing netwok. Comm. ACM, ():, 98. APPENDIX Poof of Theoem We how that the ecuive unfolding of (9) in the fom (m, n) = ρ0 n (m, n e ) + MX m k ρ k n (m + e k, n e ), give () by etting v k m k. Telecoping on cla population we get (m, n) = X d D M+ n a(d) n ρd 0 0 MY m (d k) k ρ d k k (m+d, n ne), whee (m + d, n n e ) efe to a model with cla le, d (d 0, d,..., d M ), the tem a(d) = (d d M )/(d 0 d M ) i the numbe of ecuion that teminate with (m + d, n n e ) in the ight hand ide, and = m k (m k + ) (m k + d k ) i the iing factoial. We get () noting that (d d M ) = n, and etting v k = m k in m (d k) k, i.e., m (d k) k a(d) n = d 0 MY MY m (d k) k ρ d k k = d 0 Poof of Theoem MY m (d k) k ρ d k k d k (m k + d k ) ρ d k k (m k )d k = d 0 MY v k + d k ρ d k k v. k Note that we can compute fom g each contant in uing a cla population containt of (8). Theefoe, we only need to pove that the numbe of euation (8a) and (8b) i eual to cad( + ) = +. Note that fo each of the + contant (m, n) we have population containt in (8), o the numbe of euation + (8b) i ( ). Futhe, note that we have poible way of adding a new ueue to m, hence the numbe of euation (8a) i exactly +. By the popety k n = n n k k, the total numbe of euation (8a) and (8b) i then ( + ) = = cad( + ). + Poof of Theoem + By definition, each block c ht C h ha coefficient of contant in + (P ht ). Thu, the numbe of column of c ht i h. Note that we can emove at mot h tation fom m {M + (d,..., d ) (d,..., d ) P ht } o that the eulting model i in (). Hence, the numbe of convolution expeion i cad(p ht )h = h h. The numbe of population containt with coefficient in C h C h(h+) i given by the containt (8b) multiplied by the numbe of ytem in () with aociated combination (d,..., d ) having h non-zeo, i.e., ( ) h ( ). Now the theoem follow by the elation k n = n n k k that implie that the numbe of population containt i h ( h), and o alo the numbe of ow i h + ( h) = h h h.