Performance Optimization of Markov Models in Simulating Computer Networks

Transcription

1 Proceedings of he World Congress on Engineering and Copuer Science 9 Vol I WCECS 9, Ocober -, 9, San Francisco, USA Perforance Opiizaion of Marov Models in Siulaing Copuer Newors Nisrine Sinno, Hussein Youssef, Ghaddar Ahad Absrac--This paper concerns he applicaion of Discree Marov odels [], [] for he siulaion of copuer newors where he nuber of cliens reains unchanged. Two nuerical ehods [] has been used, he Power siulaion ehod and he P-siulaion ehod. Two newor odels has been used he Gordan-Newell and he Cox-Cox odels. In siulaing hose wo odels, he asypoic perforance of he siulaion has been copared, in er of he convergence crieria oward he saionary sae of he odel. Index Ters Copuer Newors Siulaion, Discree Marov Chains, Perforance evaluaion, P-siulaion, Power siulaion. I. INTRODUCTION Copuer newors belong o a class of physical syses ha can be sudied effecively by eans of discree evens siulaion odels. The copuer newor subjec of he siulaion is he closed newor Gordan-Newell newor and Cox-Cox newor wih a consan nuber of cliens. Considering he fac ha he sae of he newor is described in ers of nuerical behavior values, copuer newor falls in he caegory of eory less behavior because he nowledge of he presen decouples he pas fro he fuure. Though exploring he behavior of copuer newors hrough Marov odels is legal, because Marov chain is a special sochasic process in which he fuure probabilisic behavior of he process is uniquely deerined by is presen sae. Newor eniies such as he access behaviors o he newor ay be described in he for of discree chain evens. Two ypes of Marov chain odels will be applied in he siulaion, he Power Marov process and he P- siulaion Marov process in a discree sae space. The wo ehod s convergence o he saionary sae will be copared in order o find which ehod is ore reliable for he siulaion of he Gordan-Newell and Cox-Cox newor odels. F. Nisrine Sinno is in he Copuer and Applied Maheeics Deparen, A he faculy of Sciences, Lebanese Universiy, e-ail: nisrine@ul.edu.lb. S. Hussein Youssef is in he deparen of Physique des aeriaux, Faculy of Sciences, Lebanese Universiy. T.Ghaddar Ahad is in he Copuer and Applied Maheeics Deparen, A he faculy of Sciences, Lebanese Universiy II. SIMULATION MODELS A. Discree Tie Marov Model A discree ie Marov odel is defined in ers of a se of ransiion probabiliies beween he discree saes of he odel. Transiions are associaed wih ransiion probabiliies beween saes. The NxN ransiion probabiliies beween N saes is called he Transiion Probabiliy Marix P. If P is he iniial probabiliy vecor hen he sae probabiliy afer seps is given by he vecor : =P* P () he arix P is he -h ransiion arix afer uliplying P o iself, ies. A seady sae no ore changes in saes probabiliies are expeced when P is uliplied by he curren probabiliy P vecor, his is because he Marov chain is ergodic []. The seady sae probabiliy vecor in all saes of he Marov Model is such ha: Π x P= Π () This equaion will be used along wih he condiion n i = i = () in order o deerine all he seady sae probabiliies for he chosen newor odels. B. Gordan-Newell Newor [4] I is a newor designed wih a consan nuber of cliens. The nuber of possible saes is liied. The newor is coposed of several nodes. Supposing ha is he nuber of nodes, and n is he consan nuber of paces raveling in he newor sen by n differen cliens, N possible saes is given by : + N= C n (4) The se of possible saes for a hree nodes newor is given by: E={(i,j,) N, i+j+ = n} (5) For =, he nuber of saes will be for n paces:

2 Proceedings of he World Congress on Engineering and Copuer Science 9 Vol I WCECS 9, Ocober -, 9, San Francisco, USA n+ ( n + )( n + ) N = E = C = (6) The Gordan-Newell newor odel is illusraed in he Fig. below. µ a -a µ The nuber of saes of he newor increase rapidly wih n. If he nuber of paces accessing he nodes of he newor is n=, he nuber of saes will be 6 disribued aong he hree nodes, he saes are: (,,),(,,),(,,),(,,),(,,),(,,). Wih he sochasic variable a such as (<a<), during he ie [,+] he Bernouilli variable wih paraeer a is he nuber of paces accessing he newor. Defining A()= as he nuber of paces accessing he node during [,+], and A()= as nuber of paces no accessing he node during ha inerval, he probabiliy o have A() paces accessing one node will be: P[A() =] = a (7) and he probabiliy no o have A() paces accessing one node is: P[A()=] = -a. (8) In Fig. in hree nodes newor, i will be he nuber of paces accessing node, wih he probabiliy μ, j will be he nuber of paces accessing node wih he probabiliy μ, will be he nuber of paces accessing node wih he probabiliy μ, The oal nuber n of paces accessing he newor is consan wih n=i+ j+. The paces in node can eiher access node wih probabiliy a or access node wih probabiliy -a, paces in node can access node wih probabiliy, paces in node can access node wih probabiliy. The saes diagra for Gordan-Newell newor, wih paces (cliens) is given Fig.. µ Figure. Gordan-Newell Model wih n=i+j+ j Figure. Saes diagra for Gordan-Newell newor wih cliens In order o consruc an algorih o eorize he sae of i node I a a cerain ie,we consider ha X is he nuber of cliens a ha oen in node I, he vecor X = ( X, X, X ) will represen he se of cliens presen a nodes,, a ie in he newor. Firs heore: X is finie, hoogeneous, irreducible [] Second heore: he seady sae probabiliy vecor for any sae I (i,j,) is given by: i j i, j, = ζ ζ ζ (9) G wih i, j, = () If D i defines he average rae of cliens accessing node i, a seady sae hen : D = D + D, D = a.d, D = (-a).d () This syse is linear, hoogeneous and irreducible, i has a leas one soluion. One soluion is: ( x, ax, (-a)x ). The seady sae probabiliy for an average rae of cliens x in node, a*x in node, (-a)*x in node is respecively : ζ = x/ μ ζ = x/ μ j G= / ζ ζ ζ. i i+ j+ = n ζ = x/ μ, () Considering he arix Q[N,N] of he seady sae probabiliies, wih N=6 saes according o he cliens Gordan-Newell newor Fig., Q will have he diensions [6,5]. In every row of he arix, one colun eleen is null. if every sae I is defined by i and j hen is deduced fro

3 Proceedings of he World Congress on Engineering and Copuer Science 9 Vol I WCECS 9, Ocober -, 9, San Francisco, USA i, j, n =n-(i+j). () In order o siulae he newor according o he Power and P-siulaion ehods, algorihs convering fro sae I ino he for (i,j) and fro he for (i,j) ino sae I has been consruced (exracing line, hen colun, hen line wih resricion, hen colun wih resricion..). We will increen I fro o N- by increening i fro o N and increening j fro o N-i. C. The Cox-Cox newor odel[5] In his odel he newor has consan nuber of cliens, bu he ransiion probabiliy arix P varies according o a cerain coefficien given by consrucion. The odel is represened in he following Fig., i is he Coaxian server represening wo copuer newors A, B wih respecively coefficien λ and he nuber of nodes for A, and for B coefficien µ wih nuber of nodes n. The siulaion of his odel will show he convergence perforance in exree cases of an organized ransiion probabiliy arix and a disurbed one by odifying he coefficiens. Theores: Seady sae probabiliy arix in A is : A = (4) Seady sae probabiliy arix in B is : B = n (5) A = µ / (λ+µ) and B = λ/ (λ+µ) (6) III. TESTS AND RESULTS A. Gordan-Newell Newor siulaion Tess have been ade using over sochasic daa enries. Fig.[4] [5] [6] [7] show ha he ehod P-siulaion is ore efficien han he Power siulaion ehod because he convergence o he saionary sae is quicer wih P-siulaion. λ. λ. i λ. Pow er ehod Error DELTA PSiulaion ehod Error DELTA A λ λ λ λ μ.μ. μ μμ μμ Er or ieraion nuber Coparison bew een pow er and PSIM Error Power Error PSIM Er or ieraion nuber Coparison bew een Dela pow er and Dela PSIM Dela PSIM Dela Power Figure. Saes Diagra for Coaxian server (wih newors A,B) E ror E ror ieraion nuber ieraion nuber Figures [4][5][6][7] Coparison beween Power and P-siulaion of he Gordan-Newell Newor wih cliens. B. Cox-Cox Model siulaion Several odels of Cox-Cox newors has been siulaed by changing he coefficiens. Three odels siulaions

4 Proceedings of he World Congress on Engineering and Copuer Science 9 Vol I WCECS 9, Ocober -, 9, San Francisco, USA are shown in he figures 8,9,: The noral newor odel Fig.9, he organized odel (Fig.) and he disurbed odel (Fig.8). The seady sae probabiliy arixes A and B are calculaed using he Marov chain P-siulaion and Power siulaion ehods [6, 7]. Figure. Error in an Organized Model Siulaion Figure 8. Error in a Disurbed Model Siulaion The firs Fig.8 shows he disurbed odel, he second Fig.9 shows he case of a noral newor, while he hird Fig. represens he resuls for he organized odel. The firs conclusion is ha he coefficiens play a ajor role: Case of he disurbed Cox-Cox odel Fig. 8 : The Power ehod sees o converge rapidly a he beginning of he ieraions, bu he P-siulaion ehod leads o a ore accurae siulaion, he error a he end of he ieraions is less han he one in he Power siulaion ehod. Case of an organized Cox-Cox odel Fig.: In his odel, he Power ehod converges rapidly o he sable sae coparing o he P-siulaion ehod. Bu coparing o he resuls wih he disurbed odel (Fig.8), he siulaion convergence is uch slower. Case of a noral Cox-Cox odel wih noralized power siulaion and p-siulaion Fig.9 : Sae conclusion as for he disurbed odel, bu wih he noralized power siulaion ehod he convergence is ore accurae han wih he P-siulaion ehod.. Figure 9. Error in noral Cox-Cox Model wih noralized power siulaion ehod Those resuls le us conclude ha: Organized odels siulaion is less convergen han disurbed and noral odels. Tha he noralized power siulaion ehod is ore accurae han he power siulaion ehod wihou noralizaion. Because we canno apply he noralizaion in p- siulaion ehods, we couldn overcoe he error as we did wih noralized power siulaion ehod. So he noralized power siulaion is beer han p- siulaion in Cox-Cox newor odel.

5 Proceedings of he World Congress on Engineering and Copuer Science 9 Vol I WCECS 9, Ocober -, 9, San Francisco, USA In order o confir his conclusion, he ess of he siulaions where done using Malab and C++ for up o ieraions in case of a noral Cox-Cox odel. We found he following resuls: Errpsi (alab) < Errpower (Malab) wihou noralizaion Errpsi (C ++ ) < Errpower (C ++ ) wihou noralizaion Errpower (C ++ ) wih noralizaion < Errpower (Malab) wih noralizaion So for cerain odels he error in noralized Power siulaion ehod is less han he one obained when using he P-siulaion ehod bu his assupion canno be generalized. V. CONCLUSION The paper has described he coparison of he asypoic perforance beween he P-siulaion ehod and he Power siulaion ehod for several cases of newors, he Gordan- Newell newor and differen inds of Cox-Cox odels. The analysis of he resuls leads o he following conclusions: The convergence acceleraion of P-siulaion depends on he sae exraced fro he se E of chosen saes. P-siulaion converges oward accurae soluions beer han wih he Power siulaion ehod. Noralizaion of he Power siulaion ehod in case of disurbed Cox-Cox odels is ore efficien han he P-siulaion ehod. Increasing he asypoic perforance in Marov chains of high diension is an open access proble. The siulaion ehods proposed in his paper help in undersanding he behavior according o several Marov chains siulaion of copuer newors. Bu ore ess on oher ind of newors are needed in order o evaluae he siulaion ehods used. REFERENCES [] Bloch, Grenier, Herann and Trivedi, «Queuing Newor and Marov Chain», J. Wiley ed [] A.Bouillard. «Opiisaion e analyse probabilise de sysèes à événeens discres». PhD hesis, ENS Lyon, France, 5. [] P.Fernandes.«Méhodes nuériques pour la soluion de sysèes Maroviens à grand espace d'éas». PhD hesis, INPG, Grenoble, France, 998. [4] Kishor, Shridharbhai and Trivedi, «Probabiliy & saisics wih reliabiliy queuing and copuer science applicaions», PRENTICE- HALL, INC., Englewood Cliffs, New Jersey 76. [5] Willia J. Sewar. «Inroducion o he nuerical soluion of Marov Chains». Princeon Universiy Press, new Jersey 994. [6] Pellauail, Boyer, Leguesdron «Réseaux ATM e P-siulaion». [7] A.Bouillard. «Opiisaion e analyse probabilise de sysèes à événeens discres». PhD hesis, ENS Lyon, France, 5.