On the Approximation Error of Mean-Field Models

Transcription

1 On the Approxmaton Error of Mean-Feld Models Le Yng School of Electrcal, Computer and Energy Engneerng Arzona State Unversty Tempe, AZ ABSTRACT Mean-feld models have been used to study large-scale and complex stochastc systems, such as large-scale data centers and dense wreless networks, usng smple determnstc models dynamcal systems. Ths paper analyzes the approxmaton error of mean-feld models for contnuous-tme Markov chans CTMC, and focuses on mean-feld models that are represented as fnte-dmensonal dynamcal systems wth a unque equlbrum pont. By applyng Sten s method and the perturbaton theory, the paper shows that under some mld condtons, f the mean-feld model s globally asymptotcally stable and locally exponentally stable, the mean square dfference between the statonary dstrbuton of the stochastc system wth sze M and the equlbrum pont of the correspondng mean-feld system s O/M. The result of ths paper establshes a general theorem for establshng the convergence and the approxmaton error.e., the rate of convergence of a large class of CTMCs to ther mean-feld lmt by manly lookng nto the stablty of the mean-feld model, whch s a determnstc system and s often easer to analyze than the CTMCs. Two applcatons of mean-feld models n data center networks are presented to demonstrate the novelty of our results.. INTRODUCTION The mean-feld method s to study large-scale and complex stochastc systems usng smple determnstc models. The dea of the mean-feld method s to assume the states of nodes n a large-scale system are ndependently and dentcally dstrbuted..d.. Based on ths..d. assumpton, n a large-scale system, the nteracton of a node to the rest of the system can be replaced wth an average nteracton, and the evoluton of the system can then be modeled as a determnstc dynamcal system, called a mean-feld model. Then the macroscopc behavors of the stochastc system can be approxmated usng the mean-feld model, e.g., the statonary dstrbuton of the stochastc system may be ap- Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. Copyrghts for components of ths work owned by others than the authors must be honored. Abstractng wth credt s permtted. To copy otherwse, or republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. Request permssons from permssons@acm.org. SIGMETRICS 6, June 4-8, 26, Antbes Juan-Les-Pns, France c 26 Copyrght held by the owner/authors. Publcaton rghts lcensed to ACM. ISBN /6/6... $5. DOI: proxmated usng the equlbrum pont of the mean-feld model. The mean-feld method has mportant applcatons n varous areas ncludng statstcal physcs, epdemology, communcaton networks, queueng theory, and game theorye.g., [5, 4, 3, 32, 22, 9, 2, 7,, 2, ]. In partcular, over the last few years, t also has emerged as a powerful method for analyzng large-scale cloud computng systems and data center networks. For example, n [22, 32], the mean-feld analyss has been used to show that routng each ncomng task to the shorter of two randomly sampled servers can sgnfcantly reduce queueng delays, a phenomenon called the power-of-two-choces Po2. The result has been extended to heavy-taled servce-tme dstrbutons [9], and to heterogeneous servers [23]. In [3], a mean-feld model has been used to quantfy the sgnfcant beneft of resource poolng. [28] establshed the asymptotc optmalty of the jon-dlequeue JIQ, proposed n [2], usng the mean-feld model. In [34], a novel randomzed load balancng algorthm, named Batch-Fllng, has been developed for cloud computng systems wth batch arrvals. The algorthm acheves smlar delay performance as the power-of-two-choces wth a samplng rato slghtly larger than one.e., t only samples slghtly more than one server on average for each ncomng task. In [33], the mean-feld analyss has been used for studyng the vrtual machne placement problem n data center networks. In these applcatons, the systems under consderaton are modeled as CTMCs, and the solutons equlbrum ponts of the correspondng mean-feld models are then used to approxmate the statonary dstrbutons of the CTMCs. To justfy the mean-feld analyss, a crtcal step s to prove that the statonary dstrbuton of the CTMC ndeed converges to the equlbrum pont of the mean-feld model as the sze of the system ncreases. Consder a famly of CTMCs. The Mth CTMC s an M-dmensonal contnuoustme Markov chan W M U M, where the superscrpt M s the number of nodes or called partcles n the system and U M R M s the state space of the CTMC. We assume U s a fnte state space and the CTMC s rreducble. Wthout loss of generalty, let U = {,,n}. We further defne x M t = M M m= M W m t= where s the ndcator functon, so x M t s the fracton of nodes n state at { tme t. Ths paper } focuses on the case such that x M = x M t,t s also an ndmensonal CTMC,.e., the CTMC s a populaton process

2 [7, 8]. We remark that many applcatons of the mean-feld method such as those n queueng networks and epdemology are for populaton processes. Now let x M denote the statonary dstrbuton of the Mth CTMC. Furthermore, let xt denote the soluton of an assocated mean-feld model and x denote ts equlbrum pont. Exstng approaches for provng the convergence of x M tox oftennvolvethe followng threecomponents. The frst component s to show the convergence of CTMCs to the trajectory of the mean-feld model for any fnte tme nterval [, t],.e., lm sup dx M s,xs =, M s t where d, s some measure of dstance. Ths can be proved usng dfferent technques ncludng Kurtz s theorem [7, 8, 22, 34], propagaton of chaos [3, 2, 9], or the convergence of the transton semgroup of CTMCs [32, 23]. 2 The second component s to prove the asymptotc stablty of the mean-feld model,.e., lm xt = t x. Lyapunov theorem or LaSalle nvarance prncple can often be used for provng the stablty. 3 After establshng the prevous two results, we obtaned lm lm t M xm t = lm xt = x. t The convergence of the statonary dstrbutons can then be concluded f we can prove the nterchange of the lmts,.e., lm M xm = lm lm M t xm t = a lm t lm M xm t = x, where step a s called the nterchange of the lmts. Snce these approaches are all based on the nterchange of the lmts and use the fnte-tme convergence equalty as the steppng stone, they are ndrect methods of provng lm M xm = x. Because of ths reason, these approaches can only establsh the convergence of mean-feld models and the asymptotc behavor of the systems.e., for M =. The approxmaton error or called the rate of convergence of mean-feld models for fnte-sze systems e.g., x M x for a fxed M s dffcult to obtan usng these ndrect methods. Ths paper tackles ths fundamental problem and drectly studes the approxmaton error of a large class of mean-feld models usngsten smethod[26, 27, 6], whchs amethodto bound the dstance of two probablty dstrbutons. Our use of Sten s method for the rate of convergence was nspred by the work by Braverman and Da [], n whch they developed a modular framework wth three components for steady-state dffuson approxmatons and establshed the rate of convergence to dffuson models for M/Ph/n + M queung systems. The results n ths paper also share smlar sprt wth the work by Gurvch [4], whch establshes the rate of convergence of dffuson models for steady-state approxmatons for exponentally ergodc Markovan queues. Ths paper dffers from both work n that t consders meanfeld models nstead of dffuson approxmatons. To establsh the approxmaton error, the paper dentfes a fundamental connecton between the perturbaton theory for nonlnear systems and the convergence of meanfeld models. The perturbaton theory shows that for a stable nonlnear system wth exponentally stable equlbrum pont, the error of the frst-order approxmaton of the nonlnear system s at the order of Oǫ 2, where ǫ s the scalng factor of the perturbaton. It turns out the mean-square dfference between the statonary dstrbuton of the Mth CTMC and the equlbrum pont of the mean-feld model s related to the cumulatve error ntegrated over nfnte tme horzon of the frst-order approxmaton of the mean-feld model. After quantfyng the cumulatve error, we establsh the followng results for fnte-dmensonal mean-feld models. If the mean-feld model s perfect see defnton n Secton 2, globally asymptotcally stable and locally exponentally stable, then the statonary dstrbutons of the CTMCs converges n the mean-square sense to the equlbrum pont of the mean-feld model wth rate O/M Theorem, specfcally, we have the followng result on the approxmaton error [ E x M x 2] = O. 2 M If the mean-feld model s not perfect, suffcent condtons that guarantee the convergence of the statonary dstrbutons have been obtaned n Corollary. We remark that these results are dfferent from the celebrated law of large numbers for Markov chans establshed by Kurtz [7, 8], where the convergence s establshed for sample paths of the CTMCs over a fnte tme nterval or for a sequence of t M whch ncreases M ncreases [24], not for the statonary dstrbutons of the CTMCs. The contrbutons of our results are two-fold: Frst, t provdes a drect method of studyng the convergence of statonary dstrbutons of stochastc systems to ther mean-feld lmts. The method connects the convergence of CTMCs wth the stablty of the mean-feld model. Note that the mean-feld model s a determnstc system, so t s often easer to analyze than the CTMCs. Second, the method quantfes the rate of convergence, and provdes bounds on the approxmaton error when usng the mean-feld lmt for approxmatng the performance of fnte-sze systems. We fnally comment that the convergence of statonary dstrbutons of one-dmensonal dscrete-tme Markov chans has been studed n [25]. The approxmaton error of meanfeld models for dscrete-tme Markov chans has been studed n [8], whch, however, focuses on numercal methods to compute the error bounds and does not establsh a general analytc answer lke 2. Furthermore, an approach smlar tosten smethodhasbeenusedn[29] toprovethetghtness of dffuson-scaled statonary dstrbutons for a two-queue system wth many servers. The tghtness result n [29] establshes an approxmaton error of the flud-lmt, whch s at the same order of the approxmaton error establshed n ths paper. The key dfferences are that ths paper consders mean-feld models for populaton processes.e., wth many queues nstead of two queues and establshes suffcent con-

3 dtons for the convergence and the rate of convergence of a large-class of systems nstead of only for a specfc system. 2. MEAN-FIELD MODELS Consder an M-dmensonal contnuous-tme Markov chan W M U M, where the superscrpt M s the number of nodes or called partcles n the system and U M R M s the state space of the CTMC. We assume U s a fnte state space and the CTMC s rreducble. Wthout loss of generalty, we assume U = {,,n}. We further defne X M t = M m= M W m t= where s the ndcator functon, so X M t s the number of nodes n state at tme t. We further defne x M t = XM t M, so x M t [,] represents the fracton of { nodes n state } at tme t. In ths paper, we assume x M = x M t,t s an n-dmensonal CTMC. We use x M to denote ts statonary dstrbuton. Furthermore, we have a mean-feld model descrbed by the followng autonomous dynamcal system: ẋ d dt xt = fxt x = x and xt D [,]n, 3 where D s a compact set. Here, we abuse the notaton and use x to denote the ntal condton, whch smplfes the notaton n the analyss later wthout causng too much confuson. Assume the system has a unque equlbrum pont and let x denote the equlbrum pont. The key dea of the mean-feld analyss s to use the soluton of ths determnstc dynamcal system to approxmate the behavor of the CTMC when M s large; for example, use x to approxmate x M. Let Q x M,yM denote the transton rate of the CTMC from state x M to state y M. A famly of CTMCs s called a densty-dependent famly of CTMCs f the normalzed transton rate q x M,y M = M Q x M,y M onlydependsonx M andy M buts ndependentofm see a detaled defnton n [22]. For a densty-dependent famly of CTMCs, the mean-feld model can often be obtaned by choosng fx = Q x,yy x = Mq x,yy x because q x,y s the transton rate from x to y and y x s the change of system state when such a transton occurs. We next llustrate the dea usng an SISsusceptble-nfectedsusceptble model wth an external nfecton source, whch s a varaton of the orgnal SIS model. Example: Let W m M denote the state of an ndvdual such that W m M = f the ndvdual s susceptble and W m M = f the ndvdual s nfected. So x M s the fracton of susceptble ndvduals and x M s the fracton of nfected ndvduals. We assume the recover tme of an ndvdual follows an exponental dstrbuton wth mean. Each nfected node randomly selects a node after watng for a random tme that s exponentally dstrbuted wth mean /β. If the selected node s an susceptble node, t gets nfected. Each susceptble node, after t becomes susceptble, gets nfected by an external nfecton source after a random tme perod that s exponentally dstrbuted wth mean /α. Therefore, W M, X m and x M are CTMCs. Specfcally, x M has the followng transton rates, where x M s the fracton of susceptble ndvduals and x M s the fracton of nfected ndvduals: Q x M,y M = Mαx M +Mβx M x M, f y M = x M + M Mx M, f y M = x M + M Mαx M Mβx M x M Mx M, f y M = x M otherwse. Note for a gven M, computng the statonary dstrbuton of x M s not easy because t has a large state space { } 2 M, 2,, M and the transton rates are nonlnear functons of the states. The SIS consdered above s a densty-dependent CTMC, so we consder the followng mean-feld model ẋ = fx = q ẋ x,yy x = αx +βx x +x To solve the mean-feld model above, we notce that x + x = always holds, so we only need to consder ẋ = αx βx x + x. The equlbrum pont can then be obtaned by solvng = αx βx x + x. For example, f α = β =.5, then x = 2 2 and x = 2, whch can be used to approxmate the fractons of susceptble and nfected populatons when M s large,.e., the statonary dstrbuton of x M. The smulaton results of the fracton of susceptble populaton wth M =,, and, are shown n Fgure, from whch the convergence of x M to 2 2 can be seen clearly. 3. STEIN S METHOD FOR QUANTIFYING THE APPROXIMATION ERROR In ths secton, we study the convergence and the approxmaton error the rate of convergence of the CTMCs to a mean-feld model usng Sten s method and the perturbaton

4 Normalzed Infected Populaton x nodes, nodes, nodes Tme = M q x,yxgy gx. Snce x M s rreducble and has fnte state space, x M has a statonary dstrbuton. Intalzng x M accordng to ts statonary dstrbuton, and usng E x M[ ] throughout to the expectaton takng over the statonary dstrbuton x M, we have E x M [G x Mgx] M q x,yxgy gx =. 5 =E x M Fgure : Smulaton results of the fracton of susceptble populaton wth M =,,, and,. α = β =.5 n these smulatons. The CTMCs were smulated usng the unformzaton method. The tme s the scaled dscrete-tme used n the unformzaton scaled wth M. Specfcally, each tme slot n the smulaton ncludes M jumps. theory. Throughout ths paper, denotes the 2-norm,.e., x = x2, and denotes the absolute value. For two vectors a,b R n, a b s the dot product. Furthermore, gx denotes the gradent of gx, and x t,x refers to dfferentatng wth respect to the locaton x, and ẋ s the dervatve wth respect to tme. Recall the mean-feld model defned n equaton 3: ẋ = fxt x = x and xt D [,] n. The mean-feld model s sad to be globally asymptotcally stable f gven any ntal condton x D and any ǫ >, there exsts tx, ǫ such that xt x ǫ t tx,ǫ. The mean-feld model s sad to be locally exponentally stable f there exst postve constants ǫ, α and κ such that startng from any ntal condton x x ǫ, xt x κ x exp αt. Let gx be the soluton to the Posson equaton gx ẋ = gx fx = n x x 2 4 = Then, the soluton has the followng form gx = x t,x x 2 dt when the ntegral s fnte see [5, 3], where xt,x s the trajectory of the dynamcal system wth x as the ntal condton. The ntegral s fnte when the mean-feld model s asymptotcally stable and locally exponentally stable, whch wll become clear n Secton 5. Note that gx can be vewed as the cumulatve square-devaton of the system state from the equlbrum pont when the ntal condton s x. Now let G x M denote the generator for the Mth CTMC, then G x Mgx = Q x,yxgy gx Then by takng expectaton of the Posson equaton 4 over the statonary dstrbuton x M and then addng 5 to the equaton, we obtan [ n E x M = x x 2 ] = E x M gx fx M q x,ygy gx Now addng and subtractng gx qx,ymy x yelds [ n E x M = x x 2 ] = E x M gx fx gx q x,ymy x M q x,ygy gx gx y x = E x M gx fx q x,ymy x 6 q x,ym gy gx gx y x.7 From the equalty above, ntutvely, that [ n ] E x M x x 2 = converges to zero as M can be establshed f the followngs are true: Bounded gradent of gx : gx s bounded by a constant ndependent of M. Convergence of the generator: lm E x M M fx q x,ymy x =. Bounded [ transton-rate ] of the CTMC: E x M qx,y s bounded.

5 Dmnshng frst-order approxmaton error: gy gx gx y x = O. M 2 Note that gx+ gx y x s the frst-order Taylor approxmaton of gy. For many CTMCs and the assocated mean-feld models, the frst three condtons mentoned above can be easly verfed. In the followng theorem, we wll prove that the last condton holds when the mean-feld model s globally asymptotcally stable and locally exponentally stable see nequalty, and then establsh the rate of convergence based on that. The followng theorem presents the man result of ths paper. Theorem. The statonary dstrbutons of the CTMCs x M, defned n Secton 2, converge to the equlbrum pont x of the mean-feld model 3 n the mean-square sense wth rate /M,.e., [ n ] E x M x x 2 = O M = when the followng condtons hold: Bounded transton-rate condton: There exsts a constant c > ndependent of M such that c. E x M q x,y Bounded state transton condton: There exsts a constant c ndependent of M such that x y c M for any x and y such that q x,y. Perfect mean-feld model condton: The meanfeld model 3 s gven by fx = q x,ymy x x. Partal dervatve condton: The functon fx s twce contnuously dfferentable. Stablty condton: The mean-feld model s globally asymptotcally stable and s locally exponentally stable. Remark. The frst four condtons are easy to verfy, so only the stablty condton requres nontrval work. Snce a dynamcal system has an exponentally stable equlbrum pont f and only f the lnearzed system at the equlbrum s exponentally stable see Theorem 4.5 n [6], the local exponental stablty can be verfed by provng the lnearzed system s exponentally stable e.g., usng Lyapunov method or numercally verfed by calculatng the egenvalues of the state matrx of the mean-feld model. The global asymptotcal stablty n general s studed usng the Lyapunov theorem. Two applcatons of ths theorem n data center networks wll be presented n Secton 4. Remark 2. It s worth to pontng out that f the meanfeld model s unstable but the perfect mean-feld model assumpton holds. Kurtz s theorem [7, 8] ndcates that the sample paths of the CTMCs converge to the trajectory of the mean-feld model for any fnte tme nterval, whch mples that the CTMCs are unstable as well. Proof. We frst prove the theorem assumng the meanfeld model s globally exponentally stable, and then extend t to the general case. Under the perfect mean-feld model assumpton, equaton 7 becomes [ n E x M = x x 2 ] = E x M q x,ym gy gx gx y x, where gx = x t,x x 2 dt. We next focus on the followng term, = gy gx gx y x x t,y x 2 x t,x x 2 2x t,x x x t,x y x dt 8 Note that we exchanged the order of ntegraton and dfferentaton for the thrd term. Ths s can be done because 2x t,x x x t,x y xdt sfnte,whchcanbeprovedusngthefactthatbothx t,x x and x t,x decay exponentally fast to zero as t ncreases apply nequaltes 2 and 3 wth z =, and the fact that y x s bounded due to the bounded state transton condton. We next defne.e., so e t = x t,y x t,x x t,x y x, x t,y = e t+x t,x+ x t,x y x, x t,y x 2 x t,x x 2 2x t,x x x t,x y x = e t+x t,x x + x t,x y x 2 x t,x x 2 2x t,x x x t,x y x = e 2 t+ x t,x y x 2 +2e t x t,x y x +2e tx t,x x = e te t+2 x t,x y x+2x t,x x + x t,x y x 2. Accordng to the perturbaton theory, n partcular, nequalty 34, when the system s exponentally stable, we have that e t et = O. M 2 Accordng to the bounded state transton condton, x y c M. Furthermore, both x t,x and x t,x are bounded see nequaltes 2 and 3 by constants ndependent of M

6 and t. Therefore, we can choose a constant b and a suffcently large M such that for any M M, e t+2 x t,x y x+2x t,x x b, whch mples that gy gx gx y x b e t dt+ x t,x y x 2 dt, b n et dt+ 9 x t,x y x 2 dt, where the last nequalty s based on the followng relaton between -norm and 2-norm: et n et. In Secton 5 n partcular, nequalty 35, we wll show that under the exponental stablty assumpton, et dt = O/M 2. From the bounded state transton condton, y x 2 c 2. Therefore, M 2 x t,x y x 2 dt c2 M 2 x t,x 2 dt. Now accordng to nequalty 3, there exst postve constants b and b 2, both ndependent M, such that x t,x y x 2 dt c2 M 2 Therefore, we can conclude that b exp b 2tdt b b 2 M 2. gy gx gx y x = O whch mples that [ n ] E x M x x 2 = O E M x M =, M 2 q x,y.2 Fnally, usng the bounded transton rate condton, we conclude [ n ] E x M x x 2 = O. 3 M = Now consder the case that the mean-feld model s not globally exponentally stable, but s globally asymptotcally stable and locally exponentally stable. Recall that D [,] n s compact. Accordng to the defnton of global asymptotc stablty Defnton 4.4 n [6], gven any ǫ >, there exsts a fnte tme t such that xt x ǫ for any t t. For any fnte t, followng a smlar analyss as n Secton 5 or Secton. n [6], et,x = O/M 2 holds. Therefore, we can bound the term 8 by separatng the ntegraton nto two ntervals: from to t, and from t to, where t s chosen such that xt converges exponentally to the equlbrum pont after t. Snce et,x = O/M 2, the analyss above apples to the ntegraton over [t,. Hence, the result holds. Example: Let us go back to the SIS model ntroduced n Secton 2. A closed-form soluton can be obtaned for x t. Agan assume α = β =.5, then the soluton of the ordnary dfferental equaton s 2t x t = e 2+ 2 x x 2 e 2t x x, 2 2 whch converges to 2 2 as t ndependent of x. Therefore, t s easy to verfy that the system s globally, asymptotcally stable. Furthermore, the lnearzed system at the equlbrum s ǫ = α+β x +ǫ, where ǫ = x x and x s the equlbrum value, so the equlbrum pont s locally exponentally stable. Furthermore, the mean-feld model s perfect n ths case and t can be easly verfed that all other condtons n Theorem hold. So n the mean square sense, statonary dstrbutons converge to the x = 2 2 and x = [ 2 wth rate O/M. Numercal evaluaton of ME x M x x 2] versus M s shown n Fgure 2, where [ M vares from to,. We can see that ME n x M = x x 2] vares wthn the nterval [.2,.27] whle the sze of the system ncreases by tmes from to,. The standard devaton devaton from 2 2 s % 2 2 when M = and s.68.6% 2 2 when M =,. From ths example, we can see that the mean-feld lmt s a good approxmaton of the system when the sze of the system s moderate large and the mean-square approxmaton error of ths example s around. 4M M Mean Square Dfference System sze M [ Fgure 2: Numercal evaluaton of ME x M x x 2] versus M. Theorem requres a perfect mean-feld model and bounded state transtons. Both condtons can be relaxed, but the rate of convergence wll be dfferent. Ths holds wthout exponental stablty, but the constant n O/M 2 may be a functon of t f the system s not exponentally stable.

7 Corollary. Assume partal dervatve condton and the stablty condton n Theorem hold. The statonary dstrbutons of the CTMCs converge n the mean square sense to equlbrum pont of the mean-feld model,.e., [ n ] lm E x M M x x 2 = = when the followng condtons also hold: lm E x M M fx q x,ymy x =. 4 lm E x M M q x,ym y x 2 = 5 lm max M x,y:q x,y> y x = 6 We say that the mean-feld model s asymptotcally accurate when condton 4 holds, whch replaces the perfect meanfeld model condton. Condtons 5 and 6 replace the bounded state transton condton. Proof. Frst recall that we have [ n E x M =E x M = x x 2 ] gx fx q x,ymy x q x,ym gy gx gx y x max gx E x x M fx q x,ymy x + E x M q x,ym gy gx gx y x. 7 8 By choosng z = n Secton 5, t s easy to verfy accordngtonequalty3 thatmax x gx s upperbounded by a constant ndependent of M. Therefore, under condton 4, 7 as M. A careful examnaton of nequalty 35 shows that et dt = O y x 2 exp α 3 y x. So under condton 6, we have et dt = O y x 2. When condton 6 holds, followng the analyss that leads to nequalty, we can agan show that there exsts a constant b ndependent M such that gy gx gx y x b n et dt+ x t,x y x 2 dt. Accordng to nequalty 3, we also have x t,x y x 2 dt = O y x 2. Therefore, we have 8 = O E x M q x,ym y x 2, whch converges to zero accordng to condton 5. Hence, the corollary holds. Remark 3. When the mean-feld model s asymptotcally accurate, the convergence rate depends on the convergence rates of 4 and APPLICATIONS IN DATA CENTER NET- WORKS In ths secton, we wll demonstrate the novelty of Theorem by consderng two applcatons n data center networks: the power-of-two-choces [22, 32] and the vrtual machne placement problem [33]. For both problems, meanfeld models have been used to analyze the performance of the systems n nfnte server regme, but the approxmaton errors of the mean-feld lmts for systems wth fnte number of servers were unknown. 4. The power-of-two-choces for servers wth fnte buffer In [22, 32], the authors consdered a data center network wth M dentcal servers as shown n Fgure 3. Assume tasks arrve at the data center followng a Posson process wth rate λm and the processng tme of each task s exponentally dstrbuted wth mean processng tme µ =. Each server mantans a queue and Q mt denotes the queue sze of server m at tme t. For each ncomng task, the router or called scheduler randomly samples two servers and dspatches the task to the server wth a smaller queue sze. In ths settng, Qt s a CTMC and s a populaton process. scheduler server server 2 server M Fgure 3: The system has M servers. When a task comes n, the scheduler samples two servers and routes the task to the server wth shorter queue. In ths example, the scheduler probes server and server M and routes the task to server. Let s M k t denote the fracton of servers wth queue sze at least k. Based on the mean-feld analyss, t has been shown n [22, 32] that s M k weakly converges to s k,

8 where s k s the equlbrum pont of the followng mean-feld system: { λs 2 ṡ k = k s 2 k s k s k+, k ;, k =. The mean-feld model above s an nfnte-dmensonal system, so Theorem does not apply. We nstead consder fnte-buffer servers wth buffer sze B, for whch, the followng mean-feld model s a perfect mean-feld model for the fnte-buffer system. ṡ k =, k = ; λs 2 k s 2 k s k s k+, B k ; λs 2 k s 2 k s k, k = B. and the equlbrum pont satsfes the condtons: s = λs k 2 s k2 s k s k+ = B k λs k 2 s k2 s k = k = B. The exstence and unqueness of the soluton has been proved n [22]. Defne the Lyapunov functon to be for w k satsfes Vst = B w k s k t s k k= w k < w k+ w k + w k δ w k λ2s k + for some δ >. The exstence of such w k > and δ > for the nfnte-dmensonal mean-feld model has been proved n [22]. The same w k and δ can be used n the fntedmensonal system as well. Followng a smlar analyss n [22], we obtan whch mples that Vt δvt, st s w Vt = w B k= w k s k t s k V w e δt. So the system s globally, exponentally stable. Other condtons n Theorem can be easly verfed. So the approxmaton error n Theorem apples. 4.2 Vrtual machne placement n cloud computng systems In [33], the authors consdered a data center network wth M dentcal servers where each server have B unts of resources and can host at most B vrtual machnes VMs as shown n Fgure 4. Assume VM requests arrve accordng to a Posson process wth rate λm and the lfetme of each VM s s exponentally dstrbuted wth mean lfetme µ =. Let Q mt denotes the number of VMs hosted at server m at tme t. For each ncomng request, the router or called scheduler randomly samples two servers and dspatches to the server wth a smaller number of VMs. If both servers have already hosted B VMs, the request s blocked. In ths settng, Qt agan s a CTMC and s a populaton process. Let s M k t denote the fracton of servers wth at least k VMs. Based on the mean-feld analyss, t has been shown, scheduler server server 2 server M Fgure 4: The system has M servers, and each server can host at most three VMs. the scheduler samples two servers and routes the VM request to the server wth a smaller number of VMs. In ths example, the scheduler routes the VM request to server. The request s blocked f both servers are full. n [33] that s M k weakly converges to s k, where s k s the equlbrum pont of the followng mean-feld system:, k = ; ṡ k = λs 2 k s 2 k ks k s k+, B k ; λs 2 k s 2 k Bs k, k = B. Ths s a fnte-dmensonal mean-feld model, so Theorem can be appled. The equlbrum pont n ths case can be recursvely solved but the closed-form expresson s dffcult to obtan. The asymptotc stablty of the system has been proved n [33]. We now consder the lnearzed system at the equlbrum, whch s, k = ; ẋ k = 2λs k x k s kx k kx k x k+, B k ; 2λs k x k s kx k Bx k, k = B, where x k = s k s k. Defne the Lyapunov functon to be Note that when x k > Vt = B x k. k= x k = ẋ k, k = ; = 2λs k x k s kx k kx k x k+, B k ; 2λs k x k s kx k Bx k, k = B,, k = ; 2λs k x k s k x k k x k x k+, B k ; 2λs k x k s k x k B x k, k = B, and when x k <, x k = ẋ k, k = ; = 2λs k x k s kx k +kx k x k+, B k ; 2λs k x k s kx k +Bx k, k = B,, k = ; 2λs k x k s k x k k x k x k+, B k ; 2λs k x k s k x k B x k, k = B, It s not dffcult to verfy that the same nequaltes hold,.

9 when x k =. Therefore, we have B B B Vt = ẋ k = x k 2λs B x B x k = Vt, k= whch mples that k= k= B x k t = Vt Ve t. k= Therefore, the equlbrum pont s locally exponentally stable. Other condtons n Theorem agan can be easly checked, so the approxmaton bound apples. For both systems, the convergence of the statonary dstrbutons to the mean-feld lmts have been proved n the lterature based on the nterchange of the lmts, but the approxmaton errors or the rate of convergence were unknown. The result n ths paper not only establshes the approxmaton errors, but also sgnfcantly reduces the addtonal analyss, n partcular, both the convergence n fnte tme and the nterchange of the lmts are no longer needed. The purpose of presentng these two applcatons s to demonstrate the novelty of our result. The mean-feld analyss of these two systems has been establshed n the lterature, and s not the focus of ths paper. We therefore gnored the detals and only presented the key steps. The smulaton results that demonstrate the convergence of the fnte systems can also be found n the orgnal papers, so were not presented due to page lmt. 5. THE PERTURBATION THEORY In ths secton, we summarze the results of the perturbaton theory for nonlnear systems. These results are specal cases of those results presented n [6] because we only need to consder a perturbaton to the ntal condton. Furthermore, the mean-feld model consdered n ths paper s an autonomous system, whch agan s a specal case of the nonlnear system consdered n [6]. For these reasons, the analyss of the perturbaton results can be smplfed. On the other hand, the perturbaton method ntroduced n [6] only states that the 2-norm of the followng error s s at the order of y x 2 ndependentof t under certan condtons et = xt,y xt,x xt,x y x Our result on the rate of convergence, however, requres such an upper bound on the cumulatve error,.e., an upper bound on et dt. Therefore, t s necessary to go through the detaled analyss for the system consdered n ths paper to establsh the result for the cumulatve error. For the completeness of the paper and the easy reference of the reader, we next ntroduce the perturbaton results n [6] wth a more detaled calculaton of et, whch shows that not only the approxmaton error s bounded, but the upper bound decays exponentally to zero as t ncreases. The analyss closely follows [6]. Consder the system ẋ = fx 9 where f : D [,] n R n. Wthout the loss of generalty, we assume x =. We are nterested n comparng the soluton of ths nomnal system wth the system wth a perturbaton on the ntal condton x = x + ǫz, where z = y x and s an n-dmensonal vector. For the meanfeld analyss consdered n ths paper, ǫ = /M. Under the ǫ condton of Theorem, for any neghborng states x and y, z = y x c. ǫ Let xt,ǫ to denote the soluton of the dynamcal system wth ntal perturbaton ǫ. Note that the dependence of the soluton on y x s omtted to smplfy the notaton. The analyss holds for any y and x. We next frst repeat the assumptons on the nomnal dynamcal system. Assumpton. For any, the functon f x s twce contnuously dfferentable. Therefore, the Jacoban matrx of fx, denoted by f, s Lpschtz. In other words, there x exsts a constant L > such that f f x x x y L x y. Assumpton 2. The dynamcal system 9 has a unque equlbrum pont and s exponentally stable. In other words, there exst postve constants α and κ such that startng from any ntal condton x D, xt κ x exp αt. 2 Under ths assumpton, accordng to Theorem 4.4 n [6], there exst a Lyapunov functon Vx and postve constants c u, c l, c d, and c p such that for any x D, the followng nequaltes hold c l x 2 Vx c u x 2 Vx c d x 2 Vx c p x. Wefrst consder the fntetaylor seres for xt,ǫ nterms of ǫ : and where xt,ǫ = x t+ǫx t+et, 2 x,ǫ = x+ǫz, 22 x t = xt, and x t = dx dǫ t,ǫ. ǫ= Substtutng 2 nto the dynamcal system equaton, we get ẋt,ǫ = ẋ t+ǫẋ t+ėt = fxt,ǫ 23 = h x t+h x tǫ+r et,ǫ, 24 where x = x,x. The zero-order term h s gven by ẋ t = h x t = f x t wth x = x,

10 whch s the nomnal system wthout the perturbaton on the ntal condton. The frst-order term s gven by h x t = d dǫ ǫ= fxt,ǫ = f x xt,ǫdx dǫ t,ǫ = f x x tx t. ǫ= Recall that f s the Jacoban matrx. Therefore, we have x ẋ t = f x x tx t wth x = z. 25 Wenextstudyet = xt,ǫ x t ǫx t.combnng the results above, we have ėt = f xt,ǫ f x t ǫ f x x tx t e =. Now by defnng ρ x t,et,ǫ =f et+x t+ǫx t f f x x tet, and γ x t,ǫ =f x t+ǫx t f x t+ǫx t x t ǫ f x x tx t, we obtan ėt = f x x tet+ρ x t,et,ǫ +γ x t,ǫ. 26 Note that both ρ and γ are n-dmensonal vectors. It s easy to see that ρ x t,,ǫ =. 27 Accordng to Taylor s theorem and the mean value theorem, we have γ l x t,ǫ = ǫ 2 x t T Hf l ξx t = ǫ 2,j 2 f l ξ x tx j t x x j for ξ = x t+αǫx t for some α. Hf l s the Hessan matrx of functon f l x. Then we have γ l x t,ǫ = ǫ 2 2 f t l ξ x tx j. x x j,j Furthermore, we have ρ l x t,et,ǫ e = f l x et+x t+ǫx t f l x x t. Accordng to the mean-value theorem and27, we have that ρ x t,et,ǫ f = x ẽt+x t+ǫx t f x x t et, whereẽt = aetforsome a. AccordngtotheLpschtz condton n Assumpton and the Cauchy-Schwarz nequalty, we have ρ x t,et,ǫ L et +ǫ x t et. Now we utlze the assumpton that the nomnal system 9 converges to the equlbrum pont exponentally fast from any ntal condton n the doman. We use the Lyapunov functon n Assumpton 2 to bound et. We start from Vet = Vet ėt = Vet fet+ Vet ėt fet a c d Vet+ Vet ėt fet = c d Vet+ Vet f x x tet f f et+ x x et fet +ρ x t,et,ǫ +γ x t,ǫ c d Vet f + Vet x x tet f x et + f x et fet + ρ x + γ t,et,ǫ x t,ǫ where nequalty a s due to assumpton 2 and the last nequalty s a result of the Cauchy-Schwarz nequalty. Note that based on Assumpton and the mean-value theorem, we have f x x tet f x et L x t et f x et fet L et 2. We also know that ρ x t,et,ǫ L et +ǫ x t et γ x t,ǫ = ǫ 2 At, where we defne At = l,j 2 f 2 l ξ x tx j t x x j to smplfy the notaton. Summarzng the results above, we get Vet c d Vet +L Vet x t +ǫ x t +2 et et

11 + Vet Atǫ 2 c d Vet +Lc p x t +ǫ x t +2 et et 2 +c patǫ 2 et c d Vet +L cp x t +ǫ x t +2 et Vet c l + cp cl Atǫ 2 Vet. Defne Wt = Vt, then we have Ẇet c d 2 Wet + L c p x t +ǫ x t +2 et Wet 2 c l + cp Atǫ 2. c By the comparson lemma n [6], we have Wt φt,w+ cp ǫ 2 φt,τaτdτ 28 c = cp ǫ 2 φt,τaτdτ 29 c where the transton functon φt,τ s φt,τ = exp c d 2 t τ + L 2 c p c l τ x γ +ǫ x γ +2 eγ dγ, and the equalty holds because e =. The followng lemma proves that the frst-order system 25 converges exponentally startng from any ntal condton n D. Lemma. The frst-order system 25 s exponentally stable for any soluton x t that starts from D. Proof. That the nomnal system s exponentally stable mples that the followng lnear system ẋ = f x x sgloballyexponentallystable, and f x shurwtzcorollary 4.3 n [6], whch further mples that there exsts a postve defnte symmetrc matrx P such that Vx = x T Px s a Lyapunov functon for the lnear system such that We start from Vx t = Vx t ẋ t Vx x 2. 3 = Vx t f x x t+ Vx t f x x tx t f x x t a x t 2 +2λ maxplx t x t 2 λ maxp Vx t+ 2λmaxP x λ L t V mnp λ 2λmaxP x maxp λ L t V mnp x t x t where nequalty a s based on 3 and the defnton of Vx, Assumpton and the mean-value theorem, and λ maxp s the largest egenvalue of matrx P. By the comparson lemma, we have Vt 2λmaxP t exp t+ λ maxp λ L x τ dτ V mnp 2λmaxPLκ x exp exp αλ mnp λ t V, maxp where the last nequalty holds because the exponental convergence assumpton 2 yelds x τ dτ x τ dτ κ x. α Recall that x = z, so V = z T Pz = p and x t 2 p 2λmaxPLκ x λ exp exp mnp αλ mnp λ t. maxp 3 From the lemma above and assumpton, we have that there exsts a constant µ such that At µ x t 2 µp 2λmaxPLκ x λ exp exp mnp αλ mnp λ t. maxp Consder the set such that et c dc l 4c pl, we have φt,τ exp c d 2 t τ+ L c p 2 c l exp c d 4 t τ+ L c p 2 c l τ x γ +ǫ x γ dγ κ x α +2ǫ λmaxp p λmaxplκ x exp, λmnp αλ mnp where last nequalty yelds from 2 and 3. Recall we have nequalty 29 Wt cp ǫ 2 φt,τaτdτ. 32 c Substtutng the bounds on φt,τ and Aτ, we obtan Wt ǫ 2 t cz x exp c d 4 t τ λ τ dτ, maxp where Z x

12 = cp L c p κ x exp c 2 c l α +2ǫ λmaxp p λmaxplκ x exp λmnp αλ mnp µp 2λmaxPLκ x λ exp mnp αλ mnp = cpµp c λ mnp exp +ǫ p cpl c l κl α In other words, we have et cp + 2λmaxP 2c l λ mnp x λ maxp λmaxplκ x exp. λmnp αλ mnp exp c d λ τ maxp ǫ 2 Z x 4 t τ dτ 33 ǫ 2 Z x c d4 exp λmaxp = f c d 4 λ maxp ǫ 2 Z x texp c d 4 t, otherwse. 34 Then et dt { ǫ 2 Z x 4λmaxP c d ǫ 2 Z x 6 c 2 d = ǫ 2 Z x 4λmaxP. c d λ maxp t exp c d 4 t, f c d 4, otherwse. λ maxp It s easy to see that wth properly defned α, α 2, α 3 and α 4, we have et dt ǫ 2 x pα exp α 2 +ǫ x pα 3exp α α α Wekeeptheterms x andαtoshowthatthecumulatve error depends on the ntal condton and the convergence rate of the mean-feld model. Furthermore, p = z T Pz λ maxp z CONCLUSION Ths paper studes the approxmaton error of a large-class of mean-feld models. When the mean-feld model s perfect, the mean-square dfference also called the rate of convergence has been proved to be O/M. Based on Sten s method for boundng the dstance of probablty dstrbutons and the perturbaton theory for nonlnear systems, a fundamental connecton between the convergence to the mean-feld lmt and the stablty of the mean-feld model has been establshed. Two applcatons of mean-feld models for large-scale data center networks were dscussed to demonstrate the novelty of our results. Acknowledgement The author s very grateful to Jm Da and Anton Braverman. Jm s semnar on Sten s method for the steady-state. dffuson approxmatons nspred ths work. The dscussons wth Jm and Anton had contnuously stmulated the author durng the wrtng of ths paper. Ths work was supported n part by the NSF under Grant ECCS REFERENCES [] S. Adlakha and R. Johar. Mean feld equlbrum n dynamc games wth strategc complementartes. Operatons Research, 64:97 989, 23. [2] V. Anantharam and M. Benchekroun. A technque for computng sojourn tmes n large networks of nteractng queues. Probablty n the engneerng and nformatonal scences, 74:44 464, 993. [3] F. Baccell, F. Karpelevch, M. Y. Kelbert, A. Puhalsk, A. Rybko, and Y. M. Suhov. A mean-feld lmt for a class of queueng networks. Journal of statstcal physcs, 663-4:83 825, 992. [4] N. T. J. Baley. The mathematcal theory of nfectous dseases and ts applcatons. Hafner Press, 975. [5]. A. D. Barbour. Sten s method and Posson process convergence. J. Appl. Probab., pages 75 84, 988. [6] A. D. Barbour and L. H. Chen. An Introducton to Sten s Method, volume 4. World Scentfc, 25. [7] C. Bordenave, D. Mcdonald, and A. Proutere. A partcle system n nteracton wth a rapdly varyng envronment: Mean feld lmts and applcatons. Networks and Heterogeneous Meda NHM, 2. [8] L. Bortoluss and R. A. Hayden. Bounds on the devaton of dscrete-tme markov chans from ther mean-feld model. Performance Evaluaton, 7: , 23. [9] M. Bramson, Y. Lu, and B. Prabhakar. Asymptotc ndependence of queues under randomzed load balancng. Queueng Systems, 73: , 22. [] A. Braverman and J. Da. Sten s method for steady-state dffuson approxmatons of m/ph/n + m systems. arxv preprnt arxv:53.774, 25. [] F. Cecch, S. C. Borst, and J. S. H. van Leeuwaardena. Mean-feld analyss of ultra-dense csma networks. ACM SIGMETRICS Performance Evaluaton Revew, 432:3 5, 25. [2] A. Chantreau, J.-Y. Le Boudec, and N. Rstanovc. The age of gossp: spatal mean feld regme. In Proc. Ann. ACM SIGMETRICS Conf., pages 9 2, Seattle, Washngton, USA, 29. [3] F. Gotze. On the rate of convergence n the multvarate clt. Ann. Probab., pages , 99. [4] I. Gurvch et al. Dffuson models and steady-state approxmatons for exponentally ergodc markovan queues. Adv. n Appl. Probab., 246: , 24. [5] L. P. Kadanoff. More s the same; phase transtons and mean feld theores. Journal of Statstcal Physcs, 375-6: , 29. [6] H. K. Khall. Nonlnear systems. Prentce Hall, 2. [7] T. G. Kurtz. Lmt theorems for sequences of jump markov processes approxmatng ordnary dfferental processes. J. Appl. Probab., 82: , 97. [8] T. G. Kurtz. Approxmaton of populaton processes, volume 36. SIAM, 98. [9] J.-M. Lasry and P.-L. Lons. Mean feld games. Japanese Journal of Mathematcs, 2:229 26, 27.

13 [2] Y. Lu, Q. Xe, G. Klot, A. Geller, J. R. Larus, and A. Greenberg. Jon-Idle-Queue: A novel load balancng algorthm for dynamcally scalable web servces. Performance Evaluaton, 68:56 7, 2. [2] M. Manjrekar, V. Ramaswamy, and S. Shakkotta. A mean feld game approach to schedulng n cellular systems. In Proc. IEEE Int. Conf. Computer Communcatons INFOCOM, pages , Toronto, Canada, 24. [22] M. Mtzenmacher. The Power of Two Choces n Randomzed Load Balancng. PhD thess, Unversty of Calforna at Berkeley, 996. [23] A. Mukhopadhyay and R. R. Mazumdar. Analyss of load balancng n large heterogeneous processor sharng systems. arxv preprnt arxv:3.586, 23. [24] M. F. Norman. A central lmt theorem for Markov processes that move by small steps. Ann. Probab., pages 65 74, 974. [25] M. F. Norman. Lmt theorems for statonary dstrbutons. Ann. Appl. Probab., pages , 975. [26] C. Sten. A bound for the error n the normal approxmaton to the dstrbuton of a sum of dependent random varables. In Proc. Sxth Berkeley Symp. Math. Stat. Prob., pages , 972. [27] C. Sten. Approxmate computaton of expectatons. Lecture Notes-Monograph Seres, 7: 64, 986. [28] A. L. Stolyar. Pull-based load dstrbuton n large-scale heterogeneous servce systems. arxv preprnt arxv: , 24. [29] A. L. Stolyar. Tghtness of statonary dstrbutons of a flexble-server system n the halfn-whtt asymptotc regme. arxv preprnt arxv: , 24. [3] A.-S. Szntman. Topcs n propagaton of chaos. In Ecole d été de probabltés de Sant-Flour XIXâĂŤ989, pages [3] J. N. Tstskls and K. Xu. On the power of even a lttle resource poolng. Stochastc Systems, 2: 66, 22. [32] N. D. Vvedenskaya, R. L. Dobrushn, and F. I. Karpelevch. Queueng system wth selecton of the shortest of two queues: An asymptotc approach. Problemy Peredach Informats, 32:2 34, 996. [33] Q. Xe, X. Dong, Y. Lu, and R. Srkant. Power of d choces for large-scale bn packng: A loss model. In Proc. Ann. ACM SIGMETRICS Conf., 25. [34] L. Yng, R. Srkant, and X. Kang. The power of slghtly more than one sample n randomzed load balancng. In Proc. IEEE Int. Conf. Computer Communcatons INFOCOM, Hong Kong, 25.