Abstract
Mean-field approximation is a powerful tool to study large-scale stochastic systems such as data-centers -- one example being the famous power of two-choice paradigm. It is shown in the literature that under quite general conditions, the empirical measure of a system of N interacting objects converges at rate O(1√N) to a deterministic dynamical system, called its mean-field approximation.
In this paper, we revisit the accuracy of mean-field approximation by focusing on expected values. We show that, under almost the same general conditions, the expectation of any performance functional converges at rate O(1/N) to its mean-field approximation. Our result applies for finite and infinite-dimensional mean-field models. We also develop a new perturbation theory argument that shows that the result holds for the stationary regime if the dynamical system is asymptotically exponentially stable. We provide numerical experiments that demonstrate that this rate of convergence is tight and that illustrate the necessity of our conditions. As an example, we apply our result to the classical two-choice model. By combining our theory with numerical experiments, we claim that, as the load rho goes to 1, the average queue length of a two-choice system with N servers is log2 1/(1--ρ) + 1/(2N(1-ρ) +O(1/N2).
- F Baccelli, FI Karpelevich, M Ya Kelbert, AA Puhalskii, AN Rybko, and Yu M Suhov. 1992. A mean-field limit for a class of queueing networks. Journal of statistical physics 66, 3--4 (1992), 803--825.Google ScholarCross Ref
- Michel Benaim and Jean-Yves Le Boudec. 2008. A class of mean field interaction models for computer and communication systems. Performance Evaluation 65, 11 (2008), 823--838. Google ScholarDigital Library
- Luca Bortolussi and Richard A Hayden. 2013. Bounds on the deviation of discrete-time Markov chains from their mean-field model. Performance Evaluation 70, 10 (2013), 736--749. Google ScholarDigital Library
- Henri Cartan. 1977. Cours de calcul différentiel. Hermann.Google Scholar
- F Cecchi, SC Borst, and JSH van Leeuwaarden. 2015. Mean-field analysis of ultra-dense csma networks. ACM SIGMETRICS Performance Evaluation Review 43, 2 (2015), 13--15. Google ScholarDigital Library
- Augustin Chaintreau, Jean-Yves Le Boudec, and Nikodin Ristanovic. 2009. The Age of Gossip: Spatial Mean Field Regime. SIGMETRICS Perform. Eval. Rev. 37, 1 (June 2009), 109--120. Google ScholarDigital Library
- Jeong-woo Cho, Jean-Yves Le Boudec, and Yuming Jiang. 2010. On the validity of the fixed point equation and decoupling assumption for analyzing the 802.11 mac protocol. ACM SIGMETRICS Performance Evaluation Review 38, 2 (2010), 36--38. Google ScholarDigital Library
- Jaap Eldering. 2013. Normally Hyperbolic Invariant Manifolds-the Noncompact Case (Atlantis Series in Dynamical Systems vol 2). Berlin: Springer.Google Scholar
- Patrick Eschenfeldt and David Gamarnik. 2016. Supermarket queueing system in the heavy traffic regime. Short queue dynamics. arXiv preprint arXiv:1610.03522 (2016).Google Scholar
- Nicolas Gast and Gaujal Bruno. 2010. A Mean Field Model of Work Stealing in Large-scale Systems. SIGMETRICS Perform. Eval. Rev. 38, 1 (June 2010), 13--24. Google ScholarDigital Library
- Nicolas Gast and Bruno Gaujal. 2012. Markov chains with discontinuous drifts have differential inclusion limits. Performance Evaluation 69, 12 (2012), 623--642. Google ScholarDigital Library
- Nicolas Gast and Benny Van Houdt. 2015. Transient and steady-state regime of a family of list-based cache replacement algorithms. In Proceedings of the 2015 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems. ACM, 123--136. Google ScholarDigital Library
- Carl Graham. 2000. Chaoticity on path space for a queueing network with selection of the shortest queue among several. Journal of Applied Probability 37, 01 (2000), 198--211.Google ScholarCross Ref
- Vassili N Kolokoltsov, Jiajie Li, and Wei Yang. 2011. Mean field games and nonlinear Markov processes. arXiv preprint arXiv:1112.3744 (2011).Google Scholar
- Thomas G Kurtz. 1970. Solutions of Ordinary Differential Equations as Limits of Pure Jump Markov Processes. Journal of Applied Probability 7 (1970), 49--58.Google ScholarCross Ref
- Thomas G Kurtz. 1978. Strong approximation theorems for density dependent Markov chains. Stochastic Processes and Their Applications 6, 3 (1978), 223--240.Google ScholarCross Ref
- Yi Lu, Qiaomin Xie, Gabriel Kliot, Alan Geller, James R Larus, and Albert Greenberg. 2011. Join-Idle-Queue: A novel load balancing algorithm for dynamically scalable web services. Performance Evaluation 68, 11 (2011), 1056--1071. Google ScholarDigital Library
- Malwina J Luczak and Colin McDiarmid. 2007. Asymptotic distributions and chaos for the supermarket model. arXiv preprint arXiv:0712.2091 (2007).Google Scholar
- Wouter Minnebo and Benny Van Houdt. 2014. A fair comparison of pull and push strategies in large distributed networks. IEEE/ACM Transactions on Networking (TON) 22, 3 (2014), 996--1006. Google ScholarDigital Library
- Michael David Mitzenmacher. 1996. The Power of Two Random Choices in Randomized Load Balancing. Ph.D. Dissertation. PhD thesis, Graduate Division of the University of California at Berkley.Google Scholar
- Charles Stein. 1986. Approximate computation of expectations. Lecture Notes-Monograph Series 7 (1986), i--164.Google Scholar
- John N Tsitsiklis and Kuang Xu. 2011. On the power of (even a little) centralization in distributed processing. ACM SIGMETRICS Performance Evaluation Review 39, 1 (2011), 121--132. Google ScholarDigital Library
- Stephen RE Turner. 1998. The effect of increasing routing choice on resource pooling. Probability in the Engineering and Informational Sciences 12, 01 (1998), 109--124.Google ScholarCross Ref
- Benny Van Houdt. 2013. A mean field model for a class of garbage collection algorithms in flash-based solid state drives. In ACM SIGMETRICS Performance Evaluation Review, Vol. 41. ACM, 191--202. Google ScholarDigital Library
- Nikita Dmitrievna Vvedenskaya, Roland L'vovich Dobrushin, and Fridrikh Izrailevich Karpelevich. 1996. Queueing system with selection of the shortest of two queues: An asymptotic approach. Problemy Peredachi Informatsii 32, 1 (1996), 20--34.Google Scholar
- Lei Ying. 2016. On the Approximation Error of Mean-Field Models. In Proceedings of the 2016 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Science. ACM, 285--297. Google ScholarDigital Library
- Lei Ying. 2016. On the Rate of Convergence of the Power-of-Two-Choices to its Mean-Field Limit. CoRR abs/1605.06581 (2016). http://arxiv.org/abs/1605.06581Google Scholar
Index Terms
- Expected Values Estimated via Mean-Field Approximation are 1/N-Accurate
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