Abstract
Mean field models are a popular means to approximate large and complex stochastic models that can be represented as N interacting objects. Recently it was shown that under very general conditions the steady-state expectation of any performance functional converges at rate O(1/N) to its mean field approximation. In this paper we establish a result that expresses the constant associated with this 1/N term. This constant can be computed easily as it is expressed in terms of the Jacobian and Hessian of the drift in the fixed point and the solution of a single Lyapunov equation. This allows us to propose a refined mean field approximation. By considering a variety of applications, that include coupon collector, load balancing and bin packing problems, we illustrate that the proposed refined mean field approximation is significantly more accurate that the classic mean field approximation for small and moderate values of N: relative errors are often below 1% for systems with N=10.
- R. H. Bartels and G. W. Stewart. 1972. Solution of the Matrix Equation AX + XB = C {F4}. Commun. ACM 15, 9 (Sept. 1972), 820--826. Google Scholar
Digital Library
- 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 Scholar
Digital Library
- Luca Bortolussi, Jane Hillston, Diego Latella, and Mieke Massink. 2013. Continuous approximation of collective system behaviour: A tutorial. Performance Evaluation 70, 5 (2013), 317--349. Google Scholar
Digital Library
- Anton Braverman and Jim Dai. 2017. Stein's method for steady-state diffusion approximations of M/Ph/n + M systems. The Annals of Applied Probability 27, 1 (2017), 550--581.Google Scholar
Cross Ref
- Anton Braverman, JG Dai, and Jiekun Feng. 2017. Stein's method for steady-state diffusion approximations: an introduction through the Erlang-A and Erlang-C models. Stochastic Systems 6, 2 (2017), 301--366.Google Scholar
Cross Ref
- Jeong-woo Cho, Jean-Yves Le Boudec, and Yuming Jiang. 2012. On the asymptotic validity of the decoupling assumption for analyzing 802.11 MAC protocol. IEEE Transactions on Information Theory 58, 11 (2012), 6879--6893. Google Scholar
Digital Library
- Richard Datko. 1972. Uniform asymptotic stability of evolutionary processes in a Banach space. SIAM Journal on Mathematical Analysis 3, 3 (1972), 428--445.Google Scholar
Cross Ref
- Jaap Eldering. 2013. Normally Hyperbolic Invariant Manifolds-the Noncompact Case (Atlantis Series in Dynamical Systems vol 2). Springer, Berlin.Google Scholar
- Nicolas Gast. 2017. Expected values estimated via mean-field approximation are 1/N-accurate. Proceedings of the ACM on Measurement and Analysis of Computing Systems 1, 1 (2017), 17. Google Scholar
Digital Library
- 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 Scholar
Digital Library
- N. Gast and B. Gaujal. 2012. Markov chains with discontinuous drifts have differential inclusion limits. Performance Evaluation 69, 12 (2012), 623--642. Google Scholar
Digital Library
- Itai Gurvich et al. 2014. Diffusion models and steady-state approximations for exponentially ergodic Markovian queues. The Annals of Applied Probability 24, 6 (2014), 2527--2559.Google Scholar
Cross Ref
- Eric Jones, Travis Oliphant, Pearu Peterson, et al. 2001--. SciPy: Open source scientific tools for Python. (2001--). http://www.scipy.org/ {Online; accessed 2017-06--15}.Google Scholar
- H. K. Khalil. 1996. Nonlinear Systems (2nd ed.). Prentice-Hall, Englewood Cliffs, NJ.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 Scholar
Cross Ref
- Thomas G Kurtz. 1978. Strong approximation theorems for density dependent Markov chains. Stochastic Processes and Their Applications 6, 3 (1978), 223--240.Google Scholar
Cross Ref
- G. Latouche and V. Ramaswami. 1999. Introduction to Matrix Analytic Methods and stochastic modeling. SIAM, Philadelphia.Google Scholar
- Jean-Yves Le Boudec, David McDonald, and Jochen Mundinger. 2007. A generic mean field convergence result for systems of interacting objects. In Quantitative Evaluation of Systems, 2007. QEST 2007. Fourth International Conference on the. IEEE, 3--18. Google Scholar
Digital Library
- M. Lin, B. Fan, J.C.S. Lui, and D. Chiu. 2007. Stochastic analysis of file-swarming systems. Performance Evaluation 64, 9 (2007), 856--875. Google Scholar
Digital Library
- L. Massoulié and M. Vojnović. 2005. Coupon Replication Systems. SIGMETRICS Perform. Eval. Rev. 33, 1 (June 2005), 2--13. Google Scholar
Digital Library
- W. Minnebo and B. Van Houdt. 2014. A Fair Comparison of Pull and Push Strategies in Large Distributed Networks. IEEE/ACM Transactions on Networking 22 (2014), 996--1006. Issue 3. Google Scholar
Digital Library
- Michael Mitzenmacher. 2001. The power of two choices in randomized load balancing. IEEE Transactions on Parallel and Distributed Systems 12, 10 (2001), 1094--1104. Google Scholar
Digital 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
- Cristopher Moore. 1990. Unpredictability and undecidability in dynamical systems. Physical Review Letters 64, 20 (1990), 2354.Google Scholar
Cross Ref
- Vu Ngoc Phat and Tran Tin Kiet. 2002. On the Lyapunov equation in Banach spaces and applications to control problems. International Journal of Mathematics and Mathematical Sciences 29, 3 (2002), 155--166.Google Scholar
Cross Ref
- V. Simoncini. 2016. Computational methods for linear matrix equations. SIAM Rev. 58 (2016), 377--441. Issue 3.Google Scholar
Digital Library
- 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 Scholar
Digital Library
- 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, ACM, New York, NY, USA, 191--202. Google Scholar
Digital 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
- Q. Xie, X. Dong, Y. Lu, and R. Srikant. 2015. Power of D Choices for Large-Scale Bin Packing: A Loss Model. SIGMETRICS Perform. Eval. Rev. 43, 1 (June 2015), 321--334. Google Scholar
Digital Library
- 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 (SIGMETRICS '16). ACM, New York, NY, USA, 285--297. Google Scholar
Digital Library
- Lei Ying. 2017. Stein's Method for Mean Field Approximations in Light and Heavy Traffic Regimes. Proceedings of the ACM on Measurement and Analysis of Computing Systems 1, 1 (2017), 12. Google Scholar
Digital Library
Index Terms
A Refined Mean Field Approximation
Recommendations
Expected Values Estimated via Mean-Field Approximation are 1/N-Accurate
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 ...
On the Approximation Error of Mean-Field Models
Performance evaluation reviewMean-field models have been used to study large-scale and complex stochastic systems, such as large-scale data centers and dense wireless networks, using simple deterministic models (dynamical systems). This paper analyzes the approximation error of ...
Expected Values Estimated via Mean-Field Approximation are 1/N-Accurate: Extended Abstract
SIGMETRICS '17 Abstracts: Proceedings of the 2017 ACM SIGMETRICS / International Conference on Measurement and Modeling of Computer SystemsIn this paper, we study the accuracy of mean-field approximation. We show that, under 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-...






Comments