skip to main content
research-article

Mean Field and Refined Mean Field Approximations for Heterogeneous Systems: It Works!

Published:28 February 2022Publication History
Skip Abstract Section

Abstract

Mean field approximation is a powerful technique to study the performance of large stochastic systems represented as n interacting objects. Applications include load balancing models, epidemic spreading, cache replacement policies, or large-scale data centers. Mean field approximation is asymptotically exact for systems composed of n homogeneous objects under mild conditions. In this paper, we study what happens when objects are heterogeneous. This can represent servers with different speeds or contents with different popularities. We define an interaction model that allows obtaining asymptotic convergence results for stochastic systems with heterogeneous object behavior, and show that the error of the mean field approximation is of order $O(1/n)$. More importantly, we show how to adapt the refined mean field approximation, developed by Gast et al., and show that the error of this approximation is reduced to O(1/n^2). To illustrate the applicability of our result, we present two examples. The first addresses a list-based cache replacement model, RANDOM(m), which is an extension of the RANDOM policy. The second is a heterogeneous supermarket model. These examples show that the proposed approximations are computationally tractable and very accurate. They also show that for moderate system sizes (30) the refined mean field approximation tends to be more accurate than simulations for any reasonable simulation time.

References

  1. , Sebastian Allmeier and Nicolas Gast. 2021. rmftool-A library to Compute (Refined) Mean Field Approximation (s). In TOSME 2021 .Google ScholarGoogle Scholar
  2. , Francc ois Baccelli and Thibaud Taillefumier. 2019. Replica-mean-field limits for intensity-based neural networks. SIAM Journal on Applied Dynamical Systems , Vol. 18, 4 (2019), 1756--1797.Google ScholarGoogle ScholarCross RefCross Ref
  3. , Anton Braverman. 2017. Stein's method for steady-state diffusion approximations . Ph.,D. Dissertation. Cornell University.Google ScholarGoogle Scholar
  4. , Anton Braverman and Jim Dai. 2017. Stein's method for steady-state diffusion approximations of $ M/Ph/nGoogle ScholarGoogle Scholar
  5. M $ systems. The Annals of Applied Probability , Vol. 27, 1 (2017), 550--581.Google ScholarGoogle Scholar
  6. 017)]% braverman2017stein2, 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 , Vol. 6, 2 (2017), 301--366.Google ScholarGoogle ScholarCross RefCross Ref
  7. 020)]% bravermanHighOrderSteadystate2020, Anton Braverman, J. G. Dai, and Xiao Fang. 2020. High Order Steady-State Diffusion Approximations. arXiv:2012.02824 (Dec. 2020). arxiv: 2012.02824Google ScholarGoogle Scholar
  8. , Giuliano Casale and Nicolas Gast. 2020. Performance analysis methods for list-based caches with non-uniform access. IEEE/ACM Transactions on Networking (2020).Google ScholarGoogle Scholar
  9. 001)]% che2001analysis, Hao Che, Zhijun Wang, and Ye Tung. 2001. Analysis and design of hierarchical web caching systems. In INFOCOM 2001. Twentieth Annual Joint Conference of the IEEE Computer and Communications Societies. Proceedings. IEEE, Vol. 3. IEEE, 1416--1424.Google ScholarGoogle ScholarCross RefCross Ref
  10. , Asit Dan and Don Towsley. 1990. An approximate analysis of the LRU and FIFO buffer replacement schemes. In Proceedings of the 1990 ACM SIGMETRICS conference on Measurement and modeling of computer systems . 143--152.Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. t almbox.(2018)]% dearrudaFundamentalsSpreadingProcesses2018, Guilherme Ferraz de Arruda , Francisco A. Rodrigues, and Yamir Moreno. 2018. Fundamentals of Spreading Processes in Single and Multilayer Complex Networks. Physics Reports , Vol. 756 (Oct. 2018), 1--59. https://doi.org/10.1016/j.physrep.2018.06.007 arxiv: 1804.08777Google ScholarGoogle ScholarCross RefCross Ref
  12. , Ronald Fagin. 1977. Asymptotic miss ratios over independent references. J. Comput. System Sci. , Vol. 14, 2 (1977), 222--250.Google ScholarGoogle ScholarCross RefCross Ref
  13. 012)]% fricker2012versatile, Christine Fricker, Philippe Robert, and James Roberts. 2012. A versatile and accurate approximation for LRU cache performance. In 2012 24th international teletraffic congress (ITC 24). IEEE, 1--8.Google ScholarGoogle Scholar
  14. , 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 , Vol. 1, 1 (2017), 17.Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. 019)]% gastSizeExpansionsMean2019, Nicolas Gast, Luca Bortolussi, and Mirco Tribastone. 2019. Size Expansions of Mean Field Approximation: Transient and Steady-State Analysis. Performance Evaluation , Vol. 129 (Feb. 2019), 60--80. https://doi.org/10.1016/j.peva.2018.09.005Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. , 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 - SIGMETRICS '15. ACM Press , Portland, Oregon, USA, 123--136. https://doi.org/10.1145/2745844.2745850Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. , Nicolas Gast and Benny Van Houdt. 2017a. A Refined Mean Field Approximation. Proceedings of the ACM on Measurement and Analysis of Computing Systems , Vol. 1, 2 (Dec. 2017), 33:1--33:28. https://doi.org/10.1145/3154491Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. , Nicolas Gast and Benny Van Houdt. 2017b. A refined mean field approximation. Proceedings of the ACM on Measurement and Analysis of Computing Systems , Vol. 1, 2 (2017), 1--28.Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. 020)]% gomesIndividualVariationSusceptibility2020, M. Gabriela M. Gomes, Ricardo Aguas, Rodrigo M. Corder, Jessica G. King, Kate E. Langwig, Caetano Souto-Maior, Jorge Carneiro, Marcelo U. Ferreira, and Carlos Penha-Gonc calves. 2020. Individual Variation in Susceptibility or Exposure to SARS -CoV -2 Lowers the Herd Immunity Threshold . Preprint. Epidemiology . https://doi.org/10.1101/2020.04.27.20081893Google ScholarGoogle Scholar
  20. , Ramon Grima. 2010. An effective rate equation approach to reaction kinetics in small volumes: Theory and application to biochemical reactions in nonequilibrium steady-state conditions. The Journal of chemical physics , Vol. 133, 3 (2010), 07B604.Google ScholarGoogle ScholarCross RefCross Ref
  21. 011)]% grima2011, Ramon Grima, Philipp Thomas, and Arthur V. Straube. 2011. How accurate are the nonlinear chemical Fokker-Planck and chemical Langevin equations? The Journal of Chemical Physics , Vol. 135, 8 (2011).Google ScholarGoogle Scholar
  22. 987)]% hazewinkelStochasticAnalysisComputer1987, M Hazewinkel, F Calogero, Yu. I Manin, A. H. G Rinnooy Kan, and G.-C Rota. 1987. Stochastic Analysis of Computer Storage .Springer Netherlands , Dordrecht .Google ScholarGoogle Scholar
  23. , R. Hirade and T. Osogami. 2010. Analysis of Page Replacement Policies in the Fluid Limit. Oper. Res. , Vol. 58 (July 2010), 971--984. https://doi.org/10.1287/opre.1090.0761Google ScholarGoogle ScholarDigital LibraryDigital Library
  24. 020)]% hodgkinsonNormalApproximationsDiscretetime2018, Liam Hodgkinson, Ross McVinish, and Philip K Pollett. 2020. Normal approximations for discrete-time occupancy processes. Stochastic Processes and their Applications , Vol. 130, 10 (2020), 6414--6444.Google ScholarGoogle Scholar
  25. 018)]% jiang2018convergence, Bo Jiang, Philippe Nain, and Don Towsley. 2018. On the convergence of the TTL approximation for an LRU cache under independent stationary request processes. ACM Transactions on Modeling and Performance Evaluation of Computing Systems (TOMPECS) , Vol. 3, 4 (2018), 1--31.Google ScholarGoogle Scholar
  26. 012)]% kolokoltsovMeanFieldGames2012, Vassili N. Kolokoltsov, Jiajie Li, and Wei Yang. 2012. Mean Field Games and Nonlinear Markov Processes. arXiv:1112.3744 (April 2012). arxiv: 1112.3744Google ScholarGoogle Scholar
  27. , Thomas G. Kurtz. 1978. Strong Approximation Theorems for Density Dependent Markov Chains. Stochastic Processes and their Applications , Vol. 6, 3 (Feb. 1978), 223--240. https://doi.org/10.1016/0304--4149(78)90020-0Google ScholarGoogle Scholar
  28. rd et almbox.(1987)]% mezard1987sk, Marc Mézard, Giorgio Parisi, and Miguel Angel Virasoro. 1987. Spin glass theory and beyond: An Introduction to the Replica Method and Its Applications. World Scientific Publishing Company (1987), 232--237.Google ScholarGoogle Scholar
  29. , M. Mitzenmacher. Oct./2001. The Power of Two Choices in Randomized Load Balancing. IEEE Transactions on Parallel and Distributed Systems , Vol. 12, 10 ( Oct./2001), 1094--1104. https://doi.org/10.1109/71.963420Google ScholarGoogle ScholarDigital LibraryDigital Library
  30. et almbox.(2020)]% montalbanHerdImmunityIndividual2020, Antonio Montalbán, Rodrigo M. Corder, and M. Gabriela M. Gomes. 2020. Herd Immunity under Individual Variation and Reinfection. arXiv:2008.00098 (Nov. 2020). arxiv: 2008.00098Google ScholarGoogle Scholar
  31. , Arpan Mukhopadhyay and Ravi R. Mazumdar. 2015. Analysis of Load Balancing in Large Heterogeneous Processor Sharing Systems. arXiv:1311.5806 (Feb. 2015). arxiv: 1311.5806Google ScholarGoogle Scholar
  32. , Charles Stein. 1986. Approximate computation of expectations. Lecture Notes-Monograph Series , Vol. 7 (1986), i--164.Google ScholarGoogle Scholar
  33. 012)]% tsukada1, N. Tsukada, R. Hirade, and N. Miyoshi. 2012. Fluid Limit Analysis of FIFO and RR Caching for Independent Reference Models. Perform. Eval. , Vol. 69, 9 (Sept. 2012), 403--412. https://doi.org/10.1016/j.peva.2012.05.008Google ScholarGoogle ScholarDigital LibraryDigital Library
  34. , Benny Van Houdt. 2013. A Mean Field Model for a Class of Garbage Collection Algorithms in Flash-Based Solid State Drives. ACM SIGMETRICS Performance Evaluation Review , Vol. 41, 1 (June 2013), 191--202. https://doi.org/10.1145/2494232.2465543Google ScholarGoogle ScholarDigital LibraryDigital Library
  35. , N. G. van Kampen. 2007. Stochastic processes in physics and chemistry .Elsevier, Amsterdam; Boston; London.Google ScholarGoogle Scholar
  36. , Lei Ying. 2015. On the Rate of Convergence of Mean-Field Models: Stein's Method Meets the Perturbation Theory. arXiv preprint arXiv:1510.00761 (2015).Google ScholarGoogle Scholar
  37. , 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 ScholarGoogle ScholarDigital LibraryDigital Library

Index Terms

  1. Mean Field and Refined Mean Field Approximations for Heterogeneous Systems: It Works!

      Recommendations

      Comments

      Login options

      Check if you have access through your login credentials or your institution to get full access on this article.

      Sign in

      Full Access

      PDF Format

      View or Download as a PDF file.

      PDF

      eReader

      View online with eReader.

      eReader
      About Cookies On This Site

      We use cookies to ensure that we give you the best experience on our website.

      Learn more

      Got it!