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 the authors of Gast et al. 2019, and show that the error of this approximation is reduced to O(1/n2). 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. For moderate system sizes (n ≈ 30) the refined mean field approximation tends to be more accurate than simulations for any reasonable simulation time.
- Sebastian Allmeier and Nicolas Gast. 2021. rmftool-A library to Compute (Refined) Mean Field Approximation (s). In TOSME 2021.Google Scholar
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- 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 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 - SIGMETRICS '15. ACM Press, Portland, Oregon, USA, 123--136. https://doi.org/10.1145/2745844.2745850Google ScholarDigital Library
- Nicolas Gast and Benny Van Houdt. 2017. 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 ScholarDigital Library
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- 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
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- Mean Field and Refined Mean Field Approximations for Heterogeneous Systems: It Works!
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