Mean Field and Refined Mean Field Approximations for Heterogeneous Systems

Sebastian Allmeier, Nicolas Gast
2022 Abstract Proceedings of the 2022 ACM SIGMETRICS/IFIP PERFORMANCE Joint International Conference on Measurement and Modeling of Computer Systems  
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
more » ... 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 [5], 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. For moderate system sizes (𝑛 ≈ 30) the refined mean field approximation tends to be more accurate than simulations for any reasonable simulation time. CCS CONCEPTS • Mathematics of computing → Stochastic processes; Ordinary differential equations.
doi:10.1145/3489048.3522653 fatcat:l2lfv5r2tveh7odgvfex4yqot4