Mean Field and Refined Mean Field Approximations for Heterogeneous Systems

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.

rmf tool - A library to Compute (Refined) Mean Field Approximation(s)

“…Let the drift in state z be denoted by f het (z), then, the mean field approximation is again the solution to the ode having f het as drift with initial condition z(0). If both, a k,s,s and b k, k,s,s,s ,s are uniformly bounded, it holds, as shown in [1] that the adapted mean field and refined mean field approximation capture the probability of the objects to be in a state with an accuracy of O(1/n) and O(1/n 2 ), i.e.…”

“…In [1], the authors extend the notion of the HomPPs to deal with populations of heterogeneous objects. As before, the heterogeneous population model consists of n interacting objects which each evolve in a finite state space {1 .…”

“…The term v k,s (t) refers to the adapted refinement term whose precise definition can be found in [1,Appendix B]. Simulations of stochastic trajectories, mean field and the refinement methods can be calculated by calling the same functions as for the homogeneous case.…”

“…This is particularly useful for heterogeneous models composed of n objects that each evolve in a S dimensional state: a natural description of the model is to construct a nS-dimensional DDPP, but that evolves in a subset of dimension n(S −1) or even smaller. For instance, the cache replacement policy studied in [1,2,8] with n objects and S lists is naturally described as a n(S + 1) process but evolves in fact in a nS − S state space. Using the automated dimension reduction greatly simplifies the definition of the model in the tool.…”

“…Although they are not directly provided as examples in the repository, the tool is also used in [1,2] to analyze the performance of cache replacement policies. These cache replacement policies are examples of non-homogeneous population models.…”