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.
We analyze the error of an ODE approximation of a generic two-timescale model (X, Y), where the slow component X describes a population of interacting particles which is fully coupled with a rapidly changing environment Y. The model is parametrized by a scaling factor N, which can be the number of particles. As N grows, the jump sizes of the slow component decrease in contrast to the unchanged dynamics of the fast component. A typical example is the random access CSMA model that we study.
By using an averaging principle, one can construct an ODE approximation of X, that we call the 'average' mean field approximation. We show that under relatively mild conditions, this approximation has a bias of order O(1/N) compared to \mathbbE [X]. This holds true under any continuous performance metric in the transient regime, as well as for the steady-state if the model is exponentially stable. To go one step further, we derive a bias correction term for the steady-state, from which we define a new approximation called the refined 'average' mean field approximation whose bias is of order O(1/N2). This refined 'average' mean field approximation allows computing an accurate approximation even for small scaling factors, i.e., N ≈ 10 - 50.
Mean field approximation is a powerful technique to study the performance of large stochastic systems represented as systems of interacting objects. Applications include load balancing models, epidemic spreading, cache replacement policies, or large-scale data centers, for which mean field approximation gives very accurate estimates of the transient or steady-state behaviors. In a series of recent papers [9, 7], a new and more accurate approximation, called the refined mean field approximation is presented. Yet, computing this new approximation can be cumbersome. The purpose of this paper is to present a tool, called rmf tool, that takes the description of a mean field model, and can numerically compute its mean field approximations and refinement.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.