Bias and Refinement of Multiscale Mean Field Models

Abstract

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.