Bias and Refinement of Multiscale Mean Field Models
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Mean field approximation is a powerful technique which has been used in many settings to study large-scale stochastic systems. In the case of two-timescale systems, the approximation is obtained by a combination of scaling arguments and the use of the averaging principle. This paper analyzes the approximation error of this `average' mean field model for a 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, e.g. the population size, which as N gets large decreases the jump size of the slow component in contrast to the unchanged dynamics of the fast component. We show that under relatively mild conditions, the `average' mean field approximation has a bias of order O(1/N) compared to š¼[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/N^2). This refined `average' mean field approximation allows computing an accurate approximation even for small scaling factors, i.e., Nā 10 -50. We illustrate the developed framework and accuracy results through an application to a random access CSMA model.
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