A Refined Mean Field Approximation

Abstract

Mean field models are a popular means to approximate large and complex stochastic models that can be represented as N interacting objects. Recently it was shown that under very general conditions the steady-state expectation of any performance functional converges at rate O(1/N) to its mean field approximation. In this paper we establish a result that expresses the constant associated with this 1/N term. This constant can be computed easily as it is expressed in terms of the Jacobian and Hessian of the drift in the fixed point and the solution of a single Lyapunov equation. This allows us to propose a refined mean field approximation. By considering a variety of applications, that include coupon collector, load balancing and bin packing problems, we illustrate that the proposed refined mean field approximation is significantly more accurate that the classic mean field approximation for small and moderate values of N: relative errors are often below 1% for systems with N=10.