Expected Values Estimated via Mean-Field Approximation are 1/N-Accurate

Abstract

Mean-field approximation is a powerful tool to study large-scale stochastic systems such as data-centers -- one example being the famous power of two-choice paradigm. It is shown in the literature that under quite general conditions, the empirical measure of a system of N interacting objects converges at rate O (1√ N ) to a deterministic dynamical system, called its mean-field approximation. In this paper, we revisit the accuracy of mean-field approximation by focusing on expected values. We show that, under almost the same general conditions, the expectation of any performance functional converges at rate O(1/ N ) to its mean-field approximation. Our result applies for finite and infinite-dimensional mean-field models. We also develop a new perturbation theory argument that shows that the result holds for the stationary regime if the dynamical system is asymptotically exponentially stable. We provide numerical experiments that demonstrate that this rate of convergence is tight and that illustrate the necessity of our conditions. As an example, we apply our result to the classical two-choice model. By combining our theory with numerical experiments, we claim that, as the load rho goes to 1, the average queue length of a two-choice system with N servers is log 2 1/(1--ρ) + 1/(2 N (1-ρ) + O (1/ N 2 ).