Stein's Method for Mean-Field Approximations in Light and Heavy Traffic Regimes

Mean-field analysis is an analytical method for understanding large-scale stochastic systems such as large-scale data centers and communication networks. The idea is to approximate the stationary distribution of a large-scale stochastic system using the equilibrium point (called the mean-field limit) of a dynamical system (called the mean-field model). This approximation is often justified by proving the weak convergence of stationary distributions to its mean-field limit. Most existing mean-field models concerned the light-traffic regime where the load of the system, denote by ρ, is strictly less than one and is independent of the size of the system. This is because a traditional mean-field model represents the limit of the corresponding stochastic system. Therefore, the load of the mean-field model is ρ=lim_{N-> ∞} ρ^(N), where ρ^(N) is the load of the stochastic system of size N. Now if ρ^(N)-> 1 as N-> ∞ (i.e., in the heavy-traffic regime), then ρ=1. For most systems, the mean-field limits when ρ=1 are trivial and meaningless. To overcome this difficulty of traditional mean-field models, this paper takes a different point of view on mean-field models. Instead of regarding a mean-field model as the limiting system of large-scale stochastic system, it views the equilibrium point of the mean-field model, called a mean-field solution, simply as an approximation of the stationary distribution of the finite-size system. Therefore both mean-field models and solutions can be functions of N. The proposed method focuses on quantifying the approximation error. If the approximation error is small (as we will show in two applications), then we can conclude that the mean-field solution is a good approximation of the stationary distribution.