On the Approximation Error of Mean-Field Models

Abstract

Mean-field models have been used to study large-scale and complex stochastic systems, such as large-scale data centers and dense wireless networks, using simple deterministic models (dynamical systems). This paper analyzes the approximation error of mean-field models for continuous-time Markov chains (CTMC), and focuses on mean-field models that are represented as finite-dimensional dynamical systems with a unique equilibrium point. By applying Stein's method and the perturbation theory, the paper shows that under some mild conditions, if the mean-field model is globally asymptotically stable and locally exponentially stable , the mean square difference between the stationary distribution of the stochastic system with size M and the equilibrium point of the corresponding mean-field system is O(1/M). The result of this paper establishes a general theorem for establishing the convergence and the approximation error (i.e., the rate of convergence) of a large class of CTMCs to their mean-field limit by mainly looking into the stability of the mean-field model, which is a deterministic system and is often easier to analyze than the CTMCs. Two applications of mean-field models in data center networks are presented to demonstrate the novelty of our results.

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

“…As for Equation (1), we show that Equation (2) holds for the transient regime and can be extended to the stationary regime under the same conditions as [26]. As a byproduct, Equation (1) can be recovered from Equation (2) by using h(.)…”

“…In the transient case, [14] obtains a O (1/N ) like Equation (2) for a simplified version of our model where a transition affects at most one object. For the stationary regime, a O (1/ √ N ) like Equation (1) is obtained in [26]. Our proof is inspired by the methodology of [26] but we obtain a stronger result by working with a generic function h. Note that infinite-dimensional models arise naturally when one considers queuing systems with unbounded queues.…”

“…The rate of convergence of X (N ) i to x i has been studied by several papers, e.g. [2,15,26], that show that the expected distance between X (N ) and x is of the order of 1/ √ N :…”

“…It was originally proven for finite time-horizon and recently extended to stationary distributions in [26].…”

“…Our approach is different and uses an idea inspired by Stein's method and [14,21,26] to relate Equation (2) and the convergence of the generators of the Markov chain.…”