A Refined Mean Field Approximation

Abstract

Mean eld 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 eld 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 xed point and the solution of a single Lyapunov equation. This allows us to propose a rened mean eld approximation. By considering a variety of applications, that include coupon collector, load balancing and bin packing problems, we illustrate that the proposed rened mean eld approximation is signicantly more accurate that the classic mean eld approximation for small and moderate values of N : relative errors are often below 1% for systems with N = 10.

A Refined Mean Field Approximation for Synchronous Population Processes

“…• When the mean field approximation does not have an exponentially stable attractor, the improved accuracy only holds for a finite time horizon. Our results extend the recent results of [7]. The authors of [7] study the steady-state of asynchronous stochastic models (that therefore have a continuous-time mean field approximation).…”

“…Our results extend the recent results of [7]. The authors of [7] study the steady-state of asynchronous stochastic models (that therefore have a continuous-time mean field approximation). There are two differences in our work : First we focus on synchronous objects; Second we obtain results also for the transient regime.…”

“…There are two differences in our work : First we focus on synchronous objects; Second we obtain results also for the transient regime. Note that the results of [7] and the one of the current paper follow from a series of recent results concerning the rate of convergence of stochastic models to their mean field approximation [4,8,12].…”

“…In this paper, we have shown that it is possible to adapt the refined mean field approximation proposed in [7] to the case of synchronous population processes, both for transient and steady-state state. This approximation is more accurate than the classical mean field approximation in many cases.…”

Steady-State Analysis of the Join-the-Shortest-Queue Model in the Halfin–Whitt Regime

“…The bounds in (1.3) are then simply an alternative interpretation of these convergence rates. For other examples of Stein's method for fluid, or mean-field models, see [16,17,40,41]. Specifically, [40] was the first to make the connection between Stein's method and convergence rates to the mean-field equilibrium.…”