Size expansions of mean field approximation: Transient and steady-state analysis

Mean field approximation is a powerful tool to study the performance of large stochastic systems that is known to be exact as the system's size N goes to infinity. Recently, it has been shown that, when one wants to compute expected performance metric in steady-state, this approximation can be made more accurate by adding a term V∕N to the original approximation. This is called a refined mean field approximation in Nicolas Gast and Benny Van Houdt (2017). In this paper, we improve this result in two directions. First, we show how to obtain the same result for the transient regime. Second, we provide a further refinement by expanding the term in 1∕N2 (both for transient and steady-state regime). Our derivations are inspired by moment-closure approximation, a popular technique in theoretical biochemistry. We provide a number of examples that show: (1) that this new approximation is usable in practice for systems with up to a few tens of dimensions, and (2) that it accurately captures the transient and steady state behavior of such systems.