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## Section: New Results

### Mean Field and Refined Mean Field Methods

Mean field approximation is a popular means to approximate stochastic models that can be represented as a system of $N$ interacting objects. It is know to be exact as $N$ goes to infinity. In a recent series of paper, [24], [25], [7], we establish theoretical results and numerical methods that allows us to define an approximation that is much more accurate than the classical mean field approximation. This new approximation, that we call the refined mean field approximation, is based on the computation of an expansion term of the order $1/N$. By considering a variety of applications, that include coupon collector, load balancing and bin packing problems, we illustrate that the proposed refined mean field approximation is significantly more accurate that the classic mean field approximation for small and moderate values of $N$: relative errors are often below $1%$ for systems with $N=10$.

In [23], [8], 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 $1/{N}^{2}$ (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.