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
interacting objects. It is know to be exact as 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 . 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 : relative errors are often below for systems with
.
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 (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.