  We set out a general procedure which allows the approximation of certain
Markov chains by the solutions of differential equations. The chains considered
have some components which oscillate rapidly and randomly, while others are
close to deterministic. The limiting dynamics are obtained by averaging the
drift of the latter with respect to a local equilibrium distribution of the
former. Some general estimates are proved under a uniform mixing condition on
the fast variable which give explicit error probabilities for the fluid
approximation. Mitzenmacher, Prabhakar and Shah [In Proc. 43rd Ann. Symp.
Found. Comp. Sci. (2002) 799-808, IEEE] introduced a variant with memory of the
"join the shortest queue" or "supermarket" model, and obtained a limit picture
for the case of a stable system in which the number of queues and the total
arrival rate are large. In this limit, the empirical distribution of queue
sizes satisfies a differential equation, while the memory of the system
oscillates rapidly and randomly. We illustrate our general fluid limit estimate
by giving a proof of this limit picture.
