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We set out a general procedure which allows the approximation of certainMarkov chains by the solutions of differential equations. The chains consideredhave some components which oscillate rapidly and randomly, while others areclose to deterministic. The limiting dynamics are obtained by averaging thedrift of the latter with respect to a local equilibrium distribution of theformer. Some general estimates are proved under a uniform mixing condition onthe fast variable which give explicit error probabilities for the fluidapproximation. 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 picturefor the case of a stable system in which the number of queues and the totalarrival rate are large. In this limit, the empirical distribution of queuesizes satisfies a differential equation, while the memory of the systemoscillates rapidly and randomly. We illustrate our general fluid limit estimateby giving a proof of this limit picture.

Averaging over fast variables in the fluid limit for Markov chains: Application to the supermarket model with memory