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

TL;DR

This paper develops a general method to approximate Markov chains with fast oscillating components by differential equations, providing explicit error bounds, and applies it to analyze a supermarket model with memory.

Contribution

It introduces a new averaging technique for Markov chains with fast variables and applies it to a supermarket model with memory, deriving fluid limit results with error estimates.

Findings

Established a general averaging procedure with error bounds for Markov chains.

Proved the fluid limit for the supermarket model with memory.

Provided explicit error probabilities for the approximation.

Abstract

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.