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June 2013 Averaging over fast variables in the fluid limit for Markov chains: Application to the supermarket model with memory
M. J. Luczak, J. R. Norris
Ann. Appl. Probab. 23(3): 957-986 (June 2013). DOI: 10.1214/12-AAP861

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.

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M. J. Luczak. J. R. Norris. "Averaging over fast variables in the fluid limit for Markov chains: Application to the supermarket model with memory." Ann. Appl. Probab. 23 (3) 957 - 986, June 2013. https://doi.org/10.1214/12-AAP861

Information

Published: June 2013
First available in Project Euclid: 7 March 2013

zbMATH: 1274.60244
MathSciNet: MR3076675
Digital Object Identifier: 10.1214/12-AAP861

Subjects:
Primary: 60J28
Secondary: 60K25

Keywords: correctors , exponential martingale inequalities , fast variables , Join the shortest queue , Law of Large Numbers , Supermarket model , supermarket model with memory

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.23 • No. 3 • June 2013
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