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Asymptotic distributions and chaos for the supermarket model

By Malwina J. Luczak and Colin McDiarmid

Abstract

In the supermarket model there are n queues, each with a unit rate server. Customers arrive in a Poisson process at rate n, where 0 < < 1. Each customer chooses d 2 queues uniformly at random, and joins a shortest one. It is known that the equilibrium distribution of a typical queue length converges to a certain explicit limiting distribution as n ! 1. We quantify the rate of convergence by showing that the total variation distance between the equilibrium distribution and the limiting distribution is essentially of order n−1; and we give a corresponding result for systems starting from quite general initial conditions (not in equilibrium). Further, we quantify the result that the systems exhibit chaotic behaviour: we show that the total variation distance between the joint law of a fixed set of queue lengths and the corresponding product law is essentially of order at most n−

Topics: QA Mathematics
Publisher: Centre for Discrete and Applicable Mathematics, London School of Economics and Political Science
Year: 2006
OAI identifier: oai:eprints.lse.ac.uk:13801
Provided by: LSE Research Online
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