Asymptotic distributions and chaos for the supermarket model

Malwina J. Luczak, Colin McDiarmid

Published 2007-12-13Version 1

In the supermarket model there are n queues, each with a unit rate server. Customers arrive in a Poisson process at rate \lambda n, where 0<\lambda <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 -> oo. 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^{-1}.