Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-2006-12

October 2006

Asymptotic Distributions and Chaos for the Supermarket Model Maximal Width Learning of Binary Functions

Malwina J. Luczak and Colin McDiarmid

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 $d \geq 2$ 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 $n \to \infty$. 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 $n^{-1}$.

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