Centre for Discrete and Applicable Mathematics 

CDAM Research Report, LSECDAM200612October 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 $n^{1}$.A PDF file (265 kB) with the full contents of this report can be downloaded by clicking here.
Alternatively, if you would like to get a free hard copy of this report, please send the number of this report, LSECDAM200612, together with your name and postal address to:
CDAM Research Reports Series Centre for Discrete and Applicable Mathematics London School of Economics Houghton Street London WC2A 2AE, U.K. 

Phone: +44(0)207955 7494. Fax: +44(0)207955 6877. Email: info@maths.lse.ac.uk 
Introduction to the CDAM Research Report Series.  
CDAM Homepage. 