Asymptotic distributions and chaos for the supermarket model
Malwina J Luczak, London School of Economics
Colin McDiarmid, University of Oxford
Abstract
In the supermarket model there are n queues, each with a unit
rate server. Customers arrive in a Poisson process at rate λn,
where 0<λ<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.
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