Asymptotic distributions and chaos for the supermarket model

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 --> infinity. 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).