Dynamics of the non-homogeneous supermarket model.

Abstract

We consider the long term behavior of a Markov chain ξ(t) on ℤ N based on the N station supermarket model with general neighborhoods, arrival rates and service rates. Different routing policies for the model give different Markov chains. We show that for a broad class of local routing policies, join the least weighted queue (JLW), the N one-dimensional components ξ i (t) can be partitioned into disjoint clusters C k . Within each cluster C k the speed of each component ξ j converges to a constant V k and under certain conditions ξ is recurrent in shape on each cluster. To establish these results we have assembled methods from two distinct areas of mathematics, semi-martingale techniques used for showing stability of Markov chains together with the theory of optimal flows in networks. As corollaries to our main result we obtain the stability classification of the supermarket model under any JLW policy and can explicitly compute the C k and V k for any instance of the model and specific JLW policy.