Dynamics of the supermarket model

TL;DR

This paper analyzes the long-term behavior of the supermarket model's Markov chain, revealing cluster formations and convergence properties under local routing policies, and provides stability classification and explicit computations.

Contribution

It introduces a novel analysis of the supermarket model's Markov chain, showing cluster-based convergence and stability under join-the-least-weighted-queue policies, combining stochastic and network flow methods.

Findings

Components form disjoint clusters with converging speeds.

The model's stability can be classified under any JLW policy.

Explicit computation of clusters and speeds for specific policies.

Abstract

We consider the long term behaviour of a Markov chain \xi(t) on \Z^N based on the N station supermarket model. Different routing policies for the supermarket model give different Markov chains. We show that for a general class of local routing policies, "join the least weighted queue" (JLW), the N one-dimensional components \xi_i(t) can be partitioned into disjoint clusters C_k. Within each cluster C_k the "speed" of each component \xi_j converges to a constant V_k and under certain conditions \xi 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.