Stability of Join the Shortest Queue Networks

TL;DR

This paper proves the stability of join the shortest queue networks under general conditions, provides bounds on equilibrium distributions, and compares workloads with random assignment networks using Lyapunov functions.

Contribution

It extends stability results to all non-idling disciplines and general distributions, and introduces new bounds and comparisons using Lyapunov methods.

Findings

Networks are stable when subcritical.

Uniform tail bounds on equilibrium distributions.

Workloads can be significantly larger than in random assignment networks.

Abstract

Join the shortest queue (JSQ) refers to networks whose incoming jobs are assigned to the shortest queue from among a randomly chosen subset of the queues in the system. After completion of service at the queue, a job leaves the network. We show that, for all non- idling service disciplines and for general interarrival and service time distributions, such networks are stable when they are subcritical. We then obtain uniform bounds on the tails of the marginal distributions of the equilibria for families of such networks; these bounds are employed to show relative compactness of the marginal distributions. We also present a family of subcritical JSQ networks whose workloads in equilibrium are much larger than for the corresponding networks where each incoming job is assigned randomly to a queue. Part of this work generalizes results in Foss and Chernova [12], which applied fluid limits to study networks with the FIFO discipline. Here, we apply an appropriate Lyapunov function.