Optimal scheduling of critically loaded multiclass GI/M/n+M queues in an alternating renewal environment

TL;DR

This paper investigates optimal control of multiclass GI/M/n+M queues in an alternating renewal environment, establishing diffusion limits and proving asymptotic optimality of scheduling policies in heavy traffic.

Contribution

It introduces a novel diffusion approximation for multiclass queues in renewal environments and proves the asymptotic optimality of certain scheduling policies.

Findings

Diffusion limits are controlled jump diffusions driven by a compound Poisson process.

Asymptotic optimality of infinite-horizon discounted and ergodic problems.

Long-run average moment bounds established via Foster-Lyapunov equations.

Abstract

In this paper, we study optimal control problems for multiclass GI/M/n+M queues in an alternating renewal (up-down) random environment in the Halfin-Whitt regime. Assuming that the downtimes are asymptotically negligible and only the service processes are affected, we show that the limits of the diffusion-scaled state processes under non-anticipative, preemptive, work-conserving scheduling policies, are controlled jump diffusions driven by a compound Poisson jump process. We establish the asymptotic optimality of the infinite-horizon discounted and long-run average (ergodic) problems for the queueing dynamics. Since the process counting the number of customers in each class is not Markov, the usual martingale arguments for convergence of mean empirical measures cannot be applied. We surmount this obstacle by demonstrating the convergence of the generators of an augmented Markovian model which incorporates the age processes of the renewal interarrival times and downtimes. We also establish long-run average moment bounds of the diffusion-scaled queueing processes under some (modified) priority scheduling policies. This is accomplished via Foster-Lyapunov equations for the augmented Markovian model.