Heavy-traffic Asymptotics of Priority Polling System with Threshold Service Policy

TL;DR

This paper analyzes the heavy-traffic behavior of a three-queue priority polling system with threshold policy, revealing exponential distribution of the critically loaded queue and tail asymptotics for stable queues, supported by simulation.

Contribution

It introduces a novel heavy-traffic analysis of a priority polling system with threshold policy using singular-perturbation techniques, and derives tail asymptotics for queue lengths.

Findings

Critically loaded queue length is exponentially distributed.

Stable queues share distribution with a priority polling system with N-policy vacation.

Tail asymptotics match classical M/M/1 preemptive priority queues.

Abstract

In this paper, by the singular-perturbation technique, we investigate the heavy-traffic behavior of a priority polling system consisting of three M/M/1 queues with threshold policy. It turns out that the scaled queue-length of the critically loaded queue is exponentially distributed, independent of that of the stable queues. In addition, the queue lengths of stable queues possess the same distributions as a priority polling system with N-policy vacation. Based on this fact, we provide the exact tail asymptotics of the vacation polling system to approximate the tail distribution of the queue lengths of the stable queues, which shows that it has the same prefactors and decay rates as the classical M/M/1 preemptive priority queues. Finally, a stochastic simulation is taken to test the results aforementioned.