Critically loaded k-limited polling systems

TL;DR

This paper analyzes a two-queue polling system with k-limited service and switch-over times, focusing on heavy-traffic behavior and providing rigorous limits for queue-length distributions using singular-perturbation techniques.

Contribution

It introduces a rigorous heavy-traffic analysis for k-limited polling systems, including joint queue-length limits and moment interchange properties, applicable to multi-queue systems.

Findings

Heavy-traffic limits for joint queue-length distributions are rigorously established.

An interchange property among moments of service and switch-over times is identified.

Results extend to systems with more than two queues and relaxed distributional assumptions.

Abstract

We consider a two-queue polling model with switch-over times and $k$-limited service (serve at most $k_i$ customers during one visit period to queue $i$) in each queue. The major benefit of the $k$-limited service discipline is that it - besides bounding the cycle time - effectuates prioritization by assigning different service limits to different queues. System performance is studied in the heavy-traffic regime, in which one of the queues becomes critically loaded with the other queue remaining stable. By using a singular-perturbation technique, we rigorously prove heavy-traffic limits for the joint queue-length distribution. Moreover, it is observed that an interchange exists among the first two moments in service and switch-over times such that the HT limits remain unchanged. Not only do the rigorously proven results readily carry over to $N$($\geq2$) queue polling systems, but one can also easily relax the distributional assumptions. The results and insights of this note prove their worth in the performance analysis of Wireless Personal Area Networks (WPAN) and mobile networks, where different users compete for access to the shared scarce resources.