Heavy-traffic analysis of k-limited polling systems

TL;DR

This paper analyzes a two-queue polling system with k-limited service under heavy traffic, deriving exact limits for queue lengths and revealing independence of queues in critical conditions.

Contribution

It provides the first heavy-traffic limit results for k-limited polling systems with exponential arrivals and services, using a singular-perturbation approach.

Findings

Queue length in the stable queue matches a vacation system distribution.

Critically loaded queue's scaled length is exponentially distributed.

Queues become independent in heavy traffic.

Abstract

In this paper we study a two-queue polling model with zero switch-over times and $k$-limited service (serve at most $k_i$ customers during one visit period to queue $i$, $i=1,2$) in each queue. The arrival processes at the two queues are Poisson, and the service times are exponentially distributed. By increasing the arrival intensities until one of the queues becomes critically loaded, we derive exact heavy-traffic limits for the joint queue-length distribution using a singular-perturbation technique. It turns out that the number of customers in the stable queue has the same distribution as the number of customers in a vacation system with Erlang-$k_2$ distributed vacations. The queue-length distribution of the critically loaded queue, after applying an appropriate scaling, is exponentially distributed. Finally, we show that the two queue-length processes are independent in heavy traffic.