A unified approach for large queue asymptotics in a heterogeneous multiserver queue

TL;DR

This paper develops a unified martingale-based approach to analyze large queue asymptotics in heterogeneous multiserver queues, covering tail behavior and heavy traffic limits under different variance conditions.

Contribution

It introduces a novel martingale method that unifies tail asymptotics and heavy traffic approximations for heterogeneous multiserver queues.

Findings

Unified framework for tail and heavy traffic asymptotics

Applicable to queues with bounded or large variances

Provides new insights into queue behavior under stability conditions

Abstract

We are interested in a large queue in a $GI/G/k$ queue with heterogeneous servers. For this, we consider tail asymptotics and weak limit approximations for the stationary distribution of its queue length process in continuous time under a stability condition. Here, two weak limit approximations are considered. One is when the variances of the inter-arrival and/or service times are bounded, and the other is when they get large. Both require a heavy traffic condition. Tail asymptotics and heavy traffic approximations have been separately studied in the literature. We develop a unified approach based on a martingale produced by a good test function for a Markov process to answer both problems.