Markov-modulated M/G/1 type queue in heavy traffic and its application to time-sharing disciplines

TL;DR

This paper analyzes a Markov-modulated M/G/1 queue in heavy traffic, deriving workload distributions and demonstrating state-space collapse under discriminatory processor sharing, with implications for queue performance analysis.

Contribution

It provides new heavy-traffic limit results for workload and queue length distributions in modulated queues, including the first derivation of workload mean and exponential distribution under scaling.

Findings

Workload mean derived for general service requirements.

Workload is exponentially distributed under heavy traffic.

Queue lengths exhibit state-space collapse under DPS.

Abstract

This paper deals with a single-server queue with modulated arrivals, service requirements and service capacity. In our first result, we derive the mean of the total workload assuming generally distributed service requirements and any service discipline which does not depend on the modulating environment. We then show that the workload is exponentially distributed under heavy-traffic scaling. In our second result, we focus on the discriminatory processor sharing (DPS) discipline. Assuming exponential, class-dependent service requirements, we show that the joint distribution of the queue lengths of different customer classes under DPS undergoes a state-space collapse when subject to heavy-traffic scaling. That is, the limiting distribution of the queue length vector is shown to be exponential, times a deterministic vector. The distribution of the scaled workload, as derived for general service disciplines, is a key quantity in the proof of the state-space collapse.