A Stochastic Fluid Model Approach to the Stationary Distribution of the Maximum Priority Process

TL;DR

This paper introduces a novel stochastic fluid model approach to explicitly determine the stationary distribution of the maximum priority process in queueing systems, addressing limitations of traditional fixed-priority models.

Contribution

It develops a mapping to tandem fluid queues to derive explicit stationary distribution expressions for maximum priority processes.

Findings

Explicit stationary distribution formulas derived.

Applicable to performance analysis of priority queueing systems.

Addresses gaps in understanding maximum priority process behavior.

Abstract

In traditional priority queues, we assume that every customer upon arrival has a fixed, class-dependent priority, and that a customer may not commence service if a customer with a higher priority is present in the queue. However, in situations where a performance target in terms of the tails of the class-dependent waiting time distributions has to be met, such models of priority queueing may not be satisfactory. In fact, there could be situations where high priority classes easily meet their performance target for the maximum waiting time, while lower classes do not. Here, we are interested in the stationary distribution at the times of commencement of service of this maximum priority process. Until now, there has been no explicit expression for this distribution. We construct a mapping of the maximum priority process to a tandem fluid queue, which enables us to find expressions for this stationary distribution. We derive the results for the stationary distribution of the maximum priority process at the times of the commencement of service.