An Infinite Dimensional Model for A Single Server Priority Queue

TL;DR

This paper introduces a novel infinite-dimensional model for a single server priority queue with continuously assigned priority levels, analyzing its steady state behavior and customer wait times.

Contribution

It develops a measure-valued stochastic process model for priority queues with continuous priorities and derives key performance formulas.

Findings

Derived the average distribution of customer priority levels.

Provided formulas for expected sojourn and waiting times based on priority.

Verified theoretical results through simulation.

Abstract

We consider a Markovian single server queue in which customers are preemptively scheduled by exogenously assigned priority levels. The novelty in our model is that the priority levels are randomly assigned from a continuous probability measure rather than a discrete one. Because the priority levels are drawn from a continuum, the queue is modeled by a measure-valued stochastic process. We analyze the steady state behavior of this process and provide several results. We derive a measure that describes the average distribution of customer priority levels in the system; we provide a formula for the expected sojourn time of a customer as a function of his priority level; and we provide a formula for the expected waiting time of a customer as a function of his priority level. We interpret these quantitative results and give a qualitative understanding of how the priority levels affect individual customers as well as how they affect the system as a whole. The theoretical analysis is verified by simulation. We also discuss some directions of future work.