An Infinite Dimensional Model for a Many Server Priority Queue

TL;DR

This paper models a many-server queue with continuous priority levels using an infinite dimensional stochastic process, deriving key performance measures and verifying results with simulations.

Contribution

It introduces an infinite dimensional model for a multi-server priority queue with continuous priorities, deriving distributional and time-based performance formulas.

Findings

Derived the Radon-Nikodym derivative of customer priority distribution

Formulated expected sojourn time as a function of priority

Provided expected waiting time formulas based on priority

Abstract

We consider a Markovian many server queueing system in which customers are preemptively scheduled according to exogenously assigned priority levels. The priority levels are randomly assigned from a continuous probability measure rather than a discrete one and hence, the queue is modeled by an infinite dimensional stochastic process. We analyze the equilibrium behavior of the system and provide several results. We derive the Radon-Nikodym derivative (with respect to Lebesgue measure) of the 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 verify our theoretical analysis with discrete-event simulations. We discuss how each of our results generalizes previous work on infinite dimensional models for single server priority queues.