Explicit Results for the Distributions of Queue Lengths for a Non-Preemptive Two-Level Priority Queue

TL;DR

This paper derives explicit, exact formulas for the joint and marginal queue-length distributions in a non-preemptive two-level priority M/M/c queue, using two different mathematical approaches.

Contribution

It provides novel closed-form analytical expressions and recurrence relations for queue-length distributions in a two-priority queue, not previously available.

Findings

Exact queue-length distributions derived

Two computational approaches validated

High-accuracy numerical evaluation demonstrated

Abstract

Explicit results are derived using simple and exact methods for the joint and marginal queue-length distributions for the M/M/c queue with two non-preemptive priority levels. Equal service rates are assumed. Two approaches are considered. One is based on numerically robust quadratic recurrence relations. The other is based on a complex contour-integral representation that yields exact closed-form analytical expressions, not hitherto available in the literature, that can also be evaluated numerically with very high accuracy.