M/G/1 Priority Queues

We examine an M/G/1 queue with several queueing disciplines. After reviewing some background material in probability theory, we consider a first-come first-served queue. Next, we examine an absolute priority queue, where high priority customers are always selected for service over lower priority customers, regardless of how long the latter have been waiting. We also review a method for calculating the expected waiting times for a variety of queueing disciplines, particularly an accumulating priority queue, where a customers priority is proportional to the time they have spent waiting. We then consider a newer method for determining the full waiting time distribution of an accumulating priority queue. Each of these distributions are given as functional relations for their Laplace-Stieltjes transforms, which in generally cannot be solved analytically. Thus, we conclude by demonstrating how to numerically invert these transforms, showing the waiting time distribution of an accumulating priority queue.