Modelling extreme uncertainty: Queues with Pareto inter-arrival times and Pareto service times

Abstract

When an operational parameter presents extremely high variability, uncertainty becomes extreme. Long-tail probability distributions can be used to model such uncertainty. We present a queuing system in which extreme uncertainty is modelled using long-tail probability distributions. There have been many queuing analyses for a single server queue fed by an M/G/traffic process, in which G is a Pareto distribution, that focus on certain limiting conditions. In this paper, we present a mathematical model to solve an infinite queuing system with one server where the inter-arrival time between jobs follows a Pareto probability distribution with shape parameter α and a scale parameter A. The system service time is also a Pareto probability distribution with shape parameter β and scale parameter B. We call this the P/P/1 queuing model.