Finite-pool queues with heavy-tailed services

TL;DR

This paper analyzes a queueing model with heavy-tailed service times, showing that the scaled queue length converges to an lpha-stable process, highlighting differences from light-tailed cases.

Contribution

It establishes the convergence of the scaled queue length process to an lpha-stable process in heavy-tailed service scenarios, extending queueing theory.

Findings

Queue length converges to an lpha-stable process with negative quadratic drift.

Characterizes the initial headstart needed for prolonged activity.

Contrasts heavy-tailed results with light-tailed cases, which converge to Brownian motion.

Abstract

We consider the $\Delta_{(i)}/G/1$ queue, in which a a total of $n$ customers independently demand service after an exponential time. We focus on the case of heavy-tailed service times, and assume that the tail of the service time distribution decays like $x^{-\alpha}$, with $\alpha \in (1,2)$. We consider the asymptotic regime in which the population size grows to infinity and establish that the scaled queue length process converges to an \alpha-stable process with a negative quadratic drift. We leverage this asymptotic result to characterize the headstart that is needed to create a long period of activity. This result should be contrasted with the case of light-tailed service times, which was shown to have a similar scaling limit, but then with a Brownian motion instead of an \alpha-stable process.