Heavy-traffic analysis through uniform acceleration of queues with diminishing populations

TL;DR

This paper analyzes the heavy-traffic behavior of a finite-population queue using uniform acceleration, revealing a convergence to a Brownian motion with parabolic drift that models diminishing customer arrivals over time.

Contribution

It introduces a novel method for heavy-traffic analysis of queues with diminishing populations by employing uniform acceleration and stochastic process limits.

Findings

Queue length converges to Brownian motion with parabolic drift under exponential arrivals.

Provides insights into queue behavior and busy periods for general arrival distributions.

Models the impact of diminishing populations on queue dynamics over time.

Abstract

We consider a single server queue that serves a finite population of $n$ customers that will enter the queue (require service) only once, also known as the $\Delta_{(i)}/G/1$ queue. This paper presents a method for analyzing heavy-traffic behavior by using uniform acceleration, which simultaneously lets $n$ and the service rate grow large, while the initial resource utilization approaches one. A key feature of the model is that, as time progresses, more customers have joined the queue, and fewer customers can potentially join. This diminishing population gives rise to a class of reflected stochastic processes that vanish over time, and hence do not have a stationary distribution. We establish that, when the arrival times are exponentially distributed, by suitably rescaling space and time, the queue length process converges to a Brownian motion with parabolic drift, a stochastic-process limit that captures the effect of a diminishing population by a negative quadratic drift. When the arrival times are generally distributed, our techniques provide information on the typical queue length and the first busy period.