Heavy-Traffic Analysis Through Uniform Acceleration of Queues with Diminishing Populations

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 Δ(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 a negative quadratic drift, a stochastic-process limit that captures the effect of the diminishing population. When the arrival times are generally distributed, our techniques provide information on the typical queue length and the first busy period.