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

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We consider the ¿(i)/GI/1 queue, in which the arrival times of a fixed population of n customers are sampled independently from an identical distribution. This model recently emerged as the canonical model for so-called transitory queues that are non-stationary, time-varying and might operate only over finite time. The model assumes a finite population of customers entering the queue only once. This paper presents a method for analyzing heavy-traffic behavior by using uniform acceleration, which simultaneously lets the population 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 {\it 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, by suitably rescaling space and time, the queue length process converges to a Brownian motion on a parabola, a stochastic-process limit that captures the effect of a diminishing population by a negative quadratic drift. The stochastic-process limit provides insight into the macroscopic behavior (for n large) of the transitory queueing process, and the different phenomena occurring at different space-time scales.
Original languageEnglish
Number of pages43
Publication statusPublished - 2014

Publication series

Volume1412.5329 [math.PR]


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