2014/12/17 by Bet, Gianmarco, van der Hofstad, Remco, van Leeuwaarden, Johan S. H.
#60K25 #90B22 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1412.5329
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 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.