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 \(\Delta_{(i)}/G/1\) queue. This paper presents a method for analyzing heavy-traffic behavior by using uniform acceleration, which simultaneou...

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Bibliographic Details
Published in:arXiv.org 2015-11
Main Authors: Bet, Gianmarco, Remco van der Hofstad, Johan S H van Leeuwaarden
Format: Article
Language:English
Subjects:
Acceleration
Brownian motion
Customers
Drift
Markov analysis
Queues
Queuing theory
Rescaling
Stochastic processes
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Summary: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.
ISSN:2331-8422