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

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 ∆ (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. arXiv:1412.5329v2 [math.PR] 30 Nov 2015

Rare events of transitory queues

“…The large deviation principle derived for the 'uniform scattering' case in this paper provides the first result on the rare event behavior of the RS/GI/1 transitory queue, building on the fluid and diffusion approximation results established in [17,15,14,2].…”

“…The current paper complements this by not assuming the near balanced condition. In recent work, [2] established diffusion approximations to the queue length process of the ∆ (i) /GI/1 queue under a uniform acceleration scaling regime, where in it is assumed that the "initial load" near time zero satisfies ρ n = 1 + βn −1/3 . This, of course, contrasts with the population acceleration regime considered in this paper, where the offered load is accelerated by the population size at all time instances in the horizon [0, 1].…”

Finite-pool queueing with heavy-tailed services

“…Then the head start can be interpreted as the persons that signed up (already) as a customer. In [3] we showed that in our heavy-traffic regime, once the head start is of order n 1/3 , and you would decide to start your business, the number of customers you will serve consecutively is of the order n 2/3 . With this mental picture, our heavy-traffic regime gives insight into dimensioning rules about how large the pool n should be in comparison to the head start, how to choose the service capacity as a function of the pool size to achieve full system utilization, and how to control the first busy period, which essentially is the relevant time of operation of the system.…”

“…[11,18], where the focus lies on overloaded regimes and see [3] for a detailed overview. In [3] we introduced a new heavy-traffic regime defined by two features: The customer pool grows to infinity and the initial (at time zero) rate of newly arriving customers is such that, on average, one new customer is expected to arrive during one service time. This gives rise to a large-scale system that (initially) operates close to full utilization, and is expected to utilize its resources efficiently.…”

“…In [3] we have studied this heavy-traffic regime under the assumption that the second moment of the service time distribution is finite, and showed that the queue length process converges to a reflected Brownian motion with negative quadratic drift. The negative drift captures the effect of a pool of potential customers that diminishes with time: After some time, the activity in the queue inevitably becomes negligible.…”