Tail behavior of the queue size and waiting time in a queue with discrete autoregressive arrivals

Abstract

Autoregressive arrival models are described by a few parameters and provide a simple means to obtain analytical models for matching the first- and second-order statistics of measured data. We consider a discrete-time queueing system where the service time of a customer occupies one slot and the arrival process is governed by a discrete autoregressive process of order 1 (a DAR(1) process) which is characterized by an arbitrary stationary batch size distribution and a correlation coefficient. The tail behaviors of the queue length and the waiting time distributions are examined. In particular, it is shown that, unlike in the classical queueing models with Markovian arrival processes, the correlation in the DAR(1) model results in nongeometric tail behavior of the queue length (and the waiting time) if the stationary distribution of the DAR(1) process has infinite support. A complete characterization of the geometric tail behavior of the queue length (and the waiting time) is presented, showing the impact of the stationary distribution and the correlation coefficient when the stationary distribution of the DAR(1) process has finite support. It is also shown that the stationary distribution of the queue length (and the waiting time) has a tail of regular variation with index -β − 1, by finding an explicit expression for the tail asymptotics when the stationary distribution of the DAR(1) process has a tail of regular variation with index -β.

Mean queue size in a queue with discrete autoregressive arrivals of order p

“…For a discrete time multiserver queue with DAR(1) inputs, Choi et al (2004) obtained the stationary distributions of the queue size and the waiting time using the matrix analytic method. For a discrete time single server queue with DAR(1) inputs, Kim and Sohraby (2006) investigated the tail behaviors of the queue size and the waiting time distributions.…”

Regularly varying tails in a queue with discrete autoregressive arrivals of order p

“…In the current paper, we generalize the result in [15] on the tail behavior of the queue size and the waiting time distributions to higher order DAR source models. Specifically, for the DAR(p)/D/1 queue, we show that if the stationary distribution of DAR(p) input has a tail of regular variation with index −β − 1, then the stationary distributions of the queue size and the waiting time have tails of regular variation with index −β.…”

“…Specifically, for the DAR(p)/D/1 queue, we show that if the stationary distribution of DAR(p) input has a tail of regular variation with index −β − 1, then the stationary distributions of the queue size and the waiting time have tails of regular variation with index −β. The method for the proof of the result on the DAR(p)/D/1 queue in this paper is totally different from that on the DAR(1)/D/1 queue in the previous work [15]. The DAR(1)/D/1 queue has an explicit expression for the probability generating function of the stationary queue size, which enables us to express the distribution of the stationary queue size in terms of an infinite summation.…”

“…For a discrete time multiserver queue with DAR(1) inputs, Choi et al [3] obtained the stationary distributions of the queue size and the waiting time using the matrix analytic method. For a discrete time single server queue with DAR(1) inputs, Kim and Sohraby [15] showed that the stationary distributions of the queue size and the waiting time have asymptotically geometric tails if and only if the stationary distribution of the DAR(1) input has a finite support. They also showed that, in the same model, the stationary distributions of the queue size and the waiting time have regularly varying tails if the stationary distribution of the DAR(1) input has a regularly varying tail.…”

“…The DAR(1)/D/1 queue has an explicit expression for the probability generating function of the stationary queue size, which enables us to express the distribution of the stationary queue size in terms of an infinite summation. The expression for the distribution of the stationary queue size is a clue for the proof in the work [15]. For the DAR(p)/D/1 queue, however, it is very difficult to obtain tractable expressions for the distributions or the probability generating functions of the stationary queue size and stationary waiting time.…”