COMPUTATIONAL ANALYSIS OF STATIONARY WAITING-TIME DISTRIBUTIONS OF GI^X/R/1 AND GI^X/D/1 QUEUES

Abstract

In this article, we obtain, in a unified way, a closed-form analytic expression, in terms of roots of the so-called characteristic equation of the stationary waiting-time distribution for the GIX/R/1 queue, where R denotes the class of distributions whose Laplace–Stieltjes transforms are rational functions (ratios of a polynomial of degree at most n to a polynomial of degree n). The analysis is not restricted to generalized distributions with phases such as Coxian-n (Cn) but also covers nonphase-type distributions such as deterministic (D). In the latter case, we get approximate results. Numerical results are presented only for (1) the first two moments of waiting time and (2) the probability that waiting time is zero. It is expected that the results obtained from the present study should prove to be useful not only for practitioners but also for queuing theorists who would like to test the accuracies of inequalities, bounds, or approximations.

System-theoretical algorithmic solution to waiting times in semi-Markov queues

“…The case of cross-correlation between A k and B k leads to dependent queues which are outside the scope of the current paper; see [3] for a review of the early literature and [4,5] for more recent studies on this topic. The random variable W designates the steady-state waiting time for the queueing system described in (1) which is known to be the semi-Markov queue or the SM/SM/1 queue in short [6]. We note that the Lindley equation (1) perfectly describes the queue waiting time in a queueing system with renewal-type services using the first come first serve (FCFS) service discipline.…”

“…The random variable W designates the steady-state waiting time for the queueing system described in (1) which is known to be the semi-Markov queue or the SM/SM/1 queue in short [6]. We note that the Lindley equation (1) perfectly describes the queue waiting time in a queueing system with renewal-type services using the first come first serve (FCFS) service discipline. However, the situation is different when the service times are of the more general Markov renewal type in which case the Lindley recurrence may come short of accommodating different models based on how the service process behaves when the queue is empty [7].…”

“…However, the situation is different when the service times are of the more general Markov renewal type in which case the Lindley recurrence may come short of accommodating different models based on how the service process behaves when the queue is empty [7]. In the current paper, we focus on queues that are described solely by the Lindley equation (1) as in [6]. For the case of non-renewal service times, more general models, for instance the ones discussed in [7], with more general descriptions than that of the recurrence (1), are left for future research.…”

“…In the current paper, we focus on queues that are described solely by the Lindley equation (1) as in [6]. For the case of non-renewal service times, more general models, for instance the ones discussed in [7], with more general descriptions than that of the recurrence (1), are left for future research. Queueing systems with arrivals governed by a semi-Markov process have been studied extensively since an SM/M/1 queue was first analyzed in [8].…”

“…The goal of the current paper is to provide a computationally efficient and stable algorithmic method to compute the actual waiting time distribution for the semi-Markov queue described by the Lindley recurrence (1). This problem is the same as the one stated in [6].…”