On the Distribution of the Number of Customers in the D-BMAP/Ga,b/1/M Queue—A Simple Approach to a Complex Problem

Abstract

In this paper, we consider a single-server finite-capacity queue with general bulk service rule where customers arrive according to a discrete-time batch Markovian arrival process (D-BMAP). The model is denoted by D-BMAP/Ga,b/1/M which includes a wide class of queueing models as special cases. We give a relation between the steady-state probabilities of the number of customers in the queue at departure- and arbitrary-epochs using the concept of the mean sojourn time in the phase of the underlying Markov chain of D-BMAP before the next arrival. We use the embedded Markov chain technique to obtain the departure-epoch probability of the number of customers in the queue. The pre-arrival probability of the number of customers in the queue is also obtained. Finally, a complete solution to the distribution of the number of customers in the D-BMAP/Ga,b/1/M queue, some computational results, and performance measures such as loss probability and mean queue length are presented.