Analytical and computational studies of the BMAP/G(a,Y)/1 queue

This article studies a single server queueing system in which customers arrive according to a batch Markovian arrival process and customers are served in batches of random capacity ‘Y’ with a minimum threshold value ‘a’. The service time of each batch is arbitrarily distributed and independent of service batch size. We first construct the vector difference equations by observing two consecutive departure epochs and then convert them into the vector probability generating function. The queue-length distribution at departure epoch is extracted in terms of roots of the associated characteristic equation. A comparative study is also carried out to compare the roots method with that of the matrix-analytic method. To obtain the queue-length distribution at random epoch, we establish a relationship between the queue-length distributions at departure and random epochs with the help of Markov renewal theory. We obtain the queue-length distribution at prearrival epoch of an arriving batch and the queue-length distribution at post-arrival epoch of a random customer of an arrived batch. Further, we derive the results of some particular well-known queueing models from our model. The results are illustrated by numerical examples for different service-time distributions to show the variation of the key performance measures.