Analysis of stationary queue-length distributions of the BMAP/R(a,b)/1 queue

This paper investigates an infinite waiting space single-server queueing system in which customers arrive according to a batch Markovian arrival process. The server serves the customers in batches of maximum size ‘b’ with a minimum threshold value ‘a’. The service time of each batch follows R-type distribution, where R-type distribution represents a class of distributions whose Laplace-Stieltjes transform is rational or approximated rational function, which is independent of service batch size and the arrival process. We determine the queue-length distribution at departure epoch in terms of roots of the associated characteristic equation of the vector probability generating function. To show the strength and advantage of roots method, we also provide a comprehensive analysis of the queue-length distribution at departure epoch using the matrix-analytic method (MAM). To determine the queue-length distribution at random epoch, we obtain a relation between the queue-length distributions at departure and random epochs using the supplementary variable technique with remaining service time of a batch in service as the supplementary variable. We also derive the queue-length distributions at pre-arrival epoch of an arrived batch and post-arrival epoch of a random customer of an arrived batch. Some numerical results are demonstrated for different service-time distributions to show the key performance measures of the system.