Single server queues with a batch Markovian arrival process and bulk renewal or non-renewal service

Abstract

We first consider an infinite-buffer single server queue where arrivals occur according to a batch Markovian arrival process(BMAP). The server serves customers in batches of maximum size �b� with a minimum threshold size �a�. The service time of each batch follows general distribution independent of each other as well as the arrival process. The proposed analysis is based on the use of matrix-analytic procedure to obtain queue-length distribution at a post-departure epoch. Next we obtain queue-length distributions at various other epochs such as, pre-arrival, arbitrary and pre-service using relations with post-departure epoch. Later we also obtain the system-length distributions at post-departure and arbitrary epochs using queue-length distribution at post-departure epoch. Some important performance measures, like mean queue-lengths and mean waiting times have been obtained. Total expected cost function per unit time is also derived to determine the locally optimal values of a and b. Secondly, we perform similar analysis for the corresponding infinite-buffer single server queue where arrivals occur according to a BMAP and service process in this case follows a non-renewal one, namely, Markovian service process (MSP). � 2015, Systems Engineering Society of China and Springer-Verlag Berlin Heidelberg.