On steady-state joint distribution of an infinite buffer batch service Poisson queue with single and multiple vacation

Abstract

This article considers a single server, infinite buffer, bulk service Poisson queue with single and multiple vacation. The customers are served in batches following ‘general bulk service’ (GBS) rule. The customers are arriving according to the Poisson process, and the service time of the batches follows an exponential distribution. Using bivariate probability generating function (PGF) method the steady-state joint distributions of the queue content and server content (when server is busy), and joint distribution of the queue content and type of the vacation taken by the server (when server is in vacation) have been obtained. Here by the ‘type of the vacation’ we mean the queue length at vacation initiation epoch. The information about these joint distributions may help in increasing the system performance. Finally, several numerical examples are carried out using MAPLE software to verify the analytical results.