Study on Infinite Buffer Batch Size Dependent Bulk Service Queue with Queue Length Dependent Vacation

Queueing models with vacations have drawn the attention of researchers over several decades as a handy tool for tackling real-life congestion problems. Keeping this in mind, we pay attention to an infinite buffer single server batch-size dependent batch service queue with queue size (queue length) dependent vacation. The arrival pattern of the customers in the system follows the Poisson process where they get the service in packets/group following the general batch service (GBS) rule. An embedded Markov chain technique is used for the mathematical analysis where service (vacation) completion epochs have been taken as an embedded Markov point. We obtain the bivariate generating functions of the queue size and vacation type (queue size at vacation initiation epoch) at vacation termination epoch, and the bivariate generating function of the queue size and batch size with the server at service completion epoch, and then we successfully extract the steady-state joint probabilities of the queue size and batch size with the server and the joint probabilities of the queue size and vacation type at various epochs. Finally, various performance measures are presented. Also, the behavior of the considered model is presented by the graphs and tables.