On finite-buffer batch-size-dependent bulk service queue with queue-length dependent vacation

Abstract

ABSTRACT This paper considers a single server finite-buffer batch-size-dependent bulk service queue with queue-length dependent vacation. Customers arrive at the system according to the Poisson process and are served in batches of maximum size ‘b’ with a minimum threshold value ‘a’ following the ‘general bulk service’ rule. The service time distribution is assumed to be of general type which modulates depending on serving batch size. The server is allowed to take vacation, either single or multiple vacations, whenever permissible numbers of customers are not found in the queue at the beginning of the service. The vacation time distribution is assumed to be of general type and dynamically changes depending on the queue content at vacation initiation epoch. We use the supplementary variable and the embedded Markov chain techniques to obtain the steady-state joint distribution of the queue content service batch size, queue content and type of vacation taken by the server at various epochs. Several numerical results are presented at the end to bring out the qualitative aspect of the model, which reveals the fact that queue-length dependent vacation further reduces congestion in the batch-size-dependent bulk service queues.