Computational Analysis of a Single Server Queue with Batch Markovian Arrival and Exponential Single Working Vacation

Abstract

In this paper, an infinite buffer queue with a single server and non-renewal batch arrival is studied. The service discipline is considered as exhaustive type under single exponential working vacation policy. Further, both the service times during the working vacation and normal busy period are assumed to be generally distributed random variables. It is also assumed that the service times and the arrival process are independent of each others. Moreover, it is accepted that at the end of an exponentially distributed working vacation, the first customer in the front of the queue is likely to receive service rate as per normal busy period service rate irrespective of received service in the working vacation period as the server shifts from working vacation mode to normal period mode. The system-length distributions at different epochs, such as post-departure and arbitrary epoch are obtained. The RG-factorization technique is applied to obtain the distribution of the system length at post-departure epoch. Henceforth, the system-length distribution at arbitrary epoch is determined by supplementary variable technique along with some simple algebraic manipulations. Some useful performance measures to be particular the mean system length of the model and the mean waiting time of an arbitrary customer in the system is discussed in the numerical section. Finally, some numerical results are presented for the model, in the form of the table and graphs. A possible application of the model in communication network is outlined in the paper.