On the joint distribution of an infinite-buffer discrete-time batch-size-dependent service queue with single and multiple vacations

Abstract

ABSTRACT Due to the widespread applicability of discrete-time queues in wireless networks or telecommunication systems, this paper analyzes an infinite-buffer batch-service queue with single and multiple vacations where customers/messages arrive according to the Bernoulli process and service time varies with the batch-size. The foremost focal point of this analysis is to get the complete joint distribution of queue length and server content at the service completion epoch, for which first the bivariate probability generating function has been derived. We also acquire the joint distribution at an arbitrary slot. We also provide several marginal distributions and performance measures for the utilization of the vendor. During the transmission of data through a particular communication channel, high transmission error may take place due to several factors. For this reason, one may skip the transmission through that particular channel. However the discrete phase-type distribution plays a noteworthy role to control this transmission error which eventually motivates us to include a numerical example where service time distribution follows discrete phase-type distribution. A comparison between batch-size-dependent and -independent services has been drawn through the graphical representation of some performance measures and total system cost.