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Due to the widespread applicability of discrete-time queues in wirelessnetworks or telecommunication systems, this paper analyzes an infinite-bufferbatch-service queue with single and multiple vacation where customers/messagesarrive according to the Bernoulii process and service time varies with thebatch-size. The foremost focal point of this analysis is to get the completejoint distribution of queue length and server content at service completionepoch, for which first the bivariate probability generating function has beenderived. We also acquire the joint distribution at arbitrary slot. We alsoprovide several marginal distributions and performance measures for theutilization of the vendor. Transmission of data through a particular channel isskipped due to the high transmission error. As the discrete phase typedistribution plays a noteworthy role to control this error, we includenumerical example where service time distribution follows discrete phase typedistribution. A comparison between batch-size dependent and independent servicehas been drawn through the graphical representation of some performancemeasures and total system cost.

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