  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 vacation where customers/messages
arrive according to the Bernoulii 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 service completion
epoch, for which first the bivariate probability generating function has been
derived. We also acquire the joint distribution at arbitrary slot. We also
provide several marginal distributions and performance measures for the
utilization of the vendor. Transmission of data through a particular channel is
skipped due to the high transmission error. As the discrete phase type
distribution plays a noteworthy role to control this error, we include
numerical example where service time distribution follows discrete phase type
distribution. A comparison between batch-size dependent and independent service
has been drawn through the graphical representation of some performance
measures and total system cost.
