Stationary joint distribution of a discrete-time group-arrival and batch-size-dependent service queue with single and multiple vacation

Discrete-time queueing systems have widespread applications in packet switching networks, internet protocol, Broadband Integrated Services Digital Network (B-ISDN), circuit switched time-division multiple access etc. In this paper, we analyze an infinite-buffer discrete-time group-arrival queue with single and multiple vacation policy where processing/transmission/service times dynamically vary with batch-size under service. The bivariate probability generating function of queue length and server content distribution has been derived first, and from that we extract the joint probabilities in terms of roots of the characteristic equation. We also find the joint distribution at arbitrary and pre-arrival slot. Discrete phase type distribution plays a noteworthy role in order to regulate the high transmission error through a particular channel. In view of this, numerical illustrations include the service time distribution to follow the discrete phase type distribution.