Steady state analysis of a single server bulk queue with general vacation times and restricted admissibility of arriving batches

We analyze the steady state behavior of a single server vacation queue with variable batch size arrivals in a compound Poisson Process, exponential service in batch of fixed size b following a min(b,n) rule and general server vacations. However, an arriving batch may or may not be allowed to join the system at all the times. We obtain explicit steady state results for the probability generating functions for the number of customers in the system, the average number of customers and the average waiting time in the queue and the system. Some special cases of interest have been derived and finally a numerical example is provided.