A batch arrival Bernoulli vacation queue with a random setup time under restricted admissibility policy

We consider a batch arrival queue with a Bernoulli vacation schedule, where, after completion of a service, the server either goes for a vacation of random length with probability p (0<p<1) or may continue to serve the next unit, if any, with probability (1−p), under a Restricted Admissibility (RA) policy of arriving batches with a random Setup Time (SET). Unlike the usual batch arrival queueing system, the RA-policy differs during a busy period and a vacation period and hence all arriving batches are not allowed to join the system at all times. We derive the steady state queue size distributions at a random point of time as well as at a departure epoch. Also, we obtain some important performance measures of this model. Further, we demonstrate the existence of stochastic decomposition result for this type of model.