A batch arrival queue with a random setup time under Bernoulli vacation schedule

We consider an Mx/G/1 queueing system with a random setup time under Bernoulli vacation schedule, where the service of the first unit at the completion of each busy period or a vacation period is preceded by a random setup time, on completion of which service starts. However, after each service completion, the server may take a vacation with probability p or remain in the system to provide next service, if any, with probability (1-p). This generalizes both the Mx/G/1 queueing system with a random setup time as well as the Bernoulli vacation model. We carryout an extensive analysis for the queue size distributions at various epochs. Further, attempts have been made to unify the results of related batch arrival vacation models.