Batch arrival queue with N-policy and single vacation

Abstract We consider an M x / G /1 queueing system with N -policy and single vacation. As soon as the system becomes empty, the server leaves the system for a vacation of random length V . When he returns from the vacation, if the system size is greater than or equal to predetermined value N (threshold), he beings to serve the customers. If not, the server waits in the system until the system size reaches or exceeds N . We derive the system size distribution and show that the system size distribution decomposes into two random variables one of which is the system size of ordinary M x / G /1 queue. The interpretation of the other random variable will also be provided. We also derive the queue waiting time distribution of an arbitrary customer. Finally we develop a procedure to find the optimal stationary operating policy under a linear cost structure.