Batch arrival queue with N-policy and single vacation

We consider an Mx/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 Mx/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.