Queue length and waiting time analysis of a batch arrival queue with bilevel control

We consider an M^X/M/1 queue with bilevel control. As soon as the system empties, the server becomes idle until m or more customers accumulate, at which time the server starts a set-up. After the set-up, if N or more customers are in the system (N≥m), the server immediately starts serving the customers in first come first served fashion. If less than N customers are in the system after the set-up, the server remains dormant in the system and waits until N or more customers accumulate. We derive the distributions of the number of customers in the system and the waiting time.