Analysis of a discrete-time queue with a gate

This paper considers a discrete-time queue with a gate. The system has a single server and a gate. Customers arrive to the system according to a batch Bernoulli process and wait for the next gate opening in front of the gate. When the gate opens, all the customers who are waiting at the gate move into the inside of the gate. The server serves the customers only inside the gate. The gate closes immediately after all the customers outside the gate move into the inside. We derive the probability generating functions for the waiting time and the queue length under the assumption that the intervals between successive openings of the gate are i.i.d. and bounded.