Discrete time analysis of MAP/PH/1 vacation queue with gated time-limited service

We analyse a single-server queue in which the server goes through alternating periods of vacation and work. In each work period, the server attends to the queue for no more than a fixed length of time, T. The system is a gated one in which the server, during any visit, does not attend to customers which were not in the system before its visit. As soon as all the customers within the gate have been served or the time limit has been reached (whichever occurs first) the server goes on a vacation. The server does not wait in the queue if the system is empty at its arrival for a visit. For this system the resulting Markov chain, of the queue length and some auxiliary variables, is level-dependent. We use special techniques to carry out the steady state analysis of the system and show that when the information regarding the number of customers in the gate is not critical we are able to reduce this problem to a level-independent Markov chain problem with large number of boundary states. For this modified system we use a hybrid method which combines matrix-geometric method for the level-independent part of the system with special solution method for the large complex boundary which is level-dependent.