A discrete MAP/PH/1 queue with vacations and exhaustive time-limited service

The paper considers a single-server queue in which the server attends to a queue only for a limited periods of time and then goes on to other queues, in which case it says it proceeds on a vacation. The paper obtains results for the probability densities of the number of customers in the system at an arbitrary time, during vacation, and during service; the customer waiting time distribution; and the amount of work in the system at polling instants and at the start of a vacation. The results are extended to the case of multiple vacations during a cycle. The paper shows that the length of a visit to a time-limited service system has a phase type distribution. The work is carried out in discrete time.