The M/G/1 queueing system with vacations and timer-controlled service

The author study a M/G/1 queueing system with vacations, in which the sojourn time of the server at the queue is controlled by a timer. They consider three policies for setting the timer value: time-limited (TL), cycle-time-limited (CL) and cycle-time-limited with accumulated lateness (CLL). When the server arrives at the queue, the timer is set to a certain value depending on the policy employed. The server serves the waiting customers until either the system becomes empty, or the timer expires, whichever occurs first. In the latter case, the server completes the service on the customer being served, then takes a vacation. Customers are not served during the vacation. If the system is empty upon the server´s return from a vacation, another vacation begins immediately. Such a queueing system finds applications in the performance analysis of local area network protocols such as the IEEE 802.4 (token bus) and the Fiber Distributed Data Interface (FDDI), in which the vacation would correspond to the time the token is away at other stations. Using the matrix-analytic approach, they compute the probability distribution of the number of customers in the system at an arbitrary point in time, and the distribution of the time the server spends at the queue before he takes a vacation