System delay analysis for a nonpreemptive time-limited service queueing system with two types of vacations and its application

We consider an M/G/1 queue with time-limited service and two types of vacations, in whichever single server serves customers until the timer expires or the queue becomes empty, whichever occurs first, and then takes type 1 or type 2 vacations. A type 1 vacation is taken at a timer expiration, while a type 2 vacation in the every case of system emptiness. The customer that had its service interrupted due to the timer expiration may be attended by the nonpreemptive discipline. Applying the analysis results to the time-limited service polling system, we derive an approximate probability density function for evaluating the system delay variance, and formulate an optimal assignment problem of the total time of limited service-periods.