Sojourn Times in a Multiclass Priority Queue with Random Feedback

We consider a priority-based multiclass queue with probabilistic feed-back. There are J service stations. Each customer belongs to one of the several priority classes, and the customers of each class arrive at each station in a Poisson process. A single server serves queued customers on a priority basis with a nonpreemptive scheduling discipline. The customers who complete their services feed back to the system instantaneously and join one of the queues of the stations or depart from the system according to a given probability. In this paper, we propose a new method to simplify the analysis of these queueing systems. By the analysis of busy periods and regenerative processes, we clarify the underlying system structure, and systematically obtain the mean for the sojourn time, i.e., the time from the arrival to the departure from the system, of a customer at every station. The mean for the number of customers queued in each station at an arbitrary time is also obtained simultaneously.