MEAN WAITING-TIMES IN A GENERAL FEEDBACK QUEUE WITH PRIORITIES

A general single server time-sharing model with multiple queues and job classes, priorities and feedback is considered which includes head-of-the-line, preemptive, and round robin scheduling. In the model, there is a finite or countably infinite number K of queues and job classes. Jobs of class k arrive at queue k according to a Poisson process, and have a general service time distribution, i.e., disregarding the service discipline, this model is of type M/G/1. Jobs are served in passes, receiving a complete quantum of service on every pass, or their remaining service demand, if this is less. If a job completes its service demand during a pass, it leaves the system. Otherwise it is fed back. Jobs are allowed m(l) passes in queue l, with quantum delta-lm in the mth pass, m = 1,..., m(l). After their m(l)th pass in queue l, they join the tail of queue l + 1. If K is finite, either m(K) or delta-K,m(K) is infinite. Jobs in queue k' have priority over jobs in queue l', when k' < l'. We consider the preemptive and nonpreemptive disciplines, and in both cases derive a system of linear equations in the mean waiting times of the job passes for all classes and queues, which we show to have a unique solution.