Joint queue-lengths and sojourn-time distributions in a general preemptive feedback queue

A general single-server preemptive queueing model with multiple queues, priorities and feedback is considered, which includes the ordinary preemptive policy and Bernoulli feedback. In the model, there are K queues, jobs arrive according to a Poisson process at every queue, and the service time in every queue is exponential. Upon completion of a job in queue k, ml jobs are fed back to queue l for l = 1, 2,…, K with a probability depending on k, ml,…, mk. Jobs in queue k have preemptive priority over jobs in queue l, when k < l. We derive the generating functions of the joint queue-lengths probabilities in stationary state at departure instants, at joint departure and preemption instants, at output instants, and at random instants, along with the Laplace-Stieltjes transform of the total sojourn time of a linear sequence of jobs, for any sequence of their priorities. As a special case, we deal with the two-queue system.