On Queues with Priority Determined by Feedback

Abstract

Customers arrive to a two-priority queueing system according to a marked Poisson process. Both waiting rooms have infinite capacity. Customers are served one at a time according to FIFO discipline on priority basis: those in waiting line 1([Formula: see text]) are given priority over the ones in line 2([Formula: see text]). The service time is class-dependent phase type. After completion of service, high priority ([Formula: see text]) customers may feed back for service according to a Bernoulli process. Feed back customers are sent to the low priority ([Formula: see text]) queue. When at a service completion epoch of a [Formula: see text] customer, if there is none left behind in [Formula: see text] line, then the server goes to serve [Formula: see text] class. For the two-priority queueing system, we assume that [Formula: see text] customers are not allowed an additional feed back. Both preemptive and non-preemptive service disciplines are analysed. Waiting time distribution of both type of customers are derived. As a special case, the situation where there is no external entry to the [Formula: see text] line is discussed.