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( ${\mathcal{P}}_1$ ) are given priority over the ones in line 2( ${\mathcal{P}}_2$ ). The service time is class-dependent phase type. After completion of service, high priority ( ${\mathcal{P}}_1$ ) customers may feed back for service according to a Bernoulli process. Feed back customers are sent to the low priority ( ${\mathcal{P}}_2$ ) queue. When at a service completion epoch of a ${\mathcal{P}}_1$ customer, if there is none left behind in ${\mathcal{P}}_1$ line, then the server goes to serve ${\mathcal{P}}_2$ class. For the two-priority queueing system, we assume that ${\mathcal{P}}_2$ 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 ${\mathcal{P}}_2$ line is discussed.