The Remaining Busy Time in a Retrial System with Unreliable Servers

Abstract

In this paper, we consider a multiserver retrial queuing system with unreliable servers class-dependent retrial rates and N classes of customers following Poisson input processes. We analyze the distribution of the stationary generalized remaining service time which includes all unavailable periods (setup times) occurring during service of the customer. During service of a class-i customer, the interruptions occur according to the i-dependent Poisson process and the following i-dependent random setup time of the server. We consider two following disciplines caused by the service interruptions: preemptive repeat different and preemptive resume. Using coupling method and regenerative approach, we derive the stationary distribution of the generalized remaining service time in an arbitrary server. For each class i, this distribution is expressed as a convolution of the corresponding original service times and setup times, and in general is available in the terms of the Laplace-Stieltjes transform allowing to calculate the moments of the target distribution. Some numerical examples are included as well.