Queueing system with a constant retrial rate, non-reliable server and threshold-based recovery

In this paper, we examine a Markovian queueing system which is composed of a retrial queue with constant retrial rate and a non-reliable server. Upon arrival a customer occupies the server if it is idle, otherwise it goes to the retrial queue. The customer at the head of the retrial queue is allowed to retry for service. When the server is busy, the server is subject to breakdowns or failures that occur according to a Poisson process. In the failed state the server can be repaired at a repair facility with exponential repair time with respect to the threshold policy: the repair starts when the number of customers in the system reaches some prespecified threshold level q [greater-or-equal, slanted] 1. We perform a steady-state analysis of the corresponding continuous-time Markov chain, derive mean performance characteristics and waiting time distribution as well as calculate optimal threshold level to minimize the long-run average losses for the given cost structure.