A Retrial Queueing System with Alternating Inter-retrial Time Distribution

Abstract

We consider a single-server retrial queuing system with Markovian Arrival Process (MAP) and phase-type (PH) service time distribution. Customers which find the server busy enter the orbit of infinite size and try their luck after some random time. Concerning the retrial process, we suppose that inter-retrial times have PH distribution if the number of customers in the orbit does not exceed some threshold and have exponential distribution otherwise. Such an assumption allows to some extent take into account the realistic nature of retrial process and, at the same time, to avoid a large increase in the dimensionality of the state space of this process. We consider two different policies of repeated attempts and describe the operation of the system by two different multi-dimensional Markov chains: by quasi-Toeplitz Markov chain in the case of a constant retrial rate and by asymptotically quasi-Toeplitz Markov chain in the case of an infinitely increasing retrial rate. Both chains are successfully analyzed in this paper. We derive the ergodicity condition, calculate the stationary distribution and the main performance measures of the system.