Discrete-time Geo [ X] / GH/1 retrial queue with Bernoulli feedback

Abstract

We analyse a discrete-time Geo [ X] / G H /1 retrial queue where each call after service either immediately returns to the orbit for another service with probability θ or leaves the system forever with probability θ = 1 − θ, where 0 ≤ θ < 1. A call that is going to be served decides to receive a service of type h ( h = 1, …, H) with probability q h , where Σ h=1 H q h = 1. We study the Markov chain underlying the considered queueing system and the ergodicity condition too. We find the generating function of the number of calls in the orbit and in the system. We derive a stochastic decomposition law and as an application we give bounds for the proximity between the steady-state distributions for our queueing system and its corresponding standard system. In the special case of individual arrivals, we develop recursive formulae for calculating the steady-state distribution of the orbit size. Besides, we prove that the M [ X] / G H /1 retrial queue with Bernoulli feedback can be approximated by our corresponding discrete-time system. Finally, we give numerical examples to illustrate the effect of the parameters on several performance characteristics.