Stochastic comparison bounds for an M1, M2/G1, G2/1 retrial queue with two way communication

Abstract

The main goal in the present paper is to provide a technique that considers the stochastic comparison approach for investigating monotonicity and comparability of an $M_{1},M_{2}/G_{1},G_{2}/1$ retrial queues with two way communication. This approach is developed for comparing a non Markov process to Markov process with many possible stochastic orderings. Particularly, we show the monotonicity of the transition operator of the embedded Markov chain relative to the strong stochastic ordering and convex ordering, as well as the comparability of two transition operators. Bounds are also obtained for the stationary distribution of the number of customers at departure epochs. Additionally, the performance measures of the system considered can be estimated by those of an $M_1,M_2/M_1,M_2/1$ retrial queue with two way communication when the service time distribution is NBUE (respectively NWUE). Finally, we validate stochastic comparison results by presenting a numerical example illustrating the interest of the approach.