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Bounds of the stationary distribution in M/G/1 retrial queue with two-way communication and n types of outgoing calls

Alem Lala Maghnia (University of Bejaia, Research Unit LaMOS (Modeling and Optimization of Systems), Bejaia, Algeria + University of Bouira, Department of Mathematics, Bouira, Algeria)
Boualem Mohamed (University of Bejaia, Faculty of Technology, Research Unit LaMOS (Modeling and Optimization of Systems), Bejaia, Algeria)
Aissani Djamil (University of Bejaia, Faculty of Exact Sciences, Research Unit LaMOS (Modeling and Optimization of Systems), Bejaia, Algeria)

In this article we analyze the M=G=1 retrial queue with two-way communication and n types of outgoing calls from a stochastic comparison viewpoint. The main idea is that given a complex Markov chain that cannot be analyzed numerically, we propose to bound it by a new Markov chain, which is easier to solve by using a stochastic comparison approach. Particularly, we study the monotonicity of the transition operator of the embedded Markov chain relative to the stochastic and convex orderings. Bounds are also obtained for the stationary distribution of the embedded Markov chain at departure epochs. Additionally, the performance measures of the considered system can be estimated by those of an M=M=1 retrial queue with two-way communication and n types of outgoing calls when the service time distribution is NBUE (respectively, NWUE). Finally, we test numerically the accuracy of the proposed bounds.

Keywords: Retrial Queues, Outgoing Calls, Markov Chain, Stochastic Comparison