# A stochastic version analysis of an M/G/1 retrial queue with Bernoulli‎ ‎schedule‎

Document Type : Research Paper

Authors

1 Research Unit LaMOS (Modeling and Optimization of Systems)‎, ‎Faculty of Technology‎, ‎University of Bejaia‎, ‎06000 Bejaia‎, ‎Algeria.

2 Department of Mathematics‎, ‎University of Biskra‎, ‎Biskra 07000‎, ‎Algeria‎ ‎Research Unit LaMOS (Modeling and Optimization of Systems)‎, ‎University of Bejaia‎, ‎06000 Bejaia‎, ‎Algeria.

3 Laboratory LaPS‎, ‎Department of athematics‎, ‎University of Badji Mokhtar‎, ‎Annaba 23000‎, ‎Algeria.

4 Research Unit LaMOS (Modeling and Optimization of Systems)‎, ‎Faculty of Exact Sciences‎, ‎University of Bejaia‎, ‎06000 Bejaia‎, ‎Algeria.

Abstract

‎In this work‎, ‎we derive insensitive bounds for various performance measures of a single-server‎ ‎retrial queue with generally distributed inter-retrial times and Bernoulli schedule‎, ‎under the special‎ ‎assumption that only the customer at the head of the orbit queue (i.e.‎, ‎a‎ ‎FCFS discipline governing the flow from the orbit to the server) is allowed‎ ‎to occupy the server‎. ‎The methodology is strongly based on stochastic comparison techniques‎. ‎Instead of studying a performance measure in a quantitative fashion‎, ‎this approach attempts to reveal the relationship between the performance measures and the parameters of the system‎. ‎We prove the monotonicity of the transition operator of the embedded Markov chain relative to strong stochastic ordering and increasing convex ordering‎. ‎We obtain comparability conditions for the distribution of the number of customers in the system‎. ‎Bounds are derived for the stationary distribution and‎ ‎some simple bounds for the mean characteristics of the system‎. ‎The proofs of these results are based on the validation of some inequalities for some cumulative probabilities associated with every state $(m‎, ‎n)$ of the system‎. ‎Finally‎, ‎the effects of various parameters on the performance of the system have been examined numerically.

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Main Subjects

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### History

• Receive Date: 02 November 2015
• Revise Date: 22 June 2016
• Accept Date: 22 June 2016