Stochastic analysis of a single server unreliable queue with balking and general retrial time

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In this investigation, we consider an M/G/1 queue with general retrial times allowing balking and server subject to breakdowns and repairs. In addition, the customer whose service is interrupted can stay at the server waiting for repair or leave and return while the server is being repaired. The server is not allowed to begin service on other customers until the current customer has completed service, even if current customer is temporarily absent. This model has a potential application in various fields, such as in the cognitive radio network and the manufacturing systems, etc. The methodology is strongly based on the general theory of stochastic orders. Particularly, we derive insensitive bounds for the stationary distribution of the embedded Markov chain of the considered system.

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1. Introduction The study on queueing models have become an indispensable area due to its wide applicability in real life situations. Retrial queues occupy an intermediate situation between an Erlang model with loss and classical model with wait, which constitute their limiting models in the case of low and high retrial rates. Retrial queueing systems are characterized by the requirement that customers finding the service area busy must join the retrial group and retry for service at random intervals. Queues in which customers are allowed to retry have been extensively used to model many problems in telephone switching systems, cognitive radio network, manufacturing systems, telecommunication networks and computer systems for competing to gain service from a central processor unit [1]-[3]. Retrial queueing systems with general service times and non-exponential retrial time distributions have been received little attention because of the complexity of the known results. Indeed, in most cases, we are faced with systems of equations whose resolution is complex, or having solutions not © Boualem M., 2020 This work is licensed under a Creative Commons Attribution 4.0 International License easily interpretable. For instance, Pollaczek-Khintchine formula requires a numerical inversion of the Laplace transform to compute the distribution of the waiting time. In many cases, even the Laplace transform or probability generating functions are not available in explicit forms. To overcome these difficulties, approximation methods are often used to obtain quantitative and/or qualitative estimates for certain performance measures. For all these reasons, in this study, a particular interest is devoted to the stochastic comparison method based on the general theory of stochastic orders [4], [5]. The stochastic comparison method is a mathematical tool used to study the performance of some systems modeled by continuous or discrete time Markov chains. The general idea of this method is to bound a complex system with a new system that is simpler to solve providing qualitative bounds for these performance measures. These methods represent one of the main research activities in various scientific fields, such as economy, biology, operation research, reliability theory, decision theory, retrial queues and queueing networks [4]-[15]. In the present study, stochastic comparison analysis of an


About the authors

Mohamed Boualem

University of Bejaia

Author for correspondence.

Full Professor, Professor of Applied Mathematics at the Department of Technology



  1. J. R. Artalejo and A. Gómez-Corral, Retrial queueing system: A computational approach. Berlin: Springer, Berlin, Heidelberg, 2008, 318 pp.
  2. A. A. Nazarov, S. V. Paul, and O. D. Lizyura, “Two-way communication retrial queue with unreliable server and multiple types of outgoing calls,” Discrete and Continuous Models and Applied Computational Science, vol. 28, no. 1, pp. 49-61, 2020. doi: 10.22363/2658-4670-2020-28-1- 49-61.
  3. D. Zirem, M. Boualem, K. Adel-Aissanou, and D. Aıssani, “Analysis of a single server batch arrival unreliable queue with balking and general retrial time,” Quality Technology & Quantitative Management, vol. 16, pp. 672-695, 2019. doi: 10.1080/16843703.2018.1510359.
  4. M. Shaked and J. G. Shanthikumar, Stochastic Orders. New York: Springer-Verlag, 2007, 473 pp.
  5. D. Stoyan, Comparison methods for queues and other stochastic models. New York: Wiley, 1983, 217 pp.
  6. L. M. Alem, M. Boualem, and D. Aıssani, “Bounds of the stationary distribution in M/G/1 retrial queue with two-way communication and n types of outgoing calls,” Yugoslav Journal of Operations Research, vol. 29, no. 3, pp. 375-391, 2019. doi: 10.2298/YJOR180715012A.
  7. L. M. Alem, M. Boualem, and D. Aıssani, “Stochastic comparison bounds for an M1, M2/G1, G2/1 retrial queue with two way communication,” Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 4, pp. 1185- 1200, 2019. doi: 10.1572/HJMS.2018.629.
  8. M. Boualem, “Insensitive bounds for the stationary distribution of a single server retrial queue with server subject to active breakdowns,” Advances in Operations Research, vol. 2014, no. 1, pp. 1-12, 2014. doi: 10.1155/2014/985453.
  9. M. Boualem, A. Bareche, and M. Cherfaoui, “Approximate controllability of stochastic bounds of stationary distribution of an M/G/1 queue with repeated attempts and two-phase service,” International Journal of Management Science and Engineering Management, vol. 14, no. 2, pp. 79-85, 2019. doi: 10.1080/17509653.2018.1488634.
  10. M. Boualem, M. Cherfaoui, and D. Aıssani, “Monotonicity properties for a single server queue with classical retrial policy and service interruptions,” Proceedings of the Jangjeon Mathematical Society, vol. 19, no. 2, pp. 225-236, 2016.
  11. M. Boualem, M. Cherfaoui, N. Djellab, and D. Aıssani, “A stochastic version analysis of an M/G/1 retrial queue with Bernoulli schedule,” Bulletin of the Iranian Mathematical Society, vol. 43, no. 5, pp. 1377- 1397, 2017.
  12. M. Boualem, M. Cherfaoui, N. Djellab, and D. Aıssani, “Inégalités stochastiques pour le modèle d’attente M/G/1/1 avec rappels,” French, Afrika Matematika, vol. 28, pp. 851-868, 2017. doi: 10.1007/s13370017-0492-x.
  13. M. Boualem, N. Djellab, and D. Aıssani, “Stochastic inequalities for M/G/1 retrial queues with vacations and constant retrial policy,” Mathematical and Computer Modelling, vol. 50, no. 1-2, pp. 207-212, 2009. doi: 10.1016/j.mcm.2009.03.009.
  14. M. Boualem, N. Djellab, and D. Aıssani, “Stochastic approximations and monotonicity of a single server feedback retrial queue,” Mathematical Problems in Engineering, vol. 2012, 12 pages, 2012. doi: 10.1155/2012/ 536982.
  15. M. Boualem, N. Djellab, and D. Aıssani, “Stochastic bounds for a single server queue with general retrial times,” Bulletin of the Iranian Mathematical Society, vol. 40, no. 1, pp. 183-198, 2014.
  16. X. Wu, P. Brill, M. Hlynka, and J. Wang, “An M/G/1 retrial queue with balking and retrials during service,” International Journal of Operational Research, vol. 1, pp. 30-51, 2005. doi: 10.1504/IJOR.2005.007432.

Copyright (c) 2020 Boualem M.

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