Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia2517910.22363/2658-4670-2020-28-4-319-326Research ArticleStochastic analysis of a single server unreliable queue with balking and general retrial timeBoualemMohamed<p>Full Professor, Professor of Applied Mathematics at the Department of Technology</p>robertt15dz@yahoo.frUniversity of Bejaia1512202028431932609122020Copyright © 2020, Boualem M.2020<p>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.</p>Retrial queueMarkov chainbalkingbreakdowns and repairsstochastic ordersboundsageing classesочередь с обновлениемцепь Марковаблокировкавыход из строя и восстановлениестохастический порядокграницыклассы старения[J. R. Artalejo and A. Gómez-Corral, Retrial queueing system: A computational approach. Berlin: Springer, Berlin, Heidelberg, 2008, 318 pp.][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.][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.][M. Shaked and J. G. Shanthikumar, Stochastic Orders. New York: Springer-Verlag, 2007, 473 pp.][D. Stoyan, Comparison methods for queues and other stochastic models. New York: Wiley, 1983, 217 pp.][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.][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.][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.][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.][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.][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.][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.][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.][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.][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.][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.]