The aim of the investigation is a closed retrial queueing system with a finite source. The server can be reached from the source (primary request) or from the orbit (secondary request). If an incoming (primary or secondary) job finds the server busy, two modes are distinguished: the job is transferred to the orbit (no collision) or the job under service is interrupted and both of them are transferred to the orbit (collision). Requests in the orbit can retry reaching the server after a random waiting time. The nonreliable case when the server is subject to breakdown is also investigated. In case of breakdown, when the server is under repair, also two cases can be investigated. For the first, primary calls from the source can reach the system, and they will be sent to the orbit. For the second, the source is blocked, so primary customers are not able to step into the system. This paper focuses on the unreliable system with collision and blocking of parameters. These types of systems can be solved by numerical, asymptotical, and simulation methods. Our goal is to provide a new approach to the algorithmic solution for calculating the steady-state probabilities of the system. Using these quantities the main performance characteristics (utilization of the server, response time, etc.) can be calculated. Examples illustrate the effect of different parameters on the distribution of requests in the system.
A.A. Ali and S. Wei, “Modeling of coupled collision and congestion in finite source wireless access systems,” in: Wireless Communications and Networking Conference (WCNC), (2015), pp. 1113–1118.
B. Almási, J. Roszik, and J. Sztrik, “Homogeneous finite-source retrial queues with server subject to breakdowns and repairs,” Math. Comput. Model., 42, No. 5–6, 673–682 (2005).
J.R. Artalejo and A. Gómez-Corral, Retrial Queueing Systems: A Computational Approach, Springer, Berlin (2008).
S. Balsamo, G.L. Dei Rossi, and A. Marin, “Modelling retrial-upon-conflict systems with productform stochastic petri nets,” in: International Conference on Analytical and Stochastic Modeling Techniques and Applications, Springer (2013), pp. 52–66.
T. Bérczes, A. Tóth, A. Nazarov, and J. Sztrik, “Performance modeling of finite-source retrial queueing systems with collisions and nonreliable server using MOSEL,” Commun. Comput. Inform. Sci., 700, 248–258 (2017).
V.I. Dragieva, “Number of retrials in a finite source retrial queue with unreliable server,” Asia-Pac. J. Oper. Res., 31, No. 2 (2014).
N. Gharbi and C. Dutheillet, “An algorithmic approach for analysis of finite-source retrial systems with unreliable servers,” Comput. Math. Appl., 62, No. 6, 2535–2546 (2011).
J.S. Kim, “Retrial queueing system with collision and impatience,” Commun. Kor. Math. Soc., 25, No. 4, 647–653 (2010).
J. Kim and B. Kim, “A survey of retrial queueing systems,” Ann. Op. Res., 247, No. 1, 3–36 (2016).
A. Kvach and A. Nazarov, “Sojourn time analysis of finite source markov retrial queuing system with collision,” Commun. Comput. Inform. Sci., 564, 64–72 (2015).
T. V. Lyubina and A. A. Nazarov, “Research of the non-Markov dynamic retrial queue system with collision,” Her. Kemerovo State Univ., 1, No. 49, 38–44 (2012).
A. Nazarov, A. Kvach, and V. Yampolsky, “Asymptotic analysis of closed Markov retrial queuing system with collision,” Commun. Comput. Inform. Sci., 487, 334–341 (2014).
A. Nazarov, J. Sztrik, A. Kvach, and T. Bérczes, “Asymptotic analysis of finite-source M/M/1 retrial queueing system with collisions and server subject to breakdowns and repairs,” Ann. Op. Res., https://doi.org/10.1007/s10479-018-2894-z (2018)
A. Nazarov, J. Sztrik, and A. Kvach, “A survey of recent results in finite-source retrial queues with collisions ” Commun. Comput. Inform. Sci., 912, 1–15 (2018).
Y. Peng, Z. Liu, and J. Wu, “An M/G/1 retrial G-queue with preemptive resume priority and collisions subject to the server breakdowns and delayed repairs,” J. Appl. Math. Comput., 44, No. 1–2, 187–213 (2014).
J. Wang, L. Zhao, and F. Zhang, “Analysis of the finite source retrial queues with server breakdowns and repairs,” J. Ind. Manag. Opt., 7, No. 3, 655–676 (2011).
F. Zhang and J. Wang, “Performance analysis of the retrial queues with finite number of sources and service interruptions,” J. Kor. Stat. Soc., 42, No. 1, 117–131 (2013).
Proceedings of the XXXV International Seminar on Stability Problems for Stochastic Models, Perm, Russia, September 24–28, 2018. Part II.
Rights and permissions
About this article
Cite this article
Kuki, A., Sztrik, J., Bérczes, T. et al. Numerical Analysis of Non-Reliable Retrial Queueing Systems with Collision and Blocking of Customers. J Math Sci 248, 1–13 (2020). https://doi.org/10.1007/s10958-020-04850-w