Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)4525310.22363/2658-4670-2025-33-2-144-156BMHYOYComputer ScienceИнформатика и вычислительная техникаResearch ArticleAsymptotic analysis of multiserver retrial queueing system with \(\pi\)-defeat of negative arrivals under heavy loadАсимптотический анализ многолинейной RQ-системы с \(\pi\)-поражением в условии большой загрузкиhttps://orcid.org/0000-0002-8708-124X58304893200MTF-1866-2025MeloshnikovaNatalya P.МелошниковаН. П.

PhD-student, Junior researcher of Laboratory of queueing theory and teletraffic theory

meloshnikovana@gmail.comhttps://orcid.org/0000-0001-8933-532256439120600E-3161-2017FedorovaEkaterina A.ФёдороваЕ. А.

PhD in Physical and Mathematical Sciences, Associate Professor of Department of Probability Theory and Mathematical Statistic

ekat_fedorova@mail.ruNational Research Tomsk State UniversityНациональный исследовательский Томский государственный университет15072025332VOL 33, NO2 (2025)ТОМ 33, №2 (2025)14415625072025Copyright ©; 2025, Meloshnikova N.P., Fedorova E.A.Copyright ©; 2025, Мелошникова Н.П., Фёдорова Е.А.2025Meloshnikova N.P., Fedorova E.A.Мелошникова Н.П., Фёдорова Е.А.https://creativecommons.org/licenses/by-nc/4.0https://journals.rudn.ru/miph/article/view/45253

The paper studies a multiserver retrial queuing system with \(\pi\)-defeat as a mathematical model of cloud services. The arrival processes of “positive” calls are Poisson. The system has a finite number of servers and the service time for calls at the servers is exponentially distributed. When all servers are busy, calls entering the system transfer to an orbit, where they experience a random delay. After the delay, calls from the orbit attempt to access the service unit according to a multiple access policy. The system also receives a stream of negative calls. Negative calls do not require the service. An negative call “deletes” a random number of calls is the service unit. For the considered model, the Kolmogorov equations are written in the steady state. The method of asymptotic analysis under a heavy load condition is applied for deriving the stationary probability distribution of the number of calls in the orbit. The results of the numerical analysis are presented.

В работе исследуется многолинейная RQ-система с \(\pi\)-поражением как математическая модель облачных сервисов. На вход системы поступает простейший поток «положительных» заявок. В системе конечное число обслуживающих приборов, время обслуживания заявок на приборах распределено по экспоненциальному закону. Когда все приборы заняты, заявки поступающие в систему переходят на орбиту, где осуществляют случайную задержку. После осуществления задержки, заявки с орбиты обращаются к блоку обслуживания согласно политике множественного доступа. Также в систему поступает поток так называемых «отрицательных» заявок. Отрицательная заявка не нуждается в обслуживании: при поступлении она удаляет случайное число обслуживаемых заявок. Для рассматриваемой модели записаны уравнения Колмогорова в стационарном режиме. Предлагается метод асимптотического анализа в условии большой загрузки для нахождения стационарного распределения вероятностей числа заявок на орбите. Представлены результаты численного анализа.

mathematical modellingretrial queueing systemnegative callsasymptotic analysisheavy loadматематическое моделированиесистема массового обслуживания с повторными вызовамиотрицательные заявкиасимптотический анализбольшая загрузкаKavis, M. J. Architecting the Cloud: Design Decisions for Cloud Computing Service Models 224 pp. (Wiley; 1st edition, 2014).Boccardi, F., Heath, R. W., Lozano, A., Marzetta, T. L. & Popovski, P. Five disruptive technology directions for 5G. IEEE Communications Magazine 52, 74-80 (May 2014).Ferrus, R., Sallent, O., Pérez-Romero, J. & Agustí, R. Management of Network Slicing in 5G Radio Access Networks. Functional Framework and Information Models (2018).Wu, J., Wang, M., Chan, Y. C., Wong, E. W. M. & Kim, T. Performance Evaluation of 5G mmWave Networks with Physical-Layer and Capacity-Limited Blockingm. IEEE 21st International Conference on High Performance Switching and Routing (HPSR), 1-6 (May 2020).Artalejo, J. R. & Gomez-Corral, A. Retrial Queueing Systems 318 pp. (Springer Berlin, 2008).Falin, G. & Templeton, J. Retrial Queues 320 pp. (Taylor & Francis, 1997).Phung-Duc, T. Retrial Queueing Models: A Survey on Theory and Applications. Stochastic Operations Research in Business and Industry, 1-26 (May 2017).Gelenbe, E. Product-form queueing networks with negative and positive customers. Journal of Applied Probability 7, 656-663 (1991).Gelenbe, E., Glynn, P. & Sigman, K. Queues with Negative Arrivals. J. Appl. Probab 28, 245-250 (1991).Gelenbe, E. G-networks with signals and batch removal. Probability in the Engineering and Informational Sciences Vоl. 7, 335-343 (1993).Do, T. Bibliography on G-networks, negative customers and applications. Mathematical and Computer Modelling, 205-212 (2011).Caglayan, M. G-networks and their applications to machine learning, energy packet networks and routing: introduction to the special issue. Probability in the Engineering and Informational Sciences 31, 381-395 (2017).Bocharov, P. & Vishnevsky, V. G-networks: development of the theory of multiplicative networks. Russian. Automation and Telemechanics Vol. 5, 70-74 (2003).Shin, Y. Multi-server retrial queue with negative customers and disasters. Queueing Syst, 223-337 (2007).Pechinkin, A. & Razumchik, R. A Method for Calculating Stationary Queue Distribution in a Queuing System with Flows of Ordinary and Negative Claims and a Bunker for Superseded Claims. J. Commun. Technol. Electron. 57, 882-891 (2012).Matalytski, M. & Naumenko, V. Analysis of the queueing network with a random bounded waiting time of positive and negative customers at a non-stationary regime. Journal of Applied Mathematics and Computational Mechanics 16, 97-108 (2017).Liu, T., Hsu, H. & Chang, F. Multi-Server Two-Way Communication Retrial Queue Subject to Disaster and Synchronous Working Vacation. Algorithms 18, 24 (May 2025).Melikov, A., Poladova, L., Edayapurath, S. & Sztrik, J. Single-Server Queuing-Inventory Systems with Negative Customers and Catastrophes in the Warehouse. Mathematics 11 (2023).Lisovskaya, E., Fedorova, E., Salimzyanov, R. & Moiseeva, S. Resource Retrial Queue with Two Orbits and Negative Customers. Mathematics 10, 321 (May 2022).Meloshnikova, N. & Fedorova, E. Asymptotic Analysis of a multiserver RQ-system with disasters in a service block under heavy load Russian. in Information and telecommunication technologies and mathematical modeling of high-tech systems: Proceedings of the All-Russian conference with international participation (RUDN, Moscow, Russian Federation, April 7-11, 2025), (in print).Meloshnikova, N. Study Retrial Queue M|M|N with negative calls under heavy load Russian. in Proceedings of Tomsk State University. Physics and Mathematics Series: Mathematical and Software Support of Information, Technical, and Economic Systems: Materials of the IX International Scientific Conference. 307 (Tomsk State University Publishing House, Tomsk, Russian Federation, May 26-28, 2021), 165-168.