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)3591710.22363/2658-4670-2023-31-3-205-217LACMZUArticlesСтатьиResearch ArticleAsymptotic diffusion analysis of the retrial queuing system with feedback and batch Poisson arrivalАсимптотически диффузионный анализ RQ-системы с обратными связями и неординарным входящим потокомhttps://orcid.org/0000-0002-5097-5629NazarovAnatoly A.НазаровА. А.

Doctor of Technical Sciences, Professor of Department of Probability Theory and Mathematical Statistics, Institute of Applied Mathematics and Computer Science

nazarov.tsu@gmail.comhttps://orcid.org/0000-0002-8888-9291RozhkovaSvetlana V.РожковаС. В.

Doctor of Physical and Mathematical Sciences, Professor of Department of Mathematics and Computer Science, School of Core Engineering Education, National Research Tomsk Polytechnic University, professor of Department of Probability Theory and Mathematical Statistics, Institute of Applied Mathematics and Computer Science, National Research Tomsk State University

rozhkova@tpu.ruhttps://orcid.org/0000-0002-0478-8232TitarenkoEkaterina Yu.ТитаренкоЕ. Ю.

Lecturer of Mathematics and Computer Science, School of Core Engineering Education

teu@tpu.ruNational Research Tomsk State UniversityТомский государственный университетNational Research Tomsk Polytechnic UniversityТомский политехнический университет12092023313VOL 31, NO3 (2023)ТОМ 31, №3 (2023)20521712092023Copyright ©; 2023, Nazarov A.A., Rozhkova S.V., Titarenko E.Y.Copyright ©; 2023, Назаров А.А., Рожкова С.В., Титаренко Е.Ю.2023Nazarov A.A., Rozhkova S.V., Titarenko E.Y.Назаров А.А., Рожкова С.В., Титаренко Е.Ю.https://creativecommons.org/licenses/by-nc/4.0https://journals.rudn.ru/miph/article/view/35917

The mathematical model of the retrial queuing system \(M^{[n]}/M/1\) with feedback and batch Poisson arrival is constructed. Customers arrive in groups. If the server is free, one of the arriving customers starts his service, the rest join the orbit. The retrial and service times are exponentially distributed. The customer whose service is completed leaves the system, or reservice, or goes to the orbit. The method of asymptotic diffusion analysis is proposed for finding the probability distribution of the number of customers in orbit. The asymptotic condition is growing average waiting time in orbit. The accuracy of the diffusion approximation is obtained.

В работе исследована \(M^{[n]}/M/1\) RQ-система с неординарным пуассоновским входящим потоком. Заявки на вход системы поступают группами. В каждый момент времени обслуживается не более одной заявки, остальные попадают на орбиту. Заявка, обслуживание которой завершено, либо покидает систему, либо повторно поступает на обслуживание, либо переходит на орбиту. Методом асимптотически диффузионного анализа при асимптотическом условии растущего среднего времени ожидания на орбите построена аппроксимация распределения вероятностей числа заявок на орбите. Показано, что точность диффузионной аппроксимации превышает точность гауссовской аппроксимации.

retrial queuing systembatch arrivalfeedbackasymptotic diffusion analysisсистема массового обслуживанияRQ-системанеординарный потокобратная связьасимптотически-диффузионный анализT. Phung-Duc, Retrial queueing models: A survey on theory and applications, 2019. arXiv: 1906.09560.J. Kim and B. Kim, “A survey of retrial queueing systems,” Annals of Operations Research, vol. 247, no. 1, pp. 3-36, 2016. DOI: 10.1007/s10479-015-2038-7.Y. Barlas and O. Özgün, “Queuing systems in a feedback environment: Continuous versus discrete-event simulation,” Journal of Simulation, vol. 12, no. 2, pp. 144-161, 2018. DOI: 10.1080/17477778.2018.1465153.A. Melikov, V. Divya, and S. Aliyeva, “Analyses of feedback queue with positive server setup time and impatient calls,” in Information technologies and mathematical modelling (ITTM-2020), Proceedings of the 19th International Conference named after A.F. Terpugov (2020 December, 2-5), Tomsk: Scientific Technology Publishing House, 2021, pp. 77-81.N. Singla and H. Kaur, “A two-state retrial queueing model with feedback and balking,” Reliability: Theory & Applications, vol. 16, no. SI 2 (64), pp. 142-155, 2021. DOI: 10.24412/1932-2321-2021-264-142-155.A. A. Nazarov, S. V. Rozhkova, and E. Y. Titarenko, “Asymptotic analysis of RQ-system with feedback and batch Poisson arrival under the condition of increasing average waiting time in orbit,” Communications in Computer and Information Science, vol. 1337, pp. 327-339, 2020. DOI: 10.1007/978-3-030-66242-4_26.A. A. Moiseev, A. A. Nazarov, and S. V. Paul, “Asymptotic diffusion analysis of multi-server retrial queue with hyper-exponential service,” Mathematics, vol. 8, no. 4, 2020. DOI: 10.3390/math8040531.