Asymptotic diffusion analysis of the retrial queuing system with feedback and batch Poisson arrival

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Abstract

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.

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Introduction There are situations in practice where an arriving customer that sees the server being occupied temporarily leaves the system or goes to orbit. In some random time customer retries to occupy a server again. These situations are modeled as retrial queuing systems. In addition, there are queuing systems in which a customer that has already received service requires a second service. It depends on the quality of the received service or external factors. Classical examples are communication networks in which erroneously transmitted data is retransmitted. The functioning of such systems is described by retrial queuing systems with feedback. There are many reviews on the study of queuing systems with repeated calls, for example [1, 2]. Models with feedback, instantaneous and delayed, have also been intensively studied in the last two decades [3-5]. At the same time, classical methods do not allow us to evaluate the characteristics of such systems. The application of asymptotic analysis methods makes it possible to obtain the asymptotic characteristics of the system under various limiting conditions. For example, in [6], a stationary probability distribution of the number of customers in orbit was obtained under conditions of a large delay of customers in orbit. To perform more detailed and accurate analysis of the model a method of asymptotic diffusion analysis is applied [7]. In this paper, we study retrial queuing systems with single server, batch Poisson arrival process, instantaneous and delayed feedback. The retrial and service times are exponentially distributed. A diffusion approximation of the probability distribution of the number of customers in orbit is constructed. It is shown that the accuracy of the diffusion approximation is higher then the accuracy of Gaussian approximation obtained in [6]. 2. System description We consider the queuing system
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About the authors

Anatoly A. Nazarov

National Research Tomsk State University

Email: nazarov.tsu@gmail.com
ORCID iD: 0000-0002-5097-5629

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

36 Lenin Avenue, Tomsk, 634050, Russian Federation

Svetlana V. Rozhkova

National Research Tomsk State University; National Research Tomsk Polytechnic University

Email: rozhkova@tpu.ru
ORCID iD: 0000-0002-8888-9291

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

36 Lenin Avenue, Tomsk, 634050, Russian Federation; 30 Lenin Avenue, Tomsk, 634050, Russian Federation

Ekaterina Yu. Titarenko

National Research Tomsk Polytechnic University

Author for correspondence.
Email: teu@tpu.ru
ORCID iD: 0000-0002-0478-8232

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

30 Lenin Avenue, Tomsk, 634050, Russian Federation

References

  1. T. Phung-Duc, Retrial queueing models: A survey on theory and applications, 2019. arXiv: 1906.09560.
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. 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.
  7. 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.

Copyright (c) 2023 Nazarov A.A., Rozhkova S.V., Titarenko E.Y.

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