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.

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

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  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.

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