Marginal asymptotic diffusion analysis of two-class retrial queueing system with probabilistic priority as a model of two-modal communication networks

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Abstract

In the paper, a retrial queueing system of \(M_2/M_2/1\) type with probabilistic priority and interruptions is considered as a model of a two-modal communication network. Two classes of customers come to the system according Poisson arrival processes. There is one service device (or channel). If a customer finds the server occupying by a customer of the same class, it goes to an orbit and makes a repeated attempt after a random delay. If an arrival customer finds the other class customer on the server, it can interrupt its service with the given probability and start servicing itself. Customers from the orbit behave the same way. There is a multiply access for customers in the orbit. Service times and inter-retrial times have exponential distributions. Customers are assumed heterogeneous, so the parameters of the distributions are different for each class. In the paper, we propose the original marginal asymptotic-diffusion method for finding of the stationary probability distributions of the number of each class customers under the long delays condition.

About the authors

Anatoly A. Nazarov

National Research Tomsk State University

Author for correspondence.
Email: nazarov.tsu@gmail.com
ORCID iD: 0000-0002-5097-5629
Scopus Author ID: 7201780364
ResearcherId: O-5862-2014

Doctor of Technical Sciences, professor of Department of Probability Theory and Mathematical Statistic

Izmailova National Research Tomsk State University, 36

Ekaterina A. Fedorova

National Research Tomsk State University

Email: moiskate@mail.ru
ORCID iD: 0000-0001-8933-5322
Scopus Author ID: 56439120600
ResearcherId: E-3161-2017

PhD in Physical and Mathematical Sciences, associate professor of Department of Probability Theory and Mathematical Statistic

Izmailova National Research Tomsk State University, 36

Yana E. Izmailova

National Research Tomsk State University

Email: evgenevna.92@mail.ru
ORCID iD: 0000-0002-9132-0127
Scopus Author ID: 57191051392
ResearcherId: T-6377-2017

PhD in Physical and Mathematical Sciences, associate professor of Department of Probability Theory and Mathematical Statistic

Izmailova National Research Tomsk State University, 36

References

  1. Al Jaafreh, M. Multimodal systems, experiences, and communications: A review toward the tactile internet vision. Recent Trends in Computer Applications, 191-220 (2018).
  2. Matveev, Y. N. Technologies for biometric personal identification by voice and other modalities (in Russian). Russian. Engineering Journal: Science and Innovation 3, 46-61 (2012).
  3. Kagirov, I. A. Multimedia database of Russian sign language gestures in three-dimensional format (in Russian). Russian. Questions of Linguistics 1, 104-123 (2020).
  4. Ryndin, A., Pakulova, E., Basov, O. & Veselov, G. Modelling of multi-path transmission system of various priority multimodal information in 2020 IEEE 14th International Conference on Application of Information and Communication Technologies (AICT) (2020), 1-5. doi: 10.1109/AICT50176.2020. 9368802.
  5. Artalejo, J. R. & Gomez-Corral, A. Retrial Queueing Systems 318 pp. (Springer Berlin, 2008).
  6. Falin, G. & Templeton, J. Retrial Queues 320 pp. (Taylor & Francis, 1997).
  7. Phung-Duc, T. Retrial Queueing Models: A Survey on Theory and Applications. Stochastic Operations Research in Business and Industry, 1-26 (May 2017).
  8. Makeeva, E., Kochetkova, I. & Alkanhel, R. Retrial Queueing System for Analyzing Impact of Priority Ultra-Reliable Low-Latency Communication Transmission on Enhanced Mobile Broadband Quality of Service Degradation in 5G Networks. Mathematics, MDPI 11, 1-23. doi: 10.3390/math11183878 (2023).
  9. Avrachenkov, K., Morozov, E. & Nekrasova, R. Optimal and Equilibrium Retrial Rates in SingleServer Multi-orbit Retrial Systems in Lecture Notes in Computer Science 9305 (Springer, Cham, 2015), 135-146. doi: 10.1007/978-3-319-23440-3_11.
  10. Morozov, E., Rumyantsev, A., Dey, S. & Deepak, T. G. Performance analysis and stability of multiclass orbit queue with constant retrial rates and balking. Performance Evaluation 134, 102005. doi: 10.1016/j.peva.2019.102005 (2019).
  11. Krishnamoorthy, A., Joshua, V. & Mathew, A. A Retrial Queueing System with Multiple Hierarchial Orbits and Orbital Search in Communications in Computer and Information Science, vol. 919 (Springer, Cham, 2018), 224-233. doi: 10.1007/978-3-319-99447-5_19.
  12. Kim, B. & Kim, J. Proof of the conjecture on the stability of a multi-class retrial queue with constant retrial rates. Queueing System 104, 175-185. doi: 10.1007/s11134-023-09881-z (2023).
  13. Kim, B. & Kim, J. Stability of a multi-class multi-server retrial queueing system with service times depending on classes and servers. Queueing System 94, 129-146. doi: 10.1007/s11134-01909634-x (2020).
  14. Avrachenkov, K. Stability and partial instability of multi-class retrial queues. Queueing Systems 100, 177-179. doi: 10.1007/s11134-022-09814-2hal-03767703 (2022).
  15. Shin, Y. W. & Moon, D. H. M/M/c Retrial Queue with Multiclass of Customers. Methodology and Computing in Applied Probability 16, 931-949. doi: 10.1007/s11009-013-9340-0 (2014).
  16. White, H. C. & Christie, L. S. Queueing with preemptive priorities or with breakdown. Operations Research 6, 79-95. doi: 10.1287/opre.6.1.79 (1958).
  17. Gaver, D. A waiting line with interrupted service including priority. J. Roy. Stat. Soc. B24 24, 73 doi: 10.3390/sym11030419 (1962).
  18. Fiems, D. & Bruneel, H. Queueing systems with different types of server interruptions. Eur. J. Oper. Res. 188, 838-845 (2008).
  19. Razumchik, R. Two-priority queueing system with LCFS service, probabilistic priority and batch arrivals in AIP Conference Proceedings 2116 (July 2019), 090011. doi: 10.1063/1.5114076.
  20. Yin, M., Yan, M., Guo, Y. & Liu, M. Analysis of a Pre-Emptive Two-Priority Queuing System with Impatient Customers and Heterogeneous Servers. Mathematics 11. doi: 10.3390/math11183878 (2023).
  21. Babu, D., Krishnamoorthy, A. & Joshua, V. C. Retrial Queue with Search of Interrupted Customers from the Finite Orbit in Information Technologies and Mathematical Modelling. Queueing Theory and Applications 912 (Springer International Publishing, Cham, 2018), 360-371. doi: 10.1007/9783-319-97595-5_28.
  22. Ammar, S. I. & Rajadurai, P. Performance Analysis of Preemptive Priority Retrial Queueing System with Disaster under Working Breakdown Services. Symmetry 11. doi: 10.3390/sym11030419 (2019).
  23. Atencia, I. A Geo/G/1 retrial queueing system with priority services. Eur. J. Oper. Res. 256, 178- 186. doi: 10.1016/j.ejor.2016.07.011 (2017).
  24. Jain, M., Bhagat, A. & Shekhar, C. Double orbit finite retrial queues with priority customers and service interruptions. Appl. Math. Comput. 253, 324-344 (2015).
  25. Nazarov, A. & Izmailova, Y. Study of the RQ-system M(2)/B(x)(2)/1 with R-persistent displacement of alternative customers (in Russian). Russian. Bulletin of the Siberian State Aerospace University named after academician M.F. Reshetneva 17, 328-334 (2016).

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