Asymptotic diffusion analysis of RQ system M/M/1 with unreliable server

Cover Page

Cite item

Full Text

Abstract

The paper considers a single-line retrial queueing system with an unreliable server. Queuing systems are called unreliable if their servers may fail from time to time and require restoration (repair), only after which they can resume servicing customers. The input of the system is a simple Poisson flow of customers. The service time and uptime of the server are distributed exponentially. An incoming customer try to get service. The server can be free, busy or under repair. The customer is serviced immediately if the server is free. If it is busy or under repair, the customer goes into orbit. And after a random time it tries to get service again. The study is carried out by the method of asymptotically diffusion analysis under the condition of a large delay of requests in orbit. In this work, the transfer coefficient and diffusion coefficient were found and a diffusion approximation

Full Text

1. Introduction Queuing systems with repeated requests are quite often used in various areas of telecommunications. Modern information processing systems often encounter unstable operating conditions, such as overloads, failures, and resource limitations. Under these conditions, conventional retrial queuing (RQ) systems may not be able to process all incoming requests, resulting in lost information and poor performance [1-4]. Repetitive request systems offer a solution to this problem by providing a mechanism for processing requests that cannot be fulfilled immediately. Instead of discarding such requests, they are resubmitted to the queue after a certain time, increasing the likelihood of successful completion of service. The most complete and detailed description of RQ systems and their detailed comparison with classical queuing systems was reflected in [5-7]. There are different types of unreliability. For example, the works [8-10] consider the unreliability of the server as a breakdown. The authors in [11-14] consider an unreliable server with collisions or conflicts during simultaneous access to the server. Figure 1. Model of retrial queueing system M/M/1 with unreliable server This problem is especially relevant when it comes to unreliable servers that can fail due to software errors, hardware malfunctions or external factors. Server failures can lead to data loss, interruption of services, and decreased performance. If the server fails while servicing the request, it goes to repair. A request under maintenance goes into orbit and awaits recovery of the server. A fairly large number of works are devoted to systems with unreliable server [15-20]. To understand the behavior of systems with repeated requests and evaluate their performance, it is necessary to use analytical methods. In this paper, we consider a single-line queuing system with an unreliable server. We will conduct the study using the method of asymptotic diffusion analysis. It has been proven that the accuracy of the diffusion approximation exceeds the accuracy of the Gaussian approximation calculated in [21]. 2. System description Any data network, having generated customers, sends them to a shared resource (server). If the server is free, then the customer is served. If the server fails while servicing a customer, it is sent for repair, and the customers go into orbit. Let’s consider an RQ system with an unreliable server, the input of which receives a simple flow of customers with parameter
×

About the authors

Nataliya M. Voronina

National Research Tomsk Polytechnic University

Email: vnm@tpu.ru
ORCID iD: 0000-0001-9044-5211
Scopus Author ID: 57802914700
ResearcherId: AAD-2035-2019

Senior Lecturer of Department of Information Technology of School of Information Technology and Robotics Engineering

30 Lenina Ave, Tomsk, 634050, Russian Federation

Svetlana V. Rozhkova

National Research Tomsk Polytechnic University; National Research Tomsk State University

Author for correspondence.
Email: rozhkova@tpu.ru
ORCID iD: 0000-0002-8888-9291
Scopus Author ID: 6603581666
ResearcherId: F-5512-2017

Doctor of Physics and Mathematics Sciences, Professor of Department of Mathematics and Mathematical Physics of School of Nuclear Technology Engineering, 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

30 Lenina Ave, Tomsk, 634050, Russian Federation; 36 Lenina Ave, Tomsk, 634050, Russian Federation

References

  1. Gross, D., Shortle, J. F., Thompson, J. M. & Harris, C. M. Fundamentals of queueing theory (John wiley & sons, 2011).
  2. G.Gosztony. Repeated call attempts and their effect on trafic engineering. Budavox Telecommunication Review 2, 16-26 (1976).
  3. Cohen, J. Basic problems of telephone traffic theory and the influence of repeated calls. Philips Telecommunication Rev. 18, 49 (1957).
  4. Hanczewski, S., Stasiak, M., Weissenberg, J. & Zwierzykowski, P. Queuing model of the access system in the packet network in Computer Networks (2016), 283-293.
  5. Falin, G. I. & Templeton, J. G. C. Retrial Queues 320 pp. doi: 10.1201/9780203740767 (Chapman & Hall, London, 1997).
  6. Artalejo, J. R. & Gómez-Corral, A. Retrial Queueing Systems doi: 10.1007/978-3-540-78725-9 (Springer Berlin Heidelberg, 2008).
  7. Dimitriou, I. A queueing model with two classes of retrial customers and paired services. Annals of Operations Research 238, 123-143. doi: 10.1007/s10479-015-2059-2 (2016).
  8. Wang, J., Cao, J. & Li, Q. Reliability analysis of the retrial queue with server breakdowns and repairs. Queueing Systems 38, 363-380 (2001).
  9. Kumar, M. S. & Arumuganathan, R. An MX/G/1 retrial queue with two-phase service subject to active server breakdowns and two types of repair. International Journal of Operational Research 8, 261-291 (2010).
  10. Kim, C., Klimenok, V. I. & Orlovsky, D. S. The BMAP/PH/N retrial queue with Markovian flow of breakdowns. European Journal of Operational Research 189, 1057-1072 (2008).
  11. Lakaour, L., Aissani, D., Adel-Aissanou, K., Barkaoui, K. & Ziani, S. An unreliable single server retrial queue with collisions and transmission errors. Communications in Statistics-Theory and Methods 51, 1085-1109 (2022).
  12. Danilyuk, E. Y., Janos, S., et al. Asymptotic analysis of retrial queueing system M/M/1 with impatient customers, collisions and unreliable server. Journal of Siberian Federal University. Mathematics & Physics 13, 218-230. doi: 10.17516/1997-1397-2020-13-2-218-230 (2020).
  13. Tóth, Á. & Sztrik, J. Simulation of Finite-Source Retrial Queuing Systems With Collisions, Non-Reliable Server and Impatient Customers in the Orbit. in ICAI (2020), 408-419.
  14. Kuki, A., Bérczes, T., Sztrik, J. & Kvach, A. Numerical Analysis of Retrial Queueing Systems with Conflict of Customers and an Unreliable Server. Journal of Mathematical Sciences 237, 673-683. doi: 10.1007/s10958-019-04193-1 (2019).
  15. Nazarov, A. A., Paul, S. V. & Lizyura, O. D. Two-way communication retrial queue with unreliable server and multiple types of outgoing calls. Discrete and Continuous Models and Applied Computational Science 28, 49-61. doi: 10.22363/2658-4670-2020-28-1-49-61 (2020).
  16. Dudin, A., Dudina, O., Dudin, S. & Samouylov, K. Analysis of Single-Server Multi-Class Queue with Unreliable Service, Batch Correlated Arrivals, Customers Impatience, and Dynamical Change of Priorities. Mathematics 9, 1257. doi: 10.3390/math9111257 (2021).
  17. Chakravarthy, S. R., OZKAR, S. & SHRUTI, S. Analysis of M/M/C retrial queue with thresholds, PH distribution of retrial times and unreliable servers. Journal of applied mathematics & informatics 39, 173-196 (2021).
  18. Falin, G. An M/G/1 retrial queue with an unreliable server and general repair times. Performance Evaluation 67, 569-582 (2010).
  19. Dudin, A., Klimenok, V. & Vishnevsky, V. Analysis of unreliable single server queueing system with hot back-up server in Optimization in the Natural Sciences: 30th Euro Mini-Conference, EmC-ONS 2014, Aveiro, Portugal, February 5-9, 2014. Revised Selected Papers 30 (2015), 149-161.
  20. Nazarov, A. A., Paul, S. V., Lizyura, O. D., et al. Two-way communication retrial queue with unreliable server and multiple types of outgoing calls. Discrete and Continuous Models and Applied Computational Science. doi: 10.22363/2658-4670-2020-28-1-49-61 (2020).
  21. Voronina, N. M., Fedorova, E. A. & Rozhkova, S. V. Asymptotic analysis of the RQ-system M/M/1 with an unreliable server. Mathematical and software for information technical and economic systems, 304-309. doi: 10.1007/978-3-031-09331-9_28 (2020).

Copyright (c) 2024 Voronina N.M., Rozhkova S.V.

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies