Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

For citation:

Fedorova E. A., Nazarov A. A., Farkhadov M. P. Asymptotic Analysis of the MMРР|M|1 Retrial Queue with Negative Calls under the Heavy Load Condition. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2020, vol. 20, iss. 4, pp. 534-547. DOI: 10.18500/1816-9791-2020-20-4-534-547, EDN: SILKZS

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
30.11.2020
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Language: 
Russian
Heading: 
Computer Sciences
Article type: 
Article
UDC: 
501.1
DOI: 
10.18500/1816-9791-2020-20-4-534-547
EDN: 
SILKZS

Asymptotic Analysis of the MMРР|M|1 Retrial Queue with Negative Calls under the Heavy Load Condition

Autors: 
Fedorova Ekaterina A., Tomsk State University
Nazarov Anatoly A., Tomsk State University
Farkhadov Mais P., V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences
Abstract: 

In the paper, a single-server retrial queueing system with MMPP arrivals and an exponential law of the service time is studied. Unserviced calls go to an orbit and stay there during random time distributed exponentially, they access to the server according to a random multiple access protocol. In the system, a Poisson process of negative calls arrives, which delete servicing positive calls. The method of the asymptotic analysis under the heavy load condition for the system studying is proposed. It is proved that the asymptotic characteristic function of a number of calls on the orbit has the gamma distribution with the obtained parameters. The value of the system capacity is obtained, so, the condition of the system stationary mode is found. The results of a numerical comparison of the asymptotic distribution and the distribution obtained by simulation are presented. Conclusions about the method applicability area are made.

Key words: 
retrial queue
negative calls
MMPP
asymptotic analysis
heavy load
References: 
  1. Kuznetsov D. Y., Nazarov A. A. Analysis of a communication network governed by an adaptive random multiple access protocol in critical load. Problems of Information Transmission, 2004, vol. 40, no. 3, pp. 243–253. DOI: https://doi.org/10.1023/B:PRIT.0000044260.05898.24
  2. Tran-Gia P., Mandjes M. Modeling of customer retrial phenomenon in cellular mobile networks. IEEE Journal on Selected Areas in Communications, 1997, vol. 15, pp. 1406– 1414. DOI: https://doi.org/10.1109/49.634781
  3. Roszik J., Sztrik J., Kim C. S. Retrial queues in the performance modelling of cellular mobile networks using MOSEL. I. J. of Simulation, 2005, vol. 6, no. 1–2, pp. 38–47.
  4. Kim C. S., Klimenok V., Dudin A. Analysis and optimization of guard channel policy in cellular mobile networks with account of retrials. Computers and Operation Research, 2014, vol. 43, pp. 181–190. DOI: https://doi.org/10.1016/j.cor.2013.09.005
  5. Artalejo J. R., G´omez-Corral A. Retrial Queueing Systems. A Computational Approach. Berlin, Springer, 2008. 267 p. DOI: https://doi.org/10.1007/978-3-540-78725-9
  6. Falin G. I., Templeton J. G. C. Retrial queues. London, Chapman & Hall, 1997. 328 p.
  7. Gelenbe E. Random neural networks with positive and negative signals and product form solution. Neural Computation, 1989, vol. 1, no. 4, pp. 502–511. DOI: https://doi.org/10.1162/neco.1989.1.4.502
  8. Gelenbe E. Product-form queueing networks with negative and positive customers. Journal of Applied Probability, 1991, vol. 28, pp. 656–663. DOI: https://doi.org/10.2307/3214499
  9. Do T. V. Bibliography on G-networks, negative customers and applications. Mathematical and Computer Modelling, 2011, vol. 53, iss. 1–2, pp. 205–212. DOI: https://doi.org/10.1016/j.mcm.2010.08.006
  10. Shin Y. W. Multi-server retrial queue with negative customers and disasters. Queueing Systems, 2007, vol. 55, iss. 4, pp. 223–237. DOI: https://doi.org/10.1007/s11134-007-9018-9
  11. Anisimov V. V., Artalejo J. R. Analysis of Markov multiserver retrial queues with negative arrivals. Queueing Systems, 2001, vol. 39, no. 2/3, pp. 157–182. DOI: https://doi.org/10.1023/A:1012796517394
  12. Berdjoudj L., Aissani D. Martingale methods for analyzing the M/M/1 retrial queue with negative arrivals. Journal of Mathematical Sciences, 2005, vol. 131, no. 3, pp. 5595–5599. DOI: https://doi.org/10.1007/s10958-005-0430-6
  13. Wu J., Lian Z. A single-server retrial G-queue with priority and unreliable server under Bernoulli vacation schedule. Computers & Industrial Engineering, 2013, vol. 64, iss. 1, pp. 84–93. DOI: https://doi.org/10.1016/j.cie.2012.08.015
  14. Kirupa K., Udaya Chandrika K. Batch Arrival Retrial Queue with Negative Customers, Multi-optional Service and Feedback. Communications on Applied Electronics, 2015, vol. 2, no. 4, pp. 14–18. DOI: https://doi.org/10.5120/cae2015651707
  15. Klimenok V. I., Dudin A. N. A BMAP/PH/N queue with negative customers and partial protection of service. Communications in Statistics – Simulation and Computation, 2012, vol. 41, no. 7, pp. 1062–1082.
  16. Dimitriou I. A mixed priority retrial queue with negative arrivals, unreliable server and multiple vacations. Applied Mathematical Modelling, 2013, vol. 37, iss. 3, pp. 1295–1309. DOI: https://doi.org/10.1016/j.apm.2012.04.011
  17. Rajadurai P. A study on M/G/1 retrial queueing system with three different types of customers under working vacation policy. International Journal of Mathematical Modelling and Numerical Optimisation (IJMMNO), 2018, vol. 8, no. 4, pp. 393–417. DOI: https://doi.org/10.1504/IJMMNO.2018.094550
  18. Zidani N., Djellab N. On the multiserver retrial queues with negative arrivals. International Journal of Mathematics in Operational Research (IJMOR), 2018, vol. 13, no. 2, pp. 219–242. DOI: https://doi.org/10.1504/IJMOR.2018.094056
  19. Bertsekas D., Gallager R. Seti peredachi dannykh [Information transmission networks]. Moscow, Mir, 1989. 544 p. (in Russian).
  20. Bljek Ju. Seti EVM: protokoly, standarty, interfejsy [Networks: protocols, standards, interfaces]. Moscow, Mir, 1990. 510 p. (in Russian).
  21. G´omez-Corral A. A bibliographical guide to the analysis of retrial queues through matrix analytic techniques. Annals of Operations Research, 2006, vol. 141, iss. 1, pp. 163–191. DOI: https://doi.org/10.1007/s10479-006-5298-4
  22. Kim C. S., Mushko V. V., Dudin A. N. Computation of the steady state distribution for multi-server retrial queues with phase type service process. Annals of Operations Research, 2012, vol. 201, iss. 1, pp. 307–323. DOI: https://doi.org/10.1007/s10479-012-1254-7
  23. Artalejo J. R., Pozo M. Numerical calculation of the stationary distribution of the main multiserver retrial queue. Annals of Operations Research, 2002, vol. 116, iss. 1–4, pp. 41– 56. DOI: https://doi.org/10.1023/A:1021359709489
  24. Ridder A. Fast simulation of retrial queues. Third Workshop on Rare Event Simulation and Related Combinatorial Optimization Problems. Piza, 2000, pp. 1–5.
  25. Moiseev A., Nazarov A. Queueing network MAP−(GI/∞)K with high-rate arrivals. European Journal of Operational Research, 2016, vol. 254, iss. 1, pp. 161–168. DOI: https://doi.org/10.1016/j.ejor.2016.04.011
  26. Nazarov A. A., Fedorova E. A. Retrial queuing system MMPP|GI|1 researching by means of the second-order asymptotic analysis method under a heavy load condition. Bulletin of the Tomsk Polytechnic University. Geo Assets Engineering, 2014, vol. 325, no. 5, pp. 6–15 (in Russian).
  27. Lisovskaja E. Ju., Moiseeva S. P. Asymptotical analysis of a non-Markovian queueing system with renewal input process and random capacity of customers. Tomsk State University Journal of Control and Computer Science, 2017, no. 39, pp. 30–38 (in Russian). DOI: https://doi.org/10.17223/19988605/39/5
  28. Moiseev A., Demin A., Dorofeev V., Sorokin V. Discrete-event approach to simulation of queueing networks. Key Engineering Materials, 2016, vol. 685, pp. 939–942. DOI: https://doi.org/10.4028/www.scientific.net/KEM.685.939
Received: 
08.11.2019
Accepted: 
30.12.2019
Published: 
30.11.2020
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