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Negative Binomial Approximation in Retrial Queue M/M/1 with Collisions and Impatient Calls

  • Elena DanilyukEmail author
  • Ekaterina Fedorova
Conference paper
  • 164 Downloads
Part of the Communications in Computer and Information Science book series (CCIS, volume 1141)

Abstract

In the paper, the retrial queueing system of M/M/1 type with collisions and impatient calls is considered. The impatience of calls in the orbit is exponential distributed. The process of the number of calls in the orbit is analyzed. We propose the method of the negative binomial approximation using the first and the second moments of the distribution which were obtained asymptotically. The numerical analysis of comparison exact (obtained by simulation) and approximate distributions for different values of the system parameters are presented.

Keywords

Retrial queueing system Negative binomial distribution Collisions Impatient calls 

References

  1. 1.
    Aguir, S., Karaesmen, F., Askin, O.Z., Chauvet, F.: The impact of retrials on call center performance. OR Spektrum 26, 353–376 (2004)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Almási, B., Bérczes, T., Kuki, A., Sztrik, J., Wang, J.: Performance modeling of finite-source cognitive radio networks. Acta Cybernetica 22(3), 617–631 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cohen, J.W.: Basic problems of telephone traffic and the influence of repeated calls. Philips Telecommun. Rev. 18(2), 49–100 (1957)Google Scholar
  4. 4.
    Gosztony, G.: Repeated call attempts and their effect on traffic engineering. Budavox Telecommun. Rev. 2, 16–26 (1976)Google Scholar
  5. 5.
    Elldin, A., Lind, G.: Elementary Telephone Traffic Theory. Ericsson Public Telecommunications, Stockholm (1971)Google Scholar
  6. 6.
    Phung-Duc, T., Kawanishi, K.: Multiserver retrial queue with setup time and its application to data centers. J. Ind. Manag. Optim. 15(1), 15–35 (2017).  https://doi.org/10.1142/S0217595914400089MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Roszik, J., Sztrik, J., Kim, C.: Retrial queues in the performance modelling of cellular mobile networks using MOSEL. Int. J. Simul. 6, 38–47 (2005)Google Scholar
  8. 8.
    Wilkinson, R.I.: Theories for toll traffic engineering in the USA. Bell Syst. Tech. J. 35(2), 421–507 (1956).  https://doi.org/10.1002/j.1538-7305.1956.tb02388.xCrossRefGoogle Scholar
  9. 9.
    Artalejo, J.R., Gómez-Corral, A.: Retrial Queueing Systems. A Computational Approach. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-78725-9CrossRefzbMATHGoogle Scholar
  10. 10.
    Falin, G.I., Templeton, J.G.C.: Retrial Queues. Chapman & Hall, London (1997)CrossRefGoogle Scholar
  11. 11.
    Sudyko, E.A., Nazarov, A.A.: A study of a Markov RQ-system with call conflicts and elementary incoming stream. Vestn. Tomsk. Gos. Univ., Upravlen., Vychisl. Tekh. Inf. 3(12), 97–106 (2010)Google Scholar
  12. 12.
    Nazarov, A., Sztrik, J., Kvach, A.: Comparative analysis of methods of residual and elapsed service time in the study of the closed retrial queuing system M/GI/1//N with collision of the customers and unreliable server. In: Dudin, A., Nazarov, A., Kirpichnikov, A. (eds.) ITMM 2017. CCIS, vol. 800, pp. 97–110. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-68069-9_8CrossRefzbMATHGoogle Scholar
  13. 13.
    Danilyuk, E.Y., Fedorova, E.A., Moiseeva, S.P.: Asymptotic analysis of an retrial queueing system M/M/1 with collisions and impatient calls. Autom. Remote Control 79(12), 2136–2146 (2018).  https://doi.org/10.1134/S0005117918120044MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Yang, T., Posner, M., Templeton, J.: The M/G/1 retrial queue with non-persistent customers. Queueing Syst. 7(2), 209–218 (1990)CrossRefGoogle Scholar
  15. 15.
    Krishnamoorthy, A., Deepak, T., Joshua, V.: An M/G/1 retrial queue with non-persistent customers and orbital search. Stoch. Anal. Appl. 23, 975–997 (2005).  https://doi.org/10.1080/07362990500186753MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kim, J.: Retrial queueing system with collision and impatience. Commun. Korean Math. Soc. 4, 647–653 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Aissani, A., Taleb, S., Hamadouche, D.: An unreliable retrial queue with impatience and preventive maintenance. In: Proceedings of the 15th Applied Stochastic Models and Data Analysis (ASMDA 2013), Mataró (Barcelona), Spain, pp. 1–9 (2013)Google Scholar
  18. 18.
    Kumar, M.S., Arumuganathan, R.O.: Performance analysis of single server retrial queue with general retrial time, impatient subscribers, two phases of service and bernoulli schedule. Tamkang J. Sci. Eng. 13(2), 135–143 (2010)Google Scholar
  19. 19.
    Fedorova, E., Voytikov, K.: Retrial queue M/G/1 with impatient calls under heavy load condition. In: Dudin, A., Nazarov, A., Kirpichnikov, A. (eds.) ITMM 2017. CCIS, vol. 800, pp. 347–357. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-68069-9_28CrossRefGoogle Scholar
  20. 20.
    Danilyuk, E., Vygoskaya, O., Moiseeva, S.: Retrial queue M/M/N with impatient customer in the orbit. In: Vishnevskiy, V.M., Kozyrev, D.V. (eds.) DCCN 2018. CCIS, vol. 919, pp. 493–504. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-99447-5_42CrossRefGoogle Scholar
  21. 21.
    Vygovskaya, O., Danilyuk, E., Moiseeva, S.: Retrial queueing system of MMPP/M/2 type with impatient calls in the orbit. In: Dudin, A., Nazarov, A., Moiseev, A. (eds.) ITMM/WRQ -2018. CCIS, vol. 912, pp. 387–399. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-97595-5_30CrossRefGoogle Scholar
  22. 22.
    Dudin, A.N., Klimenok, V.I.: Queueing system BMAP/G/1 with repeated calls. Math. Comput. Modell. 30(3–4), 115–128 (1999).  https://doi.org/10.1016/S0895-7177(99)00136-3MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Artalejo, J.R., Pozo, M.: Numerical calculation of the stationary distribution of the main multiserver retrial queue. Ann. Oper. Res. 116, 41–56 (2002).  https://doi.org/10.1023/A:1021359709489MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Neuts, M.F., Rao, B.M.: Numerical investigation of a multiserver retrial model. Queueing Syst. 7(2), 169–189 (1990).  https://doi.org/10.1007/BF01158473CrossRefzbMATHGoogle Scholar
  25. 25.
    Borovkov, A.A.: Asymptotic Methods in Queueing Theory. Wiley, New York (1984)zbMATHGoogle Scholar
  26. 26.
    Fedorova, E., Nazarov, A., Paul, S.: Discrete gamma approximation in retrial queue MMPP/M/1 based on moments calculation. In: Rykov, V.V., Singpurwalla, N.D., Zubkov, A.M. (eds.) ACMPT 2017. LNCS, vol. 10684, pp. 121–131. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-71504-9_12CrossRefzbMATHGoogle Scholar
  27. 27.
    Moiseev, A., Demin, A., Dorofeev, V., Sorokin, V.: Discrete-event approach to simulation of queueing networks. Key Eng. Mater. 685, 939–942 (2016).  https://doi.org/10.4028/www.scientific.net/KEM.685.939CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.National Research Tomsk State UniversityTomskRussian Federation

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