Advertisement

Automation and Remote Control

, Volume 79, Issue 12, pp 2136–2146 | Cite as

Asymptotic Analysis of an Retrial Queueing System M|M|1 with Collisions and Impatient Calls

  • E. Yu. DanilyukEmail author
  • E. A. Fedorova
  • S. P. Moiseeva
Stochastic Systems
  • 24 Downloads

Abstract

We consider a single-line RQ-system with collisions with Poisson arrival process; the servicing time and time delay of calls on the orbit have exponential distribution laws. Each call in orbit has the “impatience” property, that is, it can leave the system after a random time. The problem is to find the stationary distribution of the number of calls on the orbit in the system under consideration. We construct Kolmogorov equations for the distribution of state probabilities in the system in steady-state mode. To find the final probabilities, we propose a numerical algorithm and an asymptotic analysis method under the assumption of a long delay and high patience of calls in orbit. We show that the number of calls in orbit is asymptotically normal. Based on this numerical analysis, we determine the range of applicability of our asymptotic results.

Keywords

retrial queueing system orbit asymptotic analysis collisions impatient calls 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Wilkinson, R.I., Theories for Toll Traffic Engineering in the USA, The Bell Syst. Techn. J., 1956, vol. 35, no. 2, pp. 421–507.CrossRefGoogle Scholar
  2. 2.
    Cohen, J.W., Basic Problems of Telephone Trafic and the Influence of Repeated Calls, Philips Telecommun. Rev., 1957, vol. 18, no. 2, pp. 49–100.Google Scholar
  3. 3.
    Gosztony, G., Repeated Call Attempts and Their Effect on Trafic Engineering, Budavox Telecommun. Rev., 1976, vol. 2, pp. 16–26.Google Scholar
  4. 4.
    Elldin, A. and Lind, G., Elementary Telephone Trafic Theory, Stockholm: Ericsson Public Telecommunications, 1971.Google Scholar
  5. 5.
    Artalejo, J.R. and Gomez-Corral, A., Retrial Queueing Systems. A Computational Approach, Stockholm: Springer, 2008.CrossRefzbMATHGoogle Scholar
  6. 6.
    Falin, G.I. and Templeton, J.G.C., Retrial Queues, London: Chapman & Hall, 1997.CrossRefzbMATHGoogle Scholar
  7. 7.
    Artalejo, J.R. and Falin, G.I., Standard and Retrial Queueing Systems: A Comparative Analysis, Revista Mat. Complut., 2002, vol. 15, pp. 101–129.MathSciNetzbMATHGoogle Scholar
  8. 8.
    Roszik, J., Sztrik, J., and Kim, C., Retrial Queues in the Performance Modelling of Cellular Mobile Networks Using MOSEL, Int. J. Simulat., 2005, no. 6, pp. 38–47.Google Scholar
  9. 9.
    Kuznetsov, D.Yu. and Nazarov, A.A., Non-Markovian Models of Communication Networks with Adaptive Random Multiple Access Protocols, Autom. Remote Control, 2001, vol. 62, no. 5, pp. 789–808.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Aguir, S., Karaesmen, F., Askin, O.Z., and Chauvet, F., The Impact of Retrials on Call Center Performance, OR Spektrum, 2004, no. 26, pp. 353–376.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Sudyko, E.A. and Nazarov, A.A., A Study of a Markov RQ-System with Call Conflicts and Elementary Incoming Stream, Vestn. Tomsk. Gos. Univ., Upravlen., Vychisl. Tekh. Informat., 2010, no. 3(12), pp. 97–106.Google Scholar
  12. 12.
    Nazarov, A., Sztrik, J., and Kvach, A., Comparative Analysis ofMethods 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, Inform. Technol. Math. Model. Queueing Theory Appl. (ITMM 2017), Commun. Comp. Inform. Sci., 2017, vol. 800, pp. 97–110.zbMATHGoogle Scholar
  13. 13.
    Berczes, T., Sztrik, J., Toth, A., and Nazarov, A., Performance Modeling of Finite-Source Retrial Queueing Systems with Collisions and Non-Reliable Server using MOSEL, Inform. Technol. Math. Model. Queueing Theory Appl. (ITMM 2017), Commun. Comp. Inform. Sci., 2017, vol. 700, pp. 248–258.Google Scholar
  14. 14.
    Yang, T., Posner, M., and Templeton, J., The M/G/1 Retrial Queue with Non-Persistent Customers, Queueing Syst., 1990, no. 7(2), pp. 209–218.CrossRefzbMATHGoogle Scholar
  15. 15.
    Krishnamoorthy, A., Deepak, T., and Joshua, V., An M/G/1 Retrial Queue with Non-Persistent Customers and Orbital Search, Stochast. Anal. Appl., 2005, no. 23, pp. 975–997.CrossRefzbMATHGoogle Scholar
  16. 16.
    Kim, J., Retrial Queueing System with Collision and Impatience, Commun. Korean Math. Soc., 2010, no. 4, pp. 647–653.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fayolle, G., and Brun, M., On a System with Impatience and Repeated Calls, in Queueing Theory and Its Applications: Liber Amicorum for J.W. Cohen, Amsterdam: North Holland, 1988, pp. 283–305.Google Scholar
  18. 18.
    Martin, M. and Artalejo, J., Analysis of an M/G/1 Queue with Two Types of Impatient units, Adv. Appl. Probab., 1995, no. 27, pp. 647–653.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Aissani, A., Taleb, S., and Hamadouche, D., An Unreliable Retrial Queue with Impatience and Preventive Maintenance, Proc. 15 Appl. Stochast. Models Data Anal. (ASMDA2013), 2013, pp. 1–9.Google Scholar
  20. 20.
    Kumar, M. and Arumuganathan, R., Performance Analysis of Single Server Retrial Queue with General Retrial Time, Impatient Subscribers, Two Phases of Service, and Bernoulli Schedule, Tamkang J. Sci. Eng., 2010, no. 13(2), pp. 135–143.Google Scholar
  21. 21.
    Fedorova, E. and Voytikov, K., Retrial Queue M/G/1 with Impatient Calls Under Heavy Load Condition, Inform. Technol. Math. Model. Queueing Theory Appl. (ITMM 2017), Commun. Comp. Inform. Sci., 2017, vol. 800, pp. 347–357.zbMATHGoogle Scholar
  22. 22.
    Nazarov, A.A. and Fedorova, E.A., Asymptotic Analysis of the RQ-System MM1 with Impatient Calls under Long Patience, Proc. 19th Conf. Distrib. Comp. and Telecomm. Networks: Control, Computation, Communication (DCCN-2016), 2016, pp. 342–348.Google Scholar
  23. 23.
    Dudin, A.N. and Klimenok, V.I., Queueing System BMAP/G/1 with Repeated Calls, Math. Comp. Model., 1999, vol. 30, no. 3–4, pp. 115–128.Google Scholar
  24. 24.
    Stepanov, S.N., Algorithms Approximate Design Syst. Repeated Calls, Autom. Remote Control, 1983, vol. 44, no. 1, pp. 63–71.zbMATHGoogle Scholar
  25. 25.
    Nazarov, A.A. and Lyubina, T.V., The Non-Markov Dynamic RQ System with the Incoming MMP Flow of Requests, Autom. Remote Control, 2013, vol. 74, no. 7, pp. 1132–1143.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Artalejo, J.R. and Pozo, M., Numerical Calculation of the Stationary Distribution of the Main Multiserver Retrial Queue, Ann. Oper. Res., 2002, no. 116, pp. 41–56.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Neuts, M.F. and Rao, B.M., Numerical Investigation of a Multiserver Retrial Model, Queueing Syst., 1990, vol. 7, no. 2, pp. 169–189.CrossRefzbMATHGoogle Scholar
  28. 28.
    Nazarov, A.A. and Moiseeva, S.P., Metod asimptoticheskogo analiza v teorii massovogo obsluzhivaniya (The Asymptotic Analysis Method in Queueing Theory), Tomsk: Tomsk. Gos. Univ., 2006.Google Scholar
  29. 29.
    Borovkov, A.A., Asymptotic Methods in Queueing Theory, New York: Wiley, 1984.zbMATHGoogle Scholar
  30. 30.
    Zadorozhnii, V.N., Asymptotic Analysis of Systems with Intensive Interrupts, Autom. Remote Control, 2008, vol. 69, no. 2, pp. 252–261.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Inc. 2018

Authors and Affiliations

  • E. Yu. Danilyuk
    • 1
    Email author
  • E. A. Fedorova
    • 1
  • S. P. Moiseeva
    • 1
  1. 1.National Research Tomsk State UniversityTomskRussia

Personalised recommendations