Skip to main content
SpringerLink
Log in

A Retrial Queueing System with Alternating Inter-retrial Time Distribution

  • Conference paper
  • First Online:
Distributed Computer and Communication Networks (DCCN 2018)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 919))

  • 714 Accesses

Abstract

We consider a single-server retrial queuing system with Markovian Arrival Process (MAP) and phase-type (PH) service time distribution. Customers which find the server busy enter the orbit of infinite size and try their luck after some random time. Concerning the retrial process, we suppose that inter-retrial times have PH distribution if the number of customers in the orbit does not exceed some threshold and have exponential distribution otherwise. Such an assumption allows to some extent take into account the realistic nature of retrial process and, at the same time, to avoid a large increase in the dimensionality of the state space of this process. We consider two different policies of repeated attempts and describe the operation of the system by two different multi-dimensional Markov chains: by quasi-Toeplitz Markov chain in the case of a constant retrial rate and by asymptotically quasi-Toeplitz Markov chain in the case of an infinitely increasing retrial rate. Both chains are successfully analyzed in this paper. We derive the ergodicity condition, calculate the stationary distribution and the main performance measures of the system.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Artalejo, J.: Accessible bibliography on retrial queues. Math. Comput. Model. 30, 223–233 (1999)

    Google Scholar 

  2. Gomez-Corral, A.: A bibliographical guide to the analysis of retrial queues through matrix analytic techniques. Ann. Oper. Res. 141, 163–191 (2006)

    Article  MathSciNet  Google Scholar 

  3. Falin, G., Templeton, J.: Retrial Queues. Chapman and Hall, London (1997)

    Book  Google Scholar 

  4. Artalejo, J.R., Gomez-Corral, A.: Retrial Queueing Systems: A Computational Approach. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78725-9

    Book  MATH  Google Scholar 

  5. Choi, B.D., Shin, Y.W., Ahn, W.C.: Retrial queues with collision arising from unslottted CDMA/CD protocol. Queueing Syst. 11, 335–356 (1992)

    Article  Google Scholar 

  6. Gomez-Corral, A.: Stochastic analysis of a single server retrial queue with general retrial time. Naval Res. Logist. 46, 561–581 (1999)

    Article  MathSciNet  Google Scholar 

  7. Moreno, P.: An \(M/G/1\) retrial queue with recurrent customers and general retrial times. Appl. Math. Comput. 159, 651–666 (2004)

    MathSciNet  MATH  Google Scholar 

  8. Atencia, I., Moreno, P.: A single server retrial queue with general retrial time and Bernoulli schedule. Appl. Math. Comput. 162, 855–880 (2005)

    MathSciNet  MATH  Google Scholar 

  9. Choudhury, G.: An \(M/G/1\) retrial queue with an additional phase of second service and general retrial time. Int. J. Inf. Manag. Sci. 20, 1–14 (2009)

    MathSciNet  MATH  Google Scholar 

  10. Kumar, B.K., Vijay Kumar, A., Arivudainambi, D.: An \(M/G/1\) retrial queueing system with two phase service and preemptive resume. Ann. Oper. Res. 113, 61–79 (2002)

    Article  MathSciNet  Google Scholar 

  11. Wu, X., Brill, P., Hlyanka, M., Wang, J.: An \(M/G/1\) retrial queue with balking and retrials during service. Int. J. Oper. Res. 1, 30–51 (2005)

    Article  MathSciNet  Google Scholar 

  12. Choudhury, G., Ke, J.-C.: A batch arrival retrial queue with general retrial times under Bernoulli vacation schedule for unreliable server and delaying repair. Appl. Math. Model. 36, 255–269 (2012)

    Article  MathSciNet  Google Scholar 

  13. Dudin, A.N., Deepak, T.G., Varghese, C.J., Krishnamoorthy, A., Vishnevsky, V.M.: On a BMAP/G/1 retrial system with two types of search of customers from the orbit. Commun. Comput. Inf. Sci. 800, 1–12 (2017)

    Google Scholar 

  14. Liang, H.M.: Retrial queues (queueing system, stability condition, K-ordering). Ph.D. thesis, University of Noth Carolina, Chapel Hill (1991)

    Google Scholar 

  15. Yang, T., Posner, M.J.M., Templeton, J.G.C., Li, H.: An approximation method for the \(M/G/1\) retrial queue with general retrial times. Eur. J. Oper. Res. 76, 110–116 (1994)

    Article  Google Scholar 

  16. Diamond, J.E., Alfa, A.S.: An approximation method for the \(M/PH/1\) retrial queue with phase type inter-retrial times. Eur. J. Oper. Res. 113, 620–631 (1999)

    Article  Google Scholar 

  17. Lucantoni, D.: New results on the single server queue with a batch Markovian arrival process. Commun. Statist.-Stoch. Models 7, 1–46 (1991)

    Article  MathSciNet  Google Scholar 

  18. Breuer, L., Dudin, A., Klimenok, V.: A retrial \(BMAP/PH/N\) system. Queueing Syst. 40, 433–457 (2002)

    Article  MathSciNet  Google Scholar 

  19. Klimenok, V., Orlovsky, D., Dudin, A.: A \(BMAP/PH/N\) system with impatient repeated calls. Asia-Pac. J. Oper. Res. 24, 293–312 (2007)

    Article  MathSciNet  Google Scholar 

  20. Breuer, L., Klimenok, V., Birukov, A., Dudin, A., Krieger, U.: Mobile networks modeling the access to a wireless network at hot spots. Eur. Trans. Telecommun. 16, 309–316 (2005)

    Article  Google Scholar 

  21. Kim, C., Klimenok, V., Mushko, V., Dudin, A.: The \(BMAP/PH/N\) retrial queueing system operating in Markovian random environment. Comput. Oper. Res. 37, 1228–1237 (2010)

    Article  MathSciNet  Google Scholar 

  22. Klimenok, V.I., Orlovsky, D.S., Kim, C.S.: The \(BMAP/PH/N\) retrial queue with Markovian flow of breakdowns. Eur. J. Oper. Res. 189, 1057–1072 (2008)

    Article  MathSciNet  Google Scholar 

  23. Kim, C.S., Park, S.H., Dudin, A., Klimenok, V., Tsarenkov, G.: Investigaton of the BMAP/G/1’./PH/1/M tandem queue with retrials and losses. Appl. Math. Model. 34, 2926–2940 (2010)

    Article  MathSciNet  Google Scholar 

  24. Neuts, M.: Structured Stochastic Matrices of \(M/G/1\) Type and Their Applications. Marcel Dekker, New York (1989)

    MATH  Google Scholar 

  25. Graham, A.: Kronecker Products and Matrix Calculus with Applications. Ellis Horwood, Cichester (1981)

    MATH  Google Scholar 

  26. Bertsekas, D., Gallager, R.: Data Network. Prentice Hall, Englewood Cliffs, New York (1987)

    MATH  Google Scholar 

  27. Neuts, M.: Matrix-Geometric Solutions in Stochastic Models. The Johns Hopkins University Press, Baltimore (1981)

    MATH  Google Scholar 

  28. Klimenok, V.I., Dudin, A.N.: Multi-dimensional asymptotically quasi-Toeplitz Markov chains and their application in queueing theory. Queueing Syst. 54, 245–259 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work has been financially supported by the Russian Science Foundation and the Department of Science and Technology (India) via grant No 16-49-02021 (INT/RUS/RSF/16) for the joint research project by the V.A. Trapeznikov Institute of Control Problems of the Russian Academy Sciences and the CMS College Kottayam.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics and Computer Science, Belarusian State University, 220030, Minsk, Belarus

    Valentina Klimenok & Alexander Dudin

  2. Institute of Control Sciences of Russian Academy of Sciences and Closed Corporation “Information and Networking Technologies”, Moscow, Russia

    Vladimir Vishnevsky

Authors
  1. Valentina Klimenok
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander Dudin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vladimir Vishnevsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vladimir Vishnevsky .

Editor information

Editors and Affiliations

  1. V.A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    Vladimir M. Vishnevskiy

  2. V.A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    Dmitry V. Kozyrev

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Klimenok, V., Dudin, A., Vishnevsky, V. (2018). A Retrial Queueing System with Alternating Inter-retrial Time Distribution. In: Vishnevskiy, V., Kozyrev, D. (eds) Distributed Computer and Communication Networks. DCCN 2018. Communications in Computer and Information Science, vol 919. Springer, Cham. https://doi.org/10.1007/978-3-319-99447-5_26

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-99447-5_26

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-99446-8

  • Online ISBN: 978-3-319-99447-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation