Skip to main content
SpringerLink
Log in

A Survey of Recent Results in Finite-Source Retrial Queues with Collisions

  • Conference paper
  • First Online:
Information Technologies and Mathematical Modelling. Queueing Theory and Applications (ITMM 2018, WRQ 2018)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 912))

Abstract

The aim of the present paper is to give a review of recent results on single server finite-source retrial queuing systems with collision of the customers. There are investigations when the server is reliable and there are models when the server is subject to random breakdowns and repairs depending on whether it is idle or busy. Tool supported, numerical, simulation and asymptotic methods are considered under the condition of unlimited growing number of sources. Several cases and examples are treated and the results of different approaches are compared to each other showing the advantages and disadvantages of the given method. In general we could prove that the steady-state distribution of the number of customers in the service facility can be approximated by a normal distribution with given mean and variance. Using asymptotic methods under certain conditions in steady-state the distribution of the sojourn time in the orbit and in the system can be approximated by a generalized exponential one. Furthermore, it is proved that the distribution of the number of retrials until the successful service in the limit is geometrically distributed. By the help of stochastic simulation several systems are analyzed showing directions for further analytic investigations. Tables and Figures are collected to illustrate some special features of these systems.

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. Ali, A.A., Wei, S.: Modeling of coupled collision and congestion in finite source wireless access systems. In: Wireless Communications and Networking Conference (WCNC), pp. 1113–1118. IEEE (2015)

    Google Scholar 

  2. Almási, B., Roszik, J., Sztrik, J.: Homogeneous finite-source retrial queues with server subject to breakdowns and repairs. Math. Comput. Model. 42(5–6), 673–682 (2005)

    Article  MathSciNet  Google Scholar 

  3. Artalejo, J., Corral, A.G.: Retrial Queueing Systems: A Computational Approach. Springer, Heidelberg (2008)

    Book  Google Scholar 

  4. Bérczes, T., Sztrik, J., Tóth, Á., Nazarov, A.: Performance modeling of finite-source retrial queueing systems with collisions and non-reliable server using MOSEL. In: Vishnevskiy, V.M., Samouylov, K.E., Kozyrev, D.V. (eds.) DCCN 2017. CCIS, vol. 700, pp. 248–258. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66836-9_21

    Chapter  Google Scholar 

  5. Bhat, U.N.: An introduction to queueing theory. Modeling and analysis in applications. 2nd edn. Birkhäuser, Boston (2015)

    Google Scholar 

  6. Bossel, H.: Modeling and simulation. Springer, Heidelberg (2013)

    MATH  Google Scholar 

  7. Choi, B.D., Shin, Y.W., Ahn, W.C.: Retrial queues with collision arising from unslotted CSMA/CD protocol. Queueing Syst. 11(4), 335–356 (1992)

    Article  Google Scholar 

  8. Dragieva, V.I.: Number of retrials in a finite source retrial queue with unreliable server. Asia-Pac. J. Oper. Res. 31(2), 23 (2014)

    Article  MathSciNet  Google Scholar 

  9. Falin, G., Artalejo, J.: A finite source retrial queue. Eur. J. Oper. Res. 108, 409–424 (1998)

    Article  Google Scholar 

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

    Book  Google Scholar 

  11. Gharbi, N., Dutheillet, C.: An algorithmic approach for analysis of finite-source retrial systems with unreliable servers. Comput. Math. Appl. 62(6), 2535–2546 (2011)

    Article  MathSciNet  Google Scholar 

  12. Gharbi, N., Ioualalen, M.: GSPN analysis of retrial systems with servers breakdowns and repairs. Appl. Math. Comput. 174(2), 1151–1168 (2006)

    MathSciNet  MATH  Google Scholar 

  13. Gharbi, N., Mokdad, L., Ben-Othman, J.: A performance study of next generation cellular networks with base stations channels vacations. In: Global Communications Conference (GLOBECOM), pp. 1–6. IEEE (2015)

    Google Scholar 

  14. Gharbi, N., Nemmouchi, B., Mokdad, L., Ben-Othman, J.: The impact of breakdowns disciplines and repeated attempts on performances of small cell networks. J. Comput. Sci. 5(4), 633–644 (2014)

    Article  Google Scholar 

  15. Gómez-Corral, A., Phung-Duc, T.: Retrial queues and related models. Ann. Oper. Res. 247(1), 1–2 (2016)

    Article  MathSciNet  Google Scholar 

  16. Harchol-Balter, M.: Performance modeling and design of computer systems. Queueing Theory in Action. Cambridge University Press, New York (2013)

    MATH  Google Scholar 

  17. Ikhlef, L., Lekadir, O., Aïssani, D.: MRSPN analysis of Semi-Markovian finite source retrial queues. Ann. Oper. Res. 247(1), 141–167 (2016)

    Article  MathSciNet  Google Scholar 

  18. Kim, J.S.: Retrial queueing system with collision and impatience. Commun. Korean Math. Soc. 25(4), 647–653 (2010)

    Article  MathSciNet  Google Scholar 

  19. Kim, J., Kim, B.: A survey of retrial queueing systems. Ann. Oper. Res. 247(1), 3–36 (2016)

    Article  MathSciNet  Google Scholar 

  20. Kobayashi, H., Mark, B.L.: System modeling and analysis: Foundations of system performance evaluation. Pearson Education, India (2009)

    Google Scholar 

  21. Krishnamoorthy, A., Pramod, P.K., Chakravarthy, S.R.: Queues with interruptions: a survey. TOP 22(1), 290–320 (2014)

    Article  MathSciNet  Google Scholar 

  22. Kuki, A., T.Bérczes, Sztrik, J., Kvach, A.: Numerical analysis of retrial queueing systems with conflict of customers. J. Math. Sci. (2017). (submitted)

    Google Scholar 

  23. Kulkarni, V.G.: Modeling and analysis of stochastic systems. CRC Press, Boca Raton (2016)

    Google Scholar 

  24. Kumar, B.K., Vijayalakshmi, G., Krishnamoorthy, A., Basha, S.S.: A single server feedback retrial queue with collisions. Comput. Oper. Res. 37(7), 1247–1255 (2010)

    Article  MathSciNet  Google Scholar 

  25. Kvach, A., Nazarov, A.: Sojourn Time analysis of finite Source Markov retrial queuing system with collision. In: Dudin, A., Nazarov, A., Yakupov, R. (eds.) ITMM 2015. CCIS, vol. 564, pp. 64–72. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-25861-4_6

    Chapter  Google Scholar 

  26. Kvach, A.: Numerical research of a Markov closed retrial queueing system without collisions and with the collision of the customers. In: Proceedings of Tomsk State University. A series of physics and mathematics. Tomsk. Materials of the II All-Russian Scientific Conference, vol. 295, pp. 105–112. TSU Publishing House (2014). (in Russian)

    Google Scholar 

  27. Kvach, A., Nazarov, A.: Numerical research of a closed retrial queueing system M/GI/1//N with collision of the customers. In: Proceedings of Tomsk State University. A series of physics and mathematics. Tomsk. Materials of the III All-Russian Scientific Conference, vol. 297, pp. 65–70. TSU Publishing House (2015). (in Russian)

    Google Scholar 

  28. Kvach, A., Nazarov, A.: The research of a closed RQ-system M/GI/1//N with collision of the customers in the condition of an unlimited increasing number of sources. In: Probability Theory, Random Processes, Mathematical Statistics and Applications: Materials of the International Scientific Conference Devoted to the 80th Anniversary of Professor Gennady Medvedev, Doctor of Physical and Mathematical Sciences, pp. 65–70 (2015). (in Russian)

    Google Scholar 

  29. Lakatos, L., Szeidl, L., Telek, M.: Introduction to queueing systems with telecommunication applications. Springer, New York (2013)

    Book  Google Scholar 

  30. Law, A.M., Kelton, W.D.: Simulation modeling and analysis. McGraw-Hill, New York (1991)

    MATH  Google Scholar 

  31. Nazarov, A., Sztrik, J., Kvach, A., Bérczes, T.: Asymptotic analysis of finite-source M/M/1 retrial queueing system with collisions and server subject to breakdowns and repairs. Ann. Oper. Res. (2017). (submitted)

    Google Scholar 

  32. Nazarov, A., Sztrik, J., Kvach, A., Tóth, A.: Asymptotic sojourn time analysis of Markov finite-source M/M/1 retrial queueing system with collisions and server subject to breakdowns and repairs. Markov Processes and Related Fields (2017). (submitted)

    Google Scholar 

  33. Nazarov, A., Sudyko, E.: Method of asymptotic semi-invariants for studying a mathematical model of a random access network. Probl. Inf. Transm. 46(1), 86–102 (2010)

    Article  MathSciNet  Google Scholar 

  34. Nazarov, A., Terpugov, A.: Theory of Mass Service. NTL Publishing House, Tomsk (2004). (in Russian)

    Google Scholar 

  35. Nazarov, A., Kvach, A., Yampolsky, V.: Asymptotic analysis of closed markov retrial queuing system with collision. In: Dudin, A., Nazarov, A., Yakupov, R., Gortsev, A. (eds.) ITMM 2014. CCIS, vol. 487, pp. 334–341. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-13671-4_38

    Chapter  Google Scholar 

  36. 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_8

    Chapter  Google Scholar 

  37. Nazarov, A., Sztrik, J., Kvach, A.: Some features of a finite-source M/GI/1 retrial queuing system with collisions of customers. In: Vishnevskiy, V.M., Samouylov, K.E., Kozyrev, D.V. (eds.) DCCN 2017. CCIS, vol. 700, pp. 186–200. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66836-9_16

    Chapter  Google Scholar 

  38. Nazarov, A., Sztrik, J., Kvach, A.: Some features of a finite-source \(M/GI/1\) retrial queuing system with collisions of customers. In: Proceedings of International Conference on Distributed Computer and Communication Networks, DCCN 2017, pp. 79–86 (2017)

    Google Scholar 

  39. Nazarov, A., Moiseeva S.P.: Methods of asymptotic analysis in queueing theory. NTL Publishing House of Tomsk University (2006). (in Russian)

    Google Scholar 

  40. Peng, Y., Liu, Z., Wu, J.: An M/G/1 retrial G-queue with preemptive resume priority and collisions subject to the server breakdowns and delayed repairs. J. Appl. Math. Comput. 44(1–2), 187–213 (2014)

    Article  MathSciNet  Google Scholar 

  41. Roszik, J.: Homogeneous finite-source retrial queues with server and sources subject to breakdowns and repairs. Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Comput. 23, 213–227 (2004)

    MathSciNet  MATH  Google Scholar 

  42. Rubinstein, R.Y., Kroese, D.P.: Simulation and the Monte Carlo method. Wiley, Hoboken (2016)

    Book  Google Scholar 

  43. Stewart, W.J.: Probability, Markov chains, queues, and simulation. the mathematical basis of performance modeling. Princeton University Press, Princeton (2009)

    MATH  Google Scholar 

  44. Sztrik, J.: Tool supported performance modelling of finite-source retrial queues with breakdowns. Publicationes Mathematicae 66, 197–211 (2005)

    MathSciNet  MATH  Google Scholar 

  45. Sztrik, J., Almási, B., Roszik, J.: Heterogeneous finite-source retrial queues with server subject to breakdowns and repairs. J. Math. Sci. 132, 677–685 (2006)

    Article  MathSciNet  Google Scholar 

  46. Tóth, A., Bérczes, T., Sztrik, J., Kuki, A.: Comparison of two operation modes of finite-source retrial queueing systems with collisions and non-reliable server by using simulation. J. Math. Sci. (2017). (submitted)

    Google Scholar 

  47. Tóth, Á., Bérczes, T., Sztrik, J., Kvach, A.: Simulation of finite-source retrial queueing systems with collisions and non-reliable server. In: Vishnevskiy, V.M., Samouylov, K.E., Kozyrev, D.V. (eds.) DCCN 2017. CCIS, vol. 700, pp. 146–158. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66836-9_13

    Chapter  Google Scholar 

  48. Wang, J., Zhao, L., Zhang, F.: Performance analysis of the finite source retrial queue with server breakdowns and repairs. In: Proceedings of the 5th International Conference on Queueing Theory and Network Applications, pp. 169–176. ACM (2010)

    Google Scholar 

  49. Wang, J., Zhao, L., Zhang, F.: Analysis of the finite source retrial queues with server breakdowns and repairs. J. Ind. Manag. Optim. 7(3), 655–676 (2011)

    Article  MathSciNet  Google Scholar 

  50. Wehrle, K., Günes, M., Gross, J.: Modeling and tools for network simulation. Springer, Heidelberg (2010)

    Book  Google Scholar 

  51. Wüchner, P., Sztrik, J., de Meer, H.: Finite-source retrial queues with applications. In: Proceedings of 8th International Conference on Applied Informatics, Eger, Hungary. vol. 2, pp. 275–285 (2010)

    Google Scholar 

  52. Yao, J.: Asymptotic Analysis of Service Systems with Congestion-Sensitive Customers. Columbia University (2016)

    Google Scholar 

  53. Zhang, F., Wang, J.: Performance analysis of the retrial queues with finite number of sources and service interruptions. J. Korean Stat. Soc. 42(1), 117–131 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The work/publication of J. Sztrik is supported by the EFOP-3.6.1-16-2016-00022 project. The project is co-financed by the European Union and the European Social Fund.

Author information

Authors and Affiliations

  1. National Research Tomsk State University, 36 Lenina ave., Tomsk, 634050, Russia

    Anatoly Nazarov & Anna Kvach

  2. University of Debrecen, Debrecen, Hungary

    János Sztrik

Authors
  1. Anatoly Nazarov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. János Sztrik
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Anna Kvach
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to János Sztrik .

Editor information

Editors and Affiliations

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

    Alexander Dudin

  2. Institute of Applied Mathematics and Computer Science, Tomsk State University, Tomsk, Russia

    Anatoly Nazarov

  3. Institute of Applied Mathematics and Computer Science, Tomsk State University, Tomsk, Russia

    Alexander Moiseev

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

Nazarov, A., Sztrik, J., Kvach, A. (2018). A Survey of Recent Results in Finite-Source Retrial Queues with Collisions. In: Dudin, A., Nazarov, A., Moiseev, A. (eds) Information Technologies and Mathematical Modelling. Queueing Theory and Applications. ITMM WRQ 2018 2018. Communications in Computer and Information Science, vol 912. Springer, Cham. https://doi.org/10.1007/978-3-319-97595-5_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-97595-5_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-97594-8

  • Online ISBN: 978-3-319-97595-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation