Skip to main content
SpringerLink
Log in

Discrete-time Geo/G/1 retrial queues with general retrial time and Bernoulli vacation

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

This paper considers a discrete-time Geo/G/1 retrial queue where the retrial time has a general distribution and the server is subject to Bernoulli vacation policy. It is assumed that the server, after each service completion, begins a process of search in order to find the following customer to be served with a certain probability, or begins a single vacation process with complementary probability. This paper analyzes the Markov chain underlying the queueing system and obtain its ergodicity condition. The generating functions of the number of customers in the orbit and in the system are also obtained along with the marginal distributions of the orbit size when the server is idle, busy or on vacation. Finally, the author gives two stochastic decomposition laws, and as an application the author gives bounds for the proximity between the system size distributions of the model and the corresponding model without retrials.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. Bruneel and B. G. Kim, Discrete-time models for communication systems including ATM, Kluwer Academic Publishers, Boston, 1993.

    Book  Google Scholar 

  2. J. J. Hunter, Discrete-time models: Techniques and applications, Mathematical Techniques of Applied Probability, Vol.2, Academic Press, New York, 1983.

    Google Scholar 

  3. H. Takagi, Queueing analysis: A foundation of performance evaluation, Discrete-Time Systems, Vol.3, North-Holland, Amsterdam, 1993.

    Google Scholar 

  4. M. E. Woodward, Communication and Computer Networks: Modelling with Discrete-time Queues, IEEE Computer Soc. Press, Los Alamitos, CA, 1994.

    Google Scholar 

  5. G. I. Falin and J. G. C. Templeton, Retrial Queues, Chapman & Hall, London, 1997.

    MATH  Google Scholar 

  6. J. R. Artalejo and A. Gómez-Corral, Retrial Queueing Systems: A Computational Approach, Berlin: Springer, 2008.

    Book  MATH  Google Scholar 

  7. T. Yang and H. Li, On the steady-state queue size distribution of the discrete-time Geo/G/1 queue with repeated customers, Queueing System, 1995, 21: 199–215.

    Article  MATH  Google Scholar 

  8. I. Atencia and P. Moreno, Discrete-time Geo [X]/G H/1 retrial queue with Bernoulli feedback, Computers and Mathematics with Applications, 2004, 47(8–9): 1273–1294.

    Article  MathSciNet  MATH  Google Scholar 

  9. B. D. Choi and J.W. Kim, Discrete-time Geo 1,Geo 2/G/1 retrial queueing system with two types of calls, Computers and Mathematics with Applications, 1997, 33(10): 79-88.

  10. H. Li and T. Yang, Geo/G/1 discrete-time retrial queue with Bernoulli schedule, European Journal of Operational Research, 1998, 111(3): 629–649.

    Article  MATH  Google Scholar 

  11. H. Li and T. Yang, Steady-state queue size distribution of discrete-time PH/Geo/1 retrial queues, Mathematical and Computer Modelling, 1999, 30: 51–63.

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Takahashi, H. Osawa, and T. Fujisawa, Geo [X]/G/1 retrial queue with non-preemptive priority, Asia-Pacific Journal of Operational Research, 1999, 16: 215–234.

    MathSciNet  MATH  Google Scholar 

  13. I. Atencia and P. Moreno, A discrete-time Geo/G/1 retrial queue with general retrial times, Queueing Systems, 2004, 48: 5–21.

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Wang and Q. Zhao, Discrete-time Geo/G/1 retrial queue with general retrial times and starting failures, Mathematical and Computer Modelling, 2007, 45: 853–863.

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Wang and Q. Zhao, A discrete-time Geo/G/1 retrial queue with starting failures and second optional service, Computers and Mathematics with Applications, 2007, 53: 115–127.

    Article  MathSciNet  MATH  Google Scholar 

  16. J. Wang and P. Zhao, A discrete-time retrial queue with negative customers and unreliable server, Computers & Industrial Engineering, 2009, 56: 1216–1222.

    Article  Google Scholar 

  17. A. K. Aboul-Hassan, S. Rabia, and F. Taboly, A discrete-time Geo/G/1 retrial queue with general retrial times and balking customers, Journal of the Korean Statistical Society, 2008, 37: 335–348.

    Article  MathSciNet  Google Scholar 

  18. A. K. Aboul-Hassan, S. I. Rabia, and F. A. Taboly, Performance evaluation of a discrete-time Geo [X]/G/1 retrial queue with general retrial times, Computers and Mathematics with Applications, 2009, 58: 548–557.

    Article  MathSciNet  MATH  Google Scholar 

  19. B. K. Kumar and D. Arivudainambi, The M/G/1 retrial queue with Bernoulli schedules and general retrial times, Computers & Mathematics with Applications, 2002, 43: 15–30.

    Article  MathSciNet  MATH  Google Scholar 

  20. B. K. Kumar, D. Arivudainambi, and A. Vijayakumar, On the M X/G/1 retrial queue with Bernoulli schedules and general retrial times, Asia-Pacific Journal of Operational Research, 2002, 19: 177–194.

    MathSciNet  MATH  Google Scholar 

  21. S. W. Fuhrmann and R. B. Cooper, Stochastic decomposition in the M/G/1 queue with generalized vacations, Operations Research, 1985, 33: 1117–1129.

    Article  MathSciNet  MATH  Google Scholar 

  22. J. R. Artalejo and G. I. Falin, Stochastic decomposition for retrial queues, TOP, 1994, 2: 329–342.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, China

    Jinting Wang

Authors
  1. Jinting Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jinting Wang.

Additional information

This research is supported by the National Natural Science Foundation of China under Grant No. 11171019, the Fundamental Research Funds for the Central Universities under Grant No. 2011JBZ012, and the Program for New Century Excellent Talents in University under Grant No. NCET-11-0568.

This paper was recommended for publication by Editor Hanqin ZHANG.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, J. Discrete-time Geo/G/1 retrial queues with general retrial time and Bernoulli vacation. J Syst Sci Complex 25, 504–513 (2012). https://doi.org/10.1007/s11424-012-0254-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-012-0254-7

Key words

Search

Navigation