Skip to main content
SpringerLink
Log in

Reliability analysis of an M/G/1 repairable queueing system with vacations and patient server

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

This paper considers an M/G/1 repairable queueing system with vacations and patient server. It is assumed that the service station may break down during service period. Whenever a busy period ends, the server is allowed to take a vacation. When the server returns from vacation and finds an idle system, he/she waits for a patience period within the system. If a customer arrives during this period, a new busy period starts at once. Otherwise, a new vacation begins immediately at the completion epoch of the patience period. Using the total probability decomposition technique and the Laplace transform tool, some crucial reliability indices of the service station are derived, including the distribution of the time to first failure, the transient and steady-state unavailability, the mean number of failures during time interval (0, t] and the steady-state failure frequency. The results of our study indicate that the reliability measures obtained satisfy the stochastic decomposition property. Finally, some numerical examples are provided to illustrate the influence of various system parameters on these reliability indices.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Levy, Y., Yechiali, U.: Utilization of idle time in an M/G/1 queueing system. Manage. Sci. 22(2), 202–211 (1975)

    Article  Google Scholar 

  2. Doshi, B.T.: Queueing systems with vacations: a survey. Queue. Syst. 1, 29–66 (1986)

    Article  MathSciNet  Google Scholar 

  3. Ke, J.C., Wu, C.H., Zhang, Z.G.: Recent developments in vacation queueing models: a short survey. Int. J. Operat. Res. 7(4), 3–8 (2010)

    Google Scholar 

  4. Upadhyaya, S.: Queueing systems with vacation: an overview. Int. J. Math. Operat. Res. 9(2), 167–213 (2016)

    Article  MathSciNet  Google Scholar 

  5. Takagi, H.: Queueing analysis: a foundation of performance evaluation, vol. 1. North-Holland, Amsterdam (1991)

  6. Tian, N., Zhang, Z.G.: Vacation queueing models: theory and applications, vol. 93. Springer, New York (2006)

  7. Boxma, O.J., Schlegel, S., Yechiali, U.: A note on an M/G/1 queue with a waiting server, timer and vacations. Am. Math. Soc. Transl. 207, 25–35 (2002)

    MathSciNet  Google Scholar 

  8. Yechiali, U.: On the MX/G/1 queue with a waiting server and vacations. Sankhyā 66, 159–174 (2004)

    MathSciNet  Google Scholar 

  9. Cao, J.H., Cheng, K.: Analysis of M/G/1 queueing system with repairable service station. Acta Math. Appl. Sin. 5(2), 113–127 (1982)

    Article  MathSciNet  Google Scholar 

  10. Tang, Y.H.: A single-server M/G/1 queueing system subject to breakdowns: some reliability and queueing problems. Microelectron. Reliab. 37(2), 315–321 (1997)

    Article  Google Scholar 

  11. Ke, J.C.: Optimal NT policies for M/G/1 system with a startup and unreliable server. Comput. Indus. Eng. 50(3), 248–262 (2006)

    Article  Google Scholar 

  12. Yang, D.Y., Wang, K.H., Pearn, W.L.: Steady-state probability of the randomized server control system with second optional service, server breakdowns and startup. J. Appl. Math. Comput. 32, 39–58 (2010)

    Article  MathSciNet  Google Scholar 

  13. Gao, S., Wang, J.: Performance and reliability analysis of an M/G/1-G retrial queue with orbital search and non-persistent customers. Eur. J. Oper. Res. 236(2), 561–572 (2014)

    Article  MathSciNet  Google Scholar 

  14. Kumar, A., Jain, M.: Cost optimization of an unreliable server queue with two stage service process under hybrid vacation policy. Math. Comput. Simul. 204, 259–281 (2023)

    Article  MathSciNet  Google Scholar 

  15. Yang, D.Y., Chang, F.M., Ke, J.C.: On an unreliable retrial queue with general repeated attempts and J optional vacations. Appl. Math. Model. 40(4), 3275–3288 (2016)

    Article  MathSciNet  Google Scholar 

  16. Choudhury, G., Deka, M.: A batch arrival unreliable server delaying repair queue with two phases of service and Bernoulli vacation under multiple vacation policy. Qual. Technol. Quantitat. Manag. 15(2), 157–186 (2018)

    Article  Google Scholar 

  17. Chang, F.M., Liu, T.H., Ke, J.C.: On an unreliable-server retrial queue with customer feedback and impatience. Appl. Math. Model. 55, 171–182 (2018)

    Article  MathSciNet  Google Scholar 

  18. Jain, M., Kaur, S.: Bernoulli vacation model for MX/G/1 unreliable server retrial queue with Bernoulli feedback, balking and optional service. RAIRO-Operat. Res. 55, 2027–2053 (2021)

    Article  Google Scholar 

  19. Meena, R.K., Jain, M., Assad, A., et al.: Performance and cost comparative analysis for M/G/1 repairable machining system with N-policy vacation. Math. Comput. Simul. 200, 315–328 (2022)

    Article  MathSciNet  Google Scholar 

  20. Liu, T.H., Chang, F.M., Ke, J.C., et al.: Optimization of retrial queue with unreliable servers subject to imperfect coverage and reboot delay. Qual. Technol. Quantitat. Manag. 19(4), 428–453 (2022)

    Article  Google Scholar 

  21. Begum, A., Choudhury, G.: Analysis of an unreliable N-policy queue with set-up period under Bernoulli schedule and re-service approach. Commun. Stat. Theory Methods. 1–33 (2022)

  22. Cao, J.H., Cheng, K.: Introduction to Reliability Mathematics. Higher Education Press, Beijing (2006)

    Google Scholar 

Download references

Acknowledgements

The authors would like to sincerely thank the anonymous reviewers and editors for their valuable comments and constructive suggestions which are very helpful in improving the presentation of this paper. This research is supported by the Opening Project of Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing (No. 2023QYJ04), the Faculty Research Grants Awarded by Principal’s Funds (CXTD2022013), and the Natural Science Foundation of Sichuan Province of China (Grant No. 2023NSFSC1021).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong, 643000, Sichuan, China

    Siyan Wan & Shaojun Lan

Authors
  1. Siyan Wan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shaojun Lan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shaojun Lan.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this work.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wan, S., Lan, S. Reliability analysis of an M/G/1 repairable queueing system with vacations and patient server. J. Appl. Math. Comput. 69, 4333–4352 (2023). https://doi.org/10.1007/s12190-023-01930-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-023-01930-3

Keywords

MSC Classification

Search

Navigation