SpringerLink
Log in

Analysis of an \(M/\begin{pmatrix}G_1\\ G_2\end{pmatrix}/1\) Queue with Bernoulli Vacation and Server Breakdown

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

In this article, an \(M/\begin{pmatrix} G_1 \\ G_2 \end{pmatrix}/1\) queue with service interruption consisting of a definite repairability is analyzed in the steady-state regime. Here, the server, after completion of a service, is allowed to take in a single vacation under the Bernoulli schedule. For this model, probability generating function (PGF) for queue length distribution at arbitrary and service completion epoch are derived. Laplace Stieltjes Transform (LST) of a busy period distribution and waiting time distribution are obtained. In addition to these, some important performance measures such as the mean queue size and the mean waiting time of a customer are achieved. The reliability indices for this model are carried out and are included in the study. An optimal operation policy of the model in terms of total expected cost is developed at a lower cost. Finally, various numerical examples are presented in support of the theory.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

Availability of data and material

Not applicable.

References

  1. Abbas, K., Aïssani, D.: Approximation of performance measures in an M/G/1 queue with breakdowns. Qual. Technol. Quant. Manage. 7(4), 353–363 (2010)

    Google Scholar 

  2. Avi-ltzhak, B., Naor, P.: Some queueing problems with the service station subject to breakdowns. Oper. Res. 11(3), 303–320 (1963)

    MATH  Google Scholar 

  3. Ayyappan, G., Gowthami, R.: Analysis of MAP/PH(1), PH(2)/2 queue with bernoulli schedule vacation, bernoulli feedback and renege of customers. Int. J. Appl. Comput. Math. 5(6), 1–47 (2019)

    MathSciNet  MATH  Google Scholar 

  4. Ayyappan, G., Gowthami, R.: A MAP/PH/1 queue with setup time, bernoulli schedule vacation, balking and bernoulli feedback. Int. J. Appl. Comput. Math. 8(2), 1–21 (2022)

    MathSciNet  MATH  Google Scholar 

  5. Ayyappan, G., Nirmala, M., Karpagam, S.: Analysis of repairable single server bulk queue with standby server, two phase heterogeneous service, starting failure and multiple vacation. Int. J. Appl. Comput. Math. 6(2), 1–22 (2020)

    MathSciNet  MATH  Google Scholar 

  6. Begum, A., Choudhury, G.: Analysis of an unreliable single server batch arrival queue with two types of services under bernoulli vacation policy. Commun. Stat. Theory Methods 50(9), 2136–2160 (2021)

    MathSciNet  MATH  Google Scholar 

  7. Bellman, R.: Dynamic Programming. R and Corporation Research Study. Princeton University Press, New York (1957)

    Google Scholar 

  8. Chaudhry, M.L., Templeton, J.G.C.: A First Course in Bulk Queues. Wiley, New York (1983)

    MATH  Google Scholar 

  9. Choudhury, G., Deka, M.: A single server queueing system with two phases of service subject to server breakdown and bernoulli vacation. Appl. Math. Model. 36(12), 6050–6060 (2012)

    MathSciNet  MATH  Google Scholar 

  10. Choudhury, G., Kalita, C.R.: An M/G/1 queue with two types of general heterogeneous service and optional repeated service subject to server’s breakdown and delayed repair. Qual. Technol. Quant. Manage. 15(5), 622–654 (2018)

    Google Scholar 

  11. Choudhury, G., Ke, J.C.: An unreliable retrial queue with delaying repair and general retrial times under bernoulli vacation schedule. Appl. Math. Comput. 230, 436–450 (2014)

    MathSciNet  MATH  Google Scholar 

  12. Choudhury, G., Madan, K.C.: A two phase batch arrival queueing system with a vacation time under bernoulli schedule. Appl. Math. Comput. 149(2), 337–349 (2004)

    MathSciNet  MATH  Google Scholar 

  13. Choudhury, G., Tadj, L.: The optimal control of an \(\mathbb{M} ^x\)/G/1 unreliable server queue with two phases of service and bernoulli vacation schedule. Math. Comput. Model. 54(1–2), 673–688 (2011)

    MATH  Google Scholar 

  14. Cox D (1962) Renewal Theory. Methuen’s Monographs on Applied Probability and Statistics, Methuen. Wiley: New York

  15. Cox, D.R.: The analysis of non-markovian stochastic processes by the inclusion of supplementary variables. Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press, Vol. 51, pp. 433–441 (1955)

  16. Deora, P., Kumari, U., Sharma, D.: Cost analysis and optimization of machine repair model with working vacation and feedback-policy. Int. J. Appl. Comput. Math. 7(6), 1–14 (2021)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  18. Gaver, D., Jr.: A waiting line with interrupted service, including priorities. J. Roy. Stat. Soc.: Ser. B (Methodol.) 24(1), 73–90 (1962)

    MathSciNet  MATH  Google Scholar 

  19. Gerasimov.: Optimum time-sharing algorithm for a servicing device in a queuing system with interruptions. Autom Control Comput Sci 7(6), 50–55 (1973)

  20. Govindan, A., Pakiradhan, T.: Transient analysis of \(\mathbb{M}^{X_{1}},\mathbb{M}^{X_{2}}\)/\({G_1},{G_2}\)/1 retrial queueing system with priority services, working vacations and vacation interruption, emergency vacation, negative arrival and delayed repair. Int. J. Appl. Comput. Math. 4 (2018)

  21. Jiang, T., **n, B.: Computational analysis of the queue with working breakdowns and delaying repair under a bernoulli-schedule-controlled policy. Commun. Stat. Theory Methods 48(4), 926–941 (2019)

    MathSciNet  MATH  Google Scholar 

  22. Kalita, C.R., Choudhury, G.: A note on reliability analysis of an n-policy unreliable \(\mathbb{M} ^x\)/\({\begin{pmatrix} G_1 \ G_2\end{pmatrix}}\)/1 queue with optional repeated service. RAIRO-Oper. Res. 52(3), 713–724 (2018)

    MathSciNet  MATH  Google Scholar 

  23. Kalita, C.R., Choudhury, G.: Analysis of an unreliable \(\mathbb{M} ^x\)/\({\begin{pmatrix} G_1 \ G_2\end{pmatrix}}\)/1/repeated service queue with delayed repair under randomized vacation policy. Commun. Stat. Theory Methods 48(21), 5336–5369 (2019)

    MATH  Google Scholar 

  24. Ke, J.C., Huang, K.B.: Analysis of an unreliable server \(\mathbb{M} ^X\)/G/1 system with a randomized vacation policy and delayed repair. Stoch. Model. 26(2), 212–241 (2010)

    MATH  Google Scholar 

  25. Ke, J.C., Huang, K.B., Pearn, W.L.: A batch arrival queue under randomised multi-vacation policy with unreliable server and repair. Int. J. Syst. Sci. 43(3), 552–565 (2012)

    MathSciNet  MATH  Google Scholar 

  26. Keilson, J., Servi, L.: Oscillating random walk models for GI/G/1 vacation systems with bernoulli schedules. J. Appl. Probab. 23(3), 790–802 (1986)

    MathSciNet  MATH  Google Scholar 

  27. Keilson, J., Servi, L.D.: The distributional form of little’s law and the fuhrmann-cooper decomposition. Oper. Res. Lett. 9(4), 239–247 (1990)

    MathSciNet  MATH  Google Scholar 

  28. Khalaf, R.F., Madan, K.C., Lukas, C.A.: An \(\mathbb{M} ^X\)/G/1 queue with bernoulli schedule, general vacation times, random breakdowns, general delay times and general repair times. Appl. Math. Sci. 5(1), 35–51 (2011)

    MathSciNet  MATH  Google Scholar 

  29. Krishnamoorthy, A., Pramod, P., Deepak, T.: On a queue with interruptions and repeat or resumption of service. Nonlinear Anal. Theory Methods Appl. 71(12), e1673–e1683 (2009)

    MathSciNet  MATH  Google Scholar 

  30. Li, T., Zhang, L., Gao, S.: An M/G/1 retrial queue with balking customers and bernoulli working vacation interruption. Qual. Technol. Quant. Manage. 16(5), 511–530 (2019)

    Google Scholar 

  31. Li, T., Zhang, L., Gao, S.: An M/G/1 retrial queue with single working vacation under bernoulli schedule. RAIRO-Oper. Res. 54(2), 471–488 (2020)

    MathSciNet  MATH  Google Scholar 

  32. Li, W., Shi, D., Chao, X.: Reliability analysis of M/G/1 queueing systems with server breakdowns and vacations. J. Appl. Probab. 34(2), 546–555 (1997)

    MathSciNet  MATH  Google Scholar 

  33. Madan, K., Al-Rawi, Z., Al-Nasser, A.: On \(\mathbb{M} ^x\)/\({\begin{pmatrix} G_1 \ G_2\end{pmatrix}}\)/1/G(BS)/Vs vacation queue with two types of general heterogeneous service. J. Appl. Math. Decis. Sci. 3, 123–135 (2005)

    Google Scholar 

  34. Madan, K.C.: An M/G/1 type queue with time-homogeneous breakdowns and deterministic repair times. Soochow J. Math. 29(1), 103–110 (2003)

    MathSciNet  MATH  Google Scholar 

  35. Madan, K.C., Al-Nasser, A.D., Al-Masri, A.Q.: On \(\mathbb{M} ^x\)/\({\begin{pmatrix} G_1 \ G_2\end{pmatrix}}\)/1 queue with optional re-service. Appl. Math. Comput. 152(1), 71–88 (2004)

    MathSciNet  Google Scholar 

  36. Maraghi, F.A., Madan, K.C., Darby-Dowman, K.: Batch arrival queueing system with random breakdowns and bernoulli schedule server vacations having general vacation time distribution. Int. J. Inf. Manage. Sci. 20(1), 55–70 (2009)

    MathSciNet  MATH  Google Scholar 

  37. Medhi, J.: Recent Developments in Bulk Queueing Models. Wiley Eastern, New Delhi (1984)

    Google Scholar 

  38. Medhi, J.: Stochastic Models in Queueing Theory, 2nd edn. Academic Press, Boston and San Diego (2003)

    MATH  Google Scholar 

  39. Tadj, L., Ke, J.: A hysteretic bulk quorum queue with a choice of service and optional re-service. Qual. Technol. Quant. Manage. 5(2), 161–178 (2008)

    MathSciNet  Google Scholar 

  40. Takács, L.: Introduction to the Theory of Queues. Oxford University Press, New York (1962)

    MATH  Google Scholar 

  41. Takagi, H.: Queueing Analysis: A Foundation of Performance Evaluation, vol. 1. Elsevier Science Publishers, North Holland, Amsterdam (1991)

    MATH  Google Scholar 

  42. Thangaraj, V., Choudhury, G.: Stochastic Modeling in Physical and Biological Sciences. Narosa Publishing House (2016)

  43. Thiruvengadam, K.: Queuing with breakdowns. Oper. Res. 11(1), 62–71 (1963)

    MathSciNet  MATH  Google Scholar 

  44. Tian, N., Zhang, ZG.: Vacation queueing models: theory and applications, vol 93. Springer Science & Business Media (2006)

  45. Upadhyaya, S.: Admission control of bulk retrial feedback queue with k-optional vacations. Int. J. Math. Oper. Res. 7(2), 215–239 (2015)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  47. Upadhyaya, S.: Investigating a general service retrial queue with damaging and licensed units: an application in local area networks. Opsearch 57(3), 716–745 (2020)

    MathSciNet  MATH  Google Scholar 

  48. Upadhyaya, S., Kushwaha, C.: Performance prediction and anfis computing for unreliable retrial queue with delayed repair under modified vacation policy. Int. J. Math. Oper. Res. 17(4), 437–466 (2020)

    MathSciNet  MATH  Google Scholar 

  49. Wang, J.: An M/G/1 queue with second optional service and server breakdowns. Comput. Math. Appl. 47(10–11), 1713–1723 (2004)

    MathSciNet  MATH  Google Scholar 

  50. White, H., Christie, L.S.: Queuing with preemptive priorities or with breakdown. Oper. Res. 6(1), 79–95 (1958)

    MathSciNet  MATH  Google Scholar 

  51. Wolff, R.W.: Poisson arrivals see time averages. Oper. Res. 30(2), 223–231 (1982)

    MathSciNet  MATH  Google Scholar 

  52. Zhang, ZG., Vickson, R., Love, E.: The optimal service policies in an m/g/1 queueing system with multiple vacation types. INFOR: Information Systems and Operational Research 39(4):357–366 (2001)

  53. Zirem, D., Boualem, M., Adel-Aissanou, K., Aïssani, D.: Analysis of a single server batch arrival unreliable queue with balking and general retrial time. Qual. Technol. Quant. Manage. 16(6), 672–695 (2019)

    Google Scholar 

Download references

Acknowledgements

The first author acknowledges the University Grants Commission (UGC), Government of India for providing financial assistance to carry out this research work through the UGC-MANF scheme vide award number F1-17.1/2017-18/MANF-2017-18-ASS-87946.

Funding

Not applicable.

Author information

Authors and Affiliations

  1. Mathematical Sciences Division, Institute of Advanced Study in Science and Technology, Paschim Boragaon, Guwahati, 781035, Assam, India

    Anjana Begum & Gautam Choudhury

Authors
  1. Anjana Begum
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gautam Choudhury
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

GC initiated and supervised the project. AB is responsible for the performance of calculations, analysis of results and preparation of the manuscript.

Corresponding author

Correspondence to Anjana Begum.

Ethics declarations

Ethics approval and consent to participate

Not applicable.

Conflict of interest

The authors declare that they have no conflict or competing interests.

Consent for publication

Not applicable.

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

Begum, A., Choudhury, G. Analysis of an \(M/\begin{pmatrix}G_1\\ G_2\end{pmatrix}/1\) Queue with Bernoulli Vacation and Server Breakdown. Int. J. Appl. Comput. Math 9, 9 (2023). https://doi.org/10.1007/s40819-022-01481-4

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40819-022-01481-4

Keywords

Mathematics Subject Classification

Search

Navigation