Skip to main content
SpringerLink
Log in

Cost optimization analysis for an \(M^{X}/M/c\) vacation queueing system with waiting servers and impatient customers

  • Published:
SeMA Journal Aims and scope Submit manuscript

Abstract

This paper deals with the study of an \(M^{X}/M/c\) Bernoulli feedback queueing system with waiting servers and two different policies of synchronous vacations (single and multiple vacation policies). During vacation period, the customers may leave the system (reneging), and using certain customer retention mechanism, the reneged customers may be retained in the system. The probability generating function (PGF) has been used to obtain the steady state probabilities of the model. Various performances measures of the system are derived. Then, a cost model is developed. Further, a cost optimization problem is considered using quadratic fit search method. Finally, a variety of numerical illustrations are discussed for the applicability of the model.

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
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18

Similar content being viewed by others

References

  1. Altman, E., Yechiali, U.: Analysis of customers’ impatience in queues with server vacation. Queue Syst. 52(4), 261–279 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Altman, E., Yechiali, U.: Infinite-server queues with system’s additional tasks and impatient customers. Probab. Eng. Inf. Sci. 22(4), 477–493 (2008)

    MathSciNet  MATH  Google Scholar 

  3. Ammar, S.I.: Transient analysis of an \(M/M/1\) queue with impatient behavior and multiple vacations. Appl. Math. Comput. 260(C), 97–105 (2015)

    MathSciNet  MATH  Google Scholar 

  4. Ammar, S.I.: Transient solution of an \(M/M/1\) vacation queue with a waiting server and impatient customers. J. Egypt. Math. Soc. 25(3), 337–342 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Baruah, M., Madan, K.C., Eldabi, T.: Balking and re-service in a vacation queue with batch arrival and two types of heterogeneous service. J. Math. Res. 4(4), 114–124 (2012)

    Article  Google Scholar 

  6. Baruah, M., Madan, K.C., Eldabi, T.: A batch arrival queue with second optional service and reneging during vacation periods. Rev. Invest. Oper. 34(3), 244–258 (2013)

    MathSciNet  MATH  Google Scholar 

  7. Bouchentouf, A.A., Yahiaoui, L.: On feedback queueing system with reneging and retention of reneged customers, multiple working vacations and Bernoulli schedule vacation interruption. Arab. J. Math. 6(1), 1–11 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chia, H.W., Jau-Chaun, K.: Computational algorithm and parameter optimization for a multi-server system with unreliable servers and impatient customers. J. Comput. Appl. Math. 235(3), 547–562 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Ke, J.-C.: Operating characteristic analysis on the \(M^{X}/G/1\) system with a variant vacation policy and balking. Appl. Math. Model. 31(7), 1321–1337 (2007)

    Article  MATH  Google Scholar 

  11. Laxmi, P.V., Goswami, V., Jyothsna, K.: Optimization of balking and reneging queue with vacation interruption under N-policy. J. Optim. 683708, 9 (2013). https://doi.org/10.1155/2013/683708

    MATH  Google Scholar 

  12. Lee, H.W., Lee, S.S., Chae, K.: A fixed-size batch service queue with vacations. Int. J. Stochas. Anal. 9(2), 205–219 (1996)

    MathSciNet  MATH  Google Scholar 

  13. Misra, C., Goswami, V.: Analysis of power saving class II traffic in IEEE 802.16E with multiple sleep state and balking. Found. Comput Decis. Sci. 40(1), 53–66 (2015)

    Article  MathSciNet  Google Scholar 

  14. Majid, S., Manoharan, P.: Analysis of a \(M/M/c\) queue with single and multiple synchronous working vacations. Appl. Appl. Math. 12(2), 671–694 (2017)

    MathSciNet  MATH  Google Scholar 

  15. Padmavathy, R., Kalidass, K., Ramanath, K.: Vacation queues with impatient customers and a waiting server. Int. J. Latest Trends Softw. Eng. 1(1), 10–19 (2011)

    Google Scholar 

  16. Panda, G., Goswami, V.: Equilibrium balking strategies in renewal input queue with Bernoulli-schedule controlled vacation and vacation interruption. J. Ind. Manag. Optim. 12(3), 851–878 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Rardin, R.L.: Optimization in Operations Research. Prentice-Hall, Upper Saddle River (1997)

    Google Scholar 

  18. Sasikala, S., Indhira, K., Chandrasekaran, V.M.: Bulk queueing system with multiple vacations, N-policy, balking and control policy on request for re-service. Int. J. Pure Appl. Math. 115(9), 459–469 (2017)

    Google Scholar 

  19. Sun, W., Li, S., Guo, E.: Equilibrium and optimal balking strategies of customers in Markovian queues with multiple vacations and N-policy. Appl. Math. Model. 40(1), 284–301 (2016)

    Article  MathSciNet  Google Scholar 

  20. Takagi, H.: Queueing Analysis: A Foundation of Performance Evaluation, Volume 1: Vacation and Priority System. Elsevier, Amsterdam (1991)

    MATH  Google Scholar 

  21. Tian, N., Zhang, Z.G.: Vacation Queueing Models: Theory and Applications. Springer, New York (2006)

    Book  MATH  Google Scholar 

  22. Wang, K.H., Chan, M.C., Ke, J.C.: Maximum entropy analysis of the \(M^{X}/M/1\) queueing system with multiple vacations and server breakdowns. Comput. Ind. Eng. 52(2), 192–202 (2007)

    Article  Google Scholar 

  23. Yue, D., Yue, W., Sun, Y.: Performance analysis of an \(M/M/c/N\) queueing system with balking, reneging and synchronous vacations of partial servers. In: Proceedings of the 6th international symposium on operations research and its applications (ISORA’06), pp. 128–143 (2006)

  24. Yue, D., Yue, W., Zhao, G.: Analysis of an \(M/M/1\) queue with vacations and impatience timers which depends on the server’s states. J. Ind. Manag. Optim. 12(2), 653–666 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yue, D., Zhang, Y., Yue, W.: Optimal performance analysis of an \(M/M/1/N\) queue system with balking, reneging and server vacation. Int. J. Pure Appl. Math. 28(1), 101–115 (2006)

    MathSciNet  MATH  Google Scholar 

  26. Yue, D., Yue, W., Zhao, G.: Analysis of an \(M/M/c\) queueing system with impatient customers and synchronous vacations. J. Appl. Math. 2014(893094), 11 (2014). https://doi.org/10.1155/2014/893094

    Google Scholar 

  27. Zhang, Y., Yue, D., Yue, W.: Analysis of an \(M/M/1/N\) queue with balking, reneging and server vacations. In: Proceedings of the 5th international symposium on or and its applications, pp. 37–47 (2005)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Laboratory of Mathematics, Djillali Liabes University of Sidi Bel Abbes, B. P. 89, 22000, Sidi Bel Abbès, Algeria

    Amina Angelika Bouchentouf

  2. Laboratory of Stochastic Models, Statistic and Applications, Dr. Tahar Moulay University of Saida, B. P. 138, En-nasr, 20000, Saida, Algeria

    Abdelhak Guendouzi

Authors
  1. Amina Angelika Bouchentouf
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abdelhak Guendouzi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Amina Angelika Bouchentouf.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bouchentouf, A.A., Guendouzi, A. Cost optimization analysis for an \(M^{X}/M/c\) vacation queueing system with waiting servers and impatient customers. SeMA 76, 309–341 (2019). https://doi.org/10.1007/s40324-018-0180-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40324-018-0180-2

Keywords

Mathematics Subject Classification

Search

Navigation