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A Recursive Algorithm for State Dependent GI/M/1/N Queue with Bernoulli-Schedule Vacation

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Journal of Mathematical Modelling and Algorithms in Operations Research

Abstract

In this paper, we study a renewal input working vacations queue with state dependent services and Bernoulli-schedule vacations. The model is analyzed with single and multiple working vacations. The server goes for exponential working vacation whenever the queue is empty and the vacation rate is state dependent. At the instant of a service completion, the vacation is interrupted and the server resumes a regular busy period with probability 1 − q (if there are customers in the queue), or continues the vacation with probability q (0 ≤ q ≤ 1). We provide a recursive algorithm using the supplementary variable technique to numerically compute the stationary queue length distribution of the system. Finally, using some numerical results, we present the parameter effect on the various performance measures.

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References

  1. Baba, Y.: Analysis of a GI/M/1 queue with multiple working vacations. Oper. Res. Lett. 33, 201–209 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Banik, A.D.: Analysis of single working vacation in GI/M/1/N and GI/M/1 queueing systems. Int. J. Oper. Res. 7, 314–333 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Banik, A.D., Gupta, U.C., Pathak, S.S.: On the GI/M/1/N queue with multiple working vacations - analytic analysis and computation. Appl. Math. Model. 31, 1701–1710 (2007)

    Article  MATH  Google Scholar 

  4. Chao, X., Rahman, A.: Analysis and computational algorithem for queues with state dependent vacations I: G/M(n)/1/K. J. Syst. Sci. Complex. 19, 36–53 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chao, X., Rahman, A.: Analysis and computational algorithem for queues with state dependent vacations II: M(n)/G/1/K. J. Syst. Sci. Complex. 19, 191–210 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Gao, S., Liu, Z.: An M/G/1 queue with single working vacation and vacation interruption under Bernoulli schedule. Appl. Math. Model. 37, 1564–1579 (2013)

    Article  MathSciNet  Google Scholar 

  9. Goswami, V., Vijaya Laxmi, P., Jyothsna, K.: Analysis of GI/M(n)/1/N queue with state-dependent multiple working vacations. Opsearch. 50, 106–124 (2013)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Kijima, M., Makimoto, N.: A unified approach to GI/M(n)/1/K and M(n)/G/1/K queues via finite quasi-birth-death processes. Stoch. Models. 8, 269–288 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  12. Li, J., Tian, N.: Performance analysis of a GI/M/1 queue with single working vacation. Appl. Math. Comput. 217, 4960–4971 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, J., Tian, N., Ma, Z.: Performance analysis of GI/M/1 queue with working vacation and vacation interruption. Appl. Math. Model. 32, 2715–2730 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Macgregor Smith, J.: State dependent queueing models in emergency evacuation networks. Trans. Res. 25, 373–389 (1991)

    Article  Google Scholar 

  15. Madan, K.C., Anabosi, R.F.: A single server queue with two types of service, Bernoulli schedule server vacations and a single vacation policy. Pak. J. Stat. 19, 331–342 (2003)

    MathSciNet  MATH  Google Scholar 

  16. Neuts, M.: Matrix-Geometric Solutions in Stochastic Models. John Hopkins University Press, Baltimore (1981)

    MATH  Google Scholar 

  17. Ramaswami, R., Servi, L.D.: The busy period of the M/G/1 vacation model with a Bernoulli schedule. Stoch. Model. 4, 507–521 (1988)

    Article  Google Scholar 

  18. Servi, L.D., Finn, S.G.: The M/M/1 queues with working vacations (M/M/1/WV). Perform. Eval. 50, 41–52 (2002)

    Article  Google Scholar 

  19. Tian, N., Zhang, D., Cao, C.: The GI/M/1 queue with exponential vacations. Queueing Syst. 5, 331–344 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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

  21. Tian, N., Zhao, X., Wang, K.: The M/M/1 queue with single working vacation. Int. J. Inform. Manage. Sci. 19, 621–634 (2008)

    MathSciNet  MATH  Google Scholar 

  22. Zhang, H., Shi, D.: The M/M/1 queue with Bernoulli-schedule-controlled vacation and vacation interruption. Int. J. Inform. Manage. Sci. 20, 579–587 (2009)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, Andhra University, Visakhapatnam, 530003, India

    P. Vijaya Laxmi & V. Suchitra

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  1. P. Vijaya Laxmi
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  2. V. Suchitra
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Correspondence to P. Vijaya Laxmi.

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Vijaya Laxmi, P., Suchitra, V. A Recursive Algorithm for State Dependent GI/M/1/N Queue with Bernoulli-Schedule Vacation. J Math Model Algor 13, 283–299 (2014). https://doi.org/10.1007/s10852-013-9236-6

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  • DOI: https://doi.org/10.1007/s10852-013-9236-6

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