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Discrete-time GI/Geo/1 queue with multiple working vacations

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Abstract

Consider the discrete time GI/Geo/1 queue with working vacations under EAS and LAS schemes. The server takes the original work at the lower rate rather than completely stopping during the vacation period. Using the matrix-geometric solution method, we obtain the steady-state distribution of the number of customers in the system and present the stochastic decomposition property of the queue length. Furthermore, we find and verify the closed property of conditional probability for negative binomial distributions. Using such property, we obtain the specific expression for the steady-state distribution of the waiting time and explain its two conditional stochastic decomposition structures. Finally, two special models are presented.

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References

  1. Alfa, A.S.: Vacation models in discrete time. Queueing Syst. 44, 5–30 (2003)

    Article  Google Scholar 

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

    Article  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. Modell. 31(9), 1701–1710 (2007)

    Google Scholar 

  4. Chatterjee, U., Mukherjee, S.: GI/M/1 queue with server vacations. J. Oper. Res. Soc. 41, 83–87 (1990)

    Article  Google Scholar 

  5. Doshi, B.T.: Single server queues with vacations. In:  Takagi, H. (ed.) Stochastic Analysis of Computer and Communication Systems, pp. 217–265. North-Holland, Amsterdam (1990)

    Google Scholar 

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

    Article  Google Scholar 

  7. Hunter, J.J.: Mathematical Techniques of Applied Probability. Discrete Time Models: Techniques and Applications, vol. 2. Academic, New York (1983)

    Google Scholar 

  8. Ke, J.C.: The analysis of a general input queue with N-policy and exponential vacations. Queueing Syst. 45, 135–160 (2003)

    Article  Google Scholar 

  9. Kim, J.D., Choi, D.W., Chae, K.C.: Analysis of queue-length distribution of the M/G/1 queue with working vacations. In: Hawaii International Conference on Statistics and Related Fields, 2003

  10. Laxmi, P., Guipta, U.: On the finite-buffer bulk service queue with general independent arrival: GI/M*/1/N. Oper. Res. Lett. 25, 241–245 (1999)

    Article  Google Scholar 

  11. Li, J., Tian, N.: Analysis of the discrete time Geo/Geo/1 queue with single working vacation QTQM. Special issue on Queueing Models with vacations (2006, accepted)

  12. Meisling, T.: Discrete time queueing theory. Oper. Res. 6, 96–105 (1958)

    Article  Google Scholar 

  13. Neuts, M.: Matrix-Geometric Solutions in Stochastic Models. Johns Hopkins Univ. Press, Baltimore (1981)

    Google Scholar 

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

    Article  Google Scholar 

  15. Takagi, H.: Vacation and Priority Systems, Part 1. Queueing Analysis: A Foundation of Performance Evaluation, vol. 1. North-Holland/Elsevier, Amsterdam (1991)

    Google Scholar 

  16. Takagi, H.: Queueing Analysis. Discrete-Time Systems, vol. 3. North-Holland/Elsevier, Amsterdam (1993)

    Google Scholar 

  17. Tian, N., Zhang, Z.G.: The discrete time GI/Geo/1 queue with multiple vacations. Queueing Syst. 40, 283–294 (2002)

    Article  Google Scholar 

  18. Tian, N., Zhang, Z.G.: A note on GI/M/1 queues with phase-type setup times or server vacations. INFOR 41(4), 341–351 (2003)

    Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

  21. Wu, D., Takagi, H.: M/G/1 queue with multiple working vacations. Perform. Eval. 63(7), 654–681 (2006)

    Article  Google Scholar 

  22. Zhang, Z.G., Tian, N.: Geo/G/1 queue with multiple adaptive vacations. Queueing Syst. 38, 419–429 (2001)

    Article  Google Scholar 

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Authors and Affiliations

  1. College of Economics and Management, YanShan University, Qinhuangdao, 066004, People’s Republic of China

    Ji-Hong Li & Wen-Yuan Liu

  2. College of Science, YanShan University, Qinhuangdao, 066004, People’s Republic of China

    Nai-Shuo Tian

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  1. Ji-Hong Li
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  2. Nai-Shuo Tian
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  3. Wen-Yuan Liu
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Correspondence to Nai-Shuo Tian.

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Li, JH., Tian, NS. & Liu, WY. Discrete-time GI/Geo/1 queue with multiple working vacations. Queueing Syst 56, 53–63 (2007). https://doi.org/10.1007/s11134-007-9030-0

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  • DOI: https://doi.org/10.1007/s11134-007-9030-0

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