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First passage times for Markov renewal processes and applications

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Abstract

This paper proposes a uniformly convergent algorithm for the joint transform of the first passage time and the first passage number of steps for general Markov renewal processes with any initial state probability vector. The uniformly convergent algorithm with arbitrarily prescribed error can be efficiently applied to compute busy periods, busy cycles, waiting times, sojourn times, and relevant indices of various generic queueing systems and queueing networks. This paper also conducts a numerical experiment to implement the proposed algorithm.

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References

  1. Neuts, M.F., Matrix-Geometric Solutions in Stochastic Models, Baltimore: Johns Hopkins Univ. Press, 1981.

    MATH  Google Scholar 

  2. Neuts, M.F., Structured Stochastic Matrices of M/G/1 Type and Their Applications, New York: Dekker, 1989.

    MATH  Google Scholar 

  3. Ramaswami, V., The busy period of queues which have a matrix-geometric steady state probability vector, Opsearch, 1982, 9: 238.

    MathSciNet  Google Scholar 

  4. Ramaswami, V., From the matrix-geometric to the matrix-exponential, Queueing Syst. , 1990, 6: 229.

    Article  MATH  MathSciNet  Google Scholar 

  5. Lucantoni, D.M., New results on the single server queue with a batch Markovian arrival process, Stoch. Models, 1991, 7:1.

    Article  MATH  MathSciNet  Google Scholar 

  6. Melamed, B., Yadin, M., Randomization procedures in the computation of cumulative-time distributions over discrete state Markov processes, Opns. Res. , 1984, 32: 926.

    Article  MATH  MathSciNet  Google Scholar 

  7. Melamed, B., Yadin, M., Numerical computation of sojourn-time distributions in queueing networks, JACM, 1984, 31:839.

    Article  MATH  MathSciNet  Google Scholar 

  8. Hsu, G.H., He, Q.M., The distribution of the first passage time for the Markov processes of GI/M/1 type, Stoch. Models, 1991, 7: 397.

    Article  MATH  MathSciNet  Google Scholar 

  9. Jensen, A., Markov chains as an aid in the study of Markoff process, Scand. Aktuarietidskr, 1953, 36: 87.

    Google Scholar 

  10. Grassmann, W.K., Transient solutions in Markovian queueing systems, Comput. Opns. Res., 1977, 4: 47.

    Article  Google Scholar 

  11. Grassmann, W.K., Transient solutions in Markovian queues, Eur. J. Opns. Res. , 1977, 1: 396.

    Article  MATH  MathSciNet  Google Scholar 

  12. Gross, D., Miller, D.R., The randomization technique as a modelling tool and solution procedure for transient Markov processes, Opns. Res., 1984, 32: 343.

    MATH  MathSciNet  Google Scholar 

  13. Tijms, H.C., Stochastic Models: An Algorithmic Approach, New York: Wiley, 1994.

    MATH  Google Scholar 

  14. Hsu, G.H., Yuan, X.M. , The first passage times and their algorithms for Markov processes, Stoch. Models, 1995, 11: 195.

    Article  MATH  MathSciNet  Google Scholar 

  15. Hsu, G.H., Yuan, X.M., First passage times: busy periods and waiting times, Science in China, Ser. A, 1995, 38: 1187.

    MATH  Google Scholar 

  16. Hsu, G.H., Xu, D.J., First passage times for multidimensional denumerable state Markov processes, Chin. Sci. Bulletin, 1999, 44: 970.

    Article  MATH  Google Scholar 

  17. Hsu, G.H., Cao, C.X. Busy periods for a matched queuing network, OR Transactions, 2000, 4: 22.

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Applied Mathematics, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, 100080, Beijing, China

    Guanghui Xu & Xueming Yuan

  2. Asian-Pacific Operations Research Center, within CAS and APORS, 100080, Beijing, China

    Guanghui Xu & Xueming Yuan

  3. National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences, 100080, Beijing, China

    Quanlin Li

Authors
  1. Guanghui Xu
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  2. Xueming Yuan
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  3. Quanlin Li
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Correspondence to Guanghui Xu.

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Xu, G., Yuan, X. & Li, Q. First passage times for Markov renewal processes and applications. Sci. China Ser. A-Math. 43, 1238–1249 (2000). https://doi.org/10.1007/BF02880061

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  • DOI: https://doi.org/10.1007/BF02880061

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