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Using Semi-Markov Chains to Solve Semi-Markov Processes

Author

Listed:
  • Bei Wu

    (Beijing Institute of Technology)

  • Brenda Ivette Garcia Maya

    (Sorbonne University)

  • Nikolaos Limnios

    (Sorbonne University)

Abstract

This article provides a novel method to solve continuous-time semi-Markov processes by algorithms from discrete-time case, based on the fact that the Markov renewal function in discrete-time case is a finite series. Bounds of approximate errors due to discretization for the transition function matrix of the continuous-time semi-Markov process are investigated. This method is applied to a reliability problem which refers to the availability analysis of the system subject to sequential cyber-attacks. Two cases where sojourn times follow exponential and Weibull distributions are considered and computed in order to verify and illustrate the proposed method.

Suggested Citation

  • Bei Wu & Brenda Ivette Garcia Maya & Nikolaos Limnios, 2021. "Using Semi-Markov Chains to Solve Semi-Markov Processes," Methodology and Computing in Applied Probability, Springer, vol. 23(4), pages 1419-1431, December.
    • Handle: RePEc:spr:metcap:v:23:y:2021:i:4:d:10.1007_s11009-020-09820-y
    DOI: 10.1007/s11009-020-09820-y
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    References listed on IDEAS

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    1. Richard L. Warr, 2014. "Numerical Approximation of Probability Mass Functions via the Inverse Discrete Fourier Transform," Methodology and Computing in Applied Probability, Springer, vol. 16(4), pages 1025-1038, December.
    2. Gianfranco Corradi & Jacques Janssen & Raimondo Manca, 2004. "Numerical Treatment of Homogeneous Semi-Markov Processes in Transient Case–a Straightforward Approach," Methodology and Computing in Applied Probability, Springer, vol. 6(2), pages 233-246, June.
    3. Xia, Jianhong (Cecilia) & Zeephongsekul, Panlop & Packer, David, 2011. "Spatial and temporal modelling of tourist movements using Semi-Markov processes," Tourism Management, Elsevier, vol. 32(4), pages 844-851.
    4. Yunhui Hou & Nikolaos Limnios & Walter Schön, 2017. "On the Existence and Uniqueness of Solution of MRE and Applications," Methodology and Computing in Applied Probability, Springer, vol. 19(4), pages 1241-1250, December.
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    Cited by:

    1. Guglielmo D’Amico & Thomas Gkelsinis, 2024. "On a Mixed Transient–Asymptotic Result for the Sequential Interval Reliability for Semi-Markov Chains," Mathematics, MDPI, vol. 12(12), pages 1-18, June.
    2. Delia Montoro-Cazorla & Rafael Pérez-Ocón, 2022. "Analysis of k-Out-of-N-Systems with Different Units under Simultaneous Failures: A Matrix-Analytic Approach," Mathematics, MDPI, vol. 10(11), pages 1-13, June.
    3. Lijun Shang & Guojun Shang & Yongjun Du & Qingan Qiu & Li Yang & Qinglai Dong, 2022. "Post-Warranty Replacement Models for the Product under a Hybrid Warranty," Mathematics, MDPI, vol. 10(10), pages 1-18, May.
    4. Bo, Yimin & Bao, Minglei & Ding, Yi & Hu, Yishuang, 2024. "A DNN-based reliability evaluation method for multi-state series-parallel systems considering semi-Markov process," Reliability Engineering and System Safety, Elsevier, vol. 242(C).

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