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On the Derivative Counting Processes of First- and Second-order Aggregated Semi-Markov Systems

Author

Listed:
  • He Yi

    (Beijing University of Chemical Technology)

  • Lirong Cui

    (Qingdao University)

  • Narayanaswamy Balakrishnan

    (McMaster University)

Abstract

In this paper, first- and second-order discrete-time semi-Markov systems are considered with their finite state space divided into three subsets as perfect functioning states, imperfect functioning states and failure states, respectively. The counting processes for one-step increasing transitions, one-step equivalent transitions and one-step decreasing transitions in working/failure periods are defined and investigated in detail. Formulas for related distributions, joint distributions, expectations, generating functions and joint generation functions are derived in their Z-transforms. Numerical examples are presented to illustrate the results established. Extended discussions on related reliability measures are also considered. Finally, some concluding remarks and discussions are presented. Applications of the results presented here can be found in different fields such as seismology, reliability, biology and finance.

Suggested Citation

  • He Yi & Lirong Cui & Narayanaswamy Balakrishnan, 2022. "On the Derivative Counting Processes of First- and Second-order Aggregated Semi-Markov Systems," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1849-1875, September.
    • Handle: RePEc:spr:metcap:v:24:y:2022:i:3:d:10.1007_s11009-021-09896-0
    DOI: 10.1007/s11009-021-09896-0
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    References listed on IDEAS

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    1. Alan Hawkes & Lirong Cui & Zhihua Zheng, 2011. "Modeling the evolution of system reliability performance under alternative environments," IISE Transactions, Taylor & Francis Journals, vol. 43(11), pages 761-772.
    2. Yi, He & Cui, Lirong & Shen, Jingyuan & Li, Yan, 2018. "Stochastic properties and reliability measures of discrete-time semi-Markovian systems," Reliability Engineering and System Safety, Elsevier, vol. 176(C), pages 162-173.
    3. G. Nuel, 2019. "Moments of the Count of a Regular Expression in a Heterogeneous Random Sequence," Methodology and Computing in Applied Probability, Springer, vol. 21(3), pages 875-887, September.
    4. Baoliang Liu & Lirong Cui & Yanqing Wen, 2014. "Interval reliability for aggregated Markov repairable system with repair time omission," Annals of Operations Research, Springer, vol. 212(1), pages 169-183, January.
    5. Wu, Bei & Cui, Lirong & Fang, Chen, 2019. "Reliability analysis of semi-Markov systems with restriction on transition times," Reliability Engineering and System Safety, Elsevier, vol. 190(C), pages 1-1.
    6. Csenki, Attila, 1995. "The number of visits to a subset of the state space by a discrete-parameter semi-Markov process," Statistics & Probability Letters, Elsevier, vol. 22(1), pages 71-77, January.
    7. Fang, Chen & Cui, Lirong, 2021. "Balanced Systems by Considering Multi-state Competing Risks Under Degradation Processes," Reliability Engineering and System Safety, Elsevier, vol. 205(C).
    8. David Landriault & Bin Li & Hongzhong Zhang, 2017. "A Unified Approach for Drawdown (Drawup) of Time-Homogeneous Markov Processes," Papers 1702.07786, arXiv.org.
    9. Vlad Stefan Barbu & Nicolas Vergne, 2019. "Reliability and Survival Analysis for Drifting Markov Models: Modeling and Estimation," Methodology and Computing in Applied Probability, Springer, vol. 21(4), pages 1407-1429, December.
    10. Irene Votsi & Nikolaos Limnios & George Tsaklidis & Eleftheria Papadimitriou, 2012. "Estimation of the Expected Number of Earthquake Occurrences Based on Semi-Markov Models," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 685-703, September.
    11. Bei Wu & Lirong Cui & Chen Fang, 2020. "Generalized phase-type distributions based on multi-state systems," IISE Transactions, Taylor & Francis Journals, vol. 52(1), pages 104-119, January.
    12. Boutsikas V. Michael & Vaggelatou Eutichia, 2020. "On the Distribution of the Number of Success Runs in a Continuous Time Markov Chain," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 969-993, September.
    13. Yi, He & Cui, Lirong, 2017. "Distribution and availability for aggregated second-order semi-Markov ternary system with working time omission," Reliability Engineering and System Safety, Elsevier, vol. 166(C), pages 50-60.
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