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Some New Concepts and Their Computational Formulae in Aggregated Stochastic Processes with Classifications Based on Sojourn Times

Author

Listed:
  • Lirong Cui

    (Beijing Institute of Technology)

  • Quan Zhang

    (Beijing Institute of Technology)

  • Dejing Kong

    (Beijing Institute of Technology)

Abstract

The system’s performance is one of most important issues in both theory and practice. The task of evaluation of system performance first needs a series of indexes which can describe the system’s performance properly and correctly. Different indexes provide different descriptions and result in different conclusions on system’s performance. On the other hand, although it has a lot of indexes for evaluation of system performance such as reliability, availability and safety and so forth, they still cannot meet the variety requirements on the evaluation of system performance. Thus it is an important work to introduce and develop some new indexes to measure the system’s performance. In this paper, two types of related probability measures, point-wise and interval-wise probabilities, including their concepts and computation formulae, are developed under an alternative renewal process and its derivative aggregated stochastic process with state classifications based on sojourn times. All limits of introduced new measures are discussed too, and the relationship among these new measures is studied. Finally, two special cases: constant and exponential, are discussed too, and some numerical examples are presented to illustrate the concepts intuitively in this work. The work for introduction of new concepts is motivated by some practical problems, specially in repairable systems. The research results can be used not only in reliability, but also may be used in finance, engineering, economy and other fields.

Suggested Citation

  • Lirong Cui & Quan Zhang & Dejing Kong, 2016. "Some New Concepts and Their Computational Formulae in Aggregated Stochastic Processes with Classifications Based on Sojourn Times," Methodology and Computing in Applied Probability, Springer, vol. 18(4), pages 999-1019, December.
    • Handle: RePEc:spr:metcap:v:18:y:2016:i:4:d:10.1007_s11009-015-9456-5
    DOI: 10.1007/s11009-015-9456-5
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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. Robert Elliott & Tak Siu, 2010. "On risk minimizing portfolios under a Markovian regime-switching Black-Scholes economy," Annals of Operations Research, Springer, vol. 176(1), pages 271-291, April.
    3. Lirong Cui & Shijia Du & Aofu Zhang, 2014. "Reliability measures for two-part partition of states for aggregated Markov repairable systems," Annals of Operations Research, Springer, vol. 212(1), pages 93-114, January.
    4. Liying Wang & Lirong Cui, 2013. "Performance Evaluation Of Aggregated Markov Repairable Systems With Multi-Operating Levels," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 30(04), pages 1-27.
    5. Lirong Cui & Shijia Du & Alan Hawkes, 2012. "A study on a single-unit repairable system with state aggregations," IISE Transactions, Taylor & Francis Journals, vol. 44(11), pages 1022-1032.
    6. 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.
    7. Liu, Baoliang & Cui, Lirong & Wen, Yanqing & Shen, Jingyuan, 2013. "A performance measure for Markov system with stochastic supply patterns and stochastic demand patterns," Reliability Engineering and System Safety, Elsevier, vol. 119(C), pages 294-299.
    8. Nikolaos Limnios, 2012. "Reliability Measures of Semi-Markov Systems with General State Space," Methodology and Computing in Applied Probability, Springer, vol. 14(4), pages 895-917, December.
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    Cited by:

    1. Cui, Lirong & Wu, Bei, 2019. "Extended Phase-type models for multistate competing risk systems," Reliability Engineering and System Safety, Elsevier, vol. 181(C), pages 1-16.
    2. Quan Zhang & Shihang Yu & Yang Han & Yanjun Li, 2022. "Research on the model of a multistate aggregated Markov repairable system," Journal of Risk and Reliability, , vol. 236(2), pages 266-276, April.
    3. He Yi & Lirong Cui & Narayanaswamy Balakrishnan & Jingyuan Shen, 2022. "Multi-Point and Multi-Interval Bounded-Covering Availability Measures for Aggregated Markovian Repairable Systems," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2427-2453, December.

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