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Replacement policy in a system under shocks following a Markovian arrival process

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  • Montoro-Cazorla, Delia
  • Pérez-Ocón, Rafael
  • del Carmen Segovia, Maria

Abstract

We present a system subject to shocks that arrive following a Markovian arrival process. The system is minimally repaired. It is replaced when a certain number of shocks arrive. A general model where the replacements are governed by a discrete phase-type distribution is studied. For this system, the Markov process governing the system is constructed, and the interarrival times between replacements and the number of replacements are calculated. A special case of this system is when it can stand a prefixed number of shocks. For this new system, the same performance measures are calculated. The systems are considered in transient and stationary regime.

Suggested Citation

  • Montoro-Cazorla, Delia & Pérez-Ocón, Rafael & del Carmen Segovia, Maria, 2009. "Replacement policy in a system under shocks following a Markovian arrival process," Reliability Engineering and System Safety, Elsevier, vol. 94(2), pages 497-502.
    • Handle: RePEc:eee:reensy:v:94:y:2009:i:2:p:497-502
    DOI: 10.1016/j.ress.2008.06.007
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    References listed on IDEAS

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    1. Montoro-Cazorla, Delia & Pérez-Ocón, Rafael, 2006. "A deteriorating two-system with two repair modes and sojourn times phase-type distributed," Reliability Engineering and System Safety, Elsevier, vol. 91(1), pages 1-9.
    2. Biswas, Atanu & Sarkar, Jyotirmoy, 2000. "Availability of a system maintained through several imperfect repairs before a replacement or a perfect repair," Statistics & Probability Letters, Elsevier, vol. 50(2), pages 105-114, November.
    3. Perez-Ocon, Rafael & Montoro-Cazorla, Delia, 2006. "A multiple warm standby system with operational and repair times following phase-type distributions," European Journal of Operational Research, Elsevier, vol. 169(1), pages 178-188, February.
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    Cited by:

    1. Pepa Ramírez-Cobo & Rosa Lillo & Michael Wiper, 2014. "Identifiability of the MAP 2 /G/1 queueing system," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(1), pages 274-289, April.
    2. Montoro-Cazorla, Delia & Pérez-Ocón, Rafael, 2012. "A shock and wear system under environmental conditions subject to internal failures, repair, and replacement," Reliability Engineering and System Safety, Elsevier, vol. 99(C), pages 55-61.
    3. Yera Mora, Yoel Gustavo & Lillo Rodríguez, Rosa Elvira & Ramírez-Cobo, Pepa, 2017. "Findings about the two-state BMMPP for modeling point processes in reliability and queueing systems," DES - Working Papers. Statistics and Econometrics. WS 24622, Universidad Carlos III de Madrid. Departamento de Estadística.
    4. Sheu, Shey-Huei & Chang, Chin-Chih & Zhang, Zhe George & Chien, Yu-Hung, 2012. "A note on replacement policy for a system subject to non-homogeneous pure birth shocks," European Journal of Operational Research, Elsevier, vol. 216(2), pages 503-508.
    5. Finkelstein, Maxim & Marais, Francois, 2010. "On terminating Poisson processes in some shock models," Reliability Engineering and System Safety, Elsevier, vol. 95(8), pages 874-879.
    6. Cha, Ji Hwan & Finkelstein, Maxim & Levitin, Gregory, 2018. "Bivariate preventive maintenance of systems with lifetimes dependent on a random shock process," European Journal of Operational Research, Elsevier, vol. 266(1), pages 122-134.
    7. Maxim Finkelstein & Gregory Levitin, 2020. "On missions’ quality of performance for systems with partially or completely observable degradation," Journal of Risk and Reliability, , vol. 234(5), pages 676-685, October.
    8. Maxim Finkelstein & Gregory Levitin, 2018. "Optimal mission duration for systems subject to shocks and internal failures," Journal of Risk and Reliability, , vol. 232(1), pages 82-91, February.
    9. Liu, Baoliang & Cui, Lirong & Wen, Yanqing & Shen, Jingyuan, 2015. "A cold standby repairable system with working vacations and vacation interruption following Markovian arrival process," Reliability Engineering and System Safety, Elsevier, vol. 142(C), pages 1-8.
    10. Montoro-Cazorla, Delia & Pérez-Ocón, Rafael, 2014. "A redundant n-system under shocks and repairs following Markovian arrival processes," Reliability Engineering and System Safety, Elsevier, vol. 130(C), pages 69-75.
    11. 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.
    12. Ji Hwan Cha & Maxim Finkelstein, 2019. "On some characteristics of quality for systems operating in a random environment," Journal of Risk and Reliability, , vol. 233(2), pages 257-267, April.
    13. Montoro-Cazorla, Delia & Pérez-Ocón, Rafael, 2014. "A reliability system under different types of shock governed by a Markovian arrival process and maintenance policy K," European Journal of Operational Research, Elsevier, vol. 235(3), pages 636-642.
    14. Cha, Ji Hwan & Finkelstein, Maxim, 2016. "On some properties of shock processes in a ‘natural’ scale," Reliability Engineering and System Safety, Elsevier, vol. 145(C), pages 104-110.
    15. Cha, Ji Hwan & Finkelstein, Maxim & Levitin, Gregory, 2017. "On preventive maintenance of systems with lifetimes dependent on a random shock process," Reliability Engineering and System Safety, Elsevier, vol. 168(C), pages 90-97.
    16. Zarezadeh, Somayeh & Ashrafi, Somayeh, 2019. "On preventive maintenance of networks with components subject to external shocks," Reliability Engineering and System Safety, Elsevier, vol. 191(C).
    17. Rodríguez, Joanna & Lillo, Rosa E. & Ramírez-Cobo, Pepa, 2015. "Failure modeling of an electrical N-component framework by the non-stationary Markovian arrival process," Reliability Engineering and System Safety, Elsevier, vol. 134(C), pages 126-133.
    18. Pepa Ramírez-Cobo & Rosa E. Lillo, 2012. "New Results About Weakly Equivalent MAP 2 and MAP 3 Processes," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 421-444, September.
    19. Ji Cha & Maxim Finkelstein & Francois Marais, 2014. "Survival of systems with protection subject to two types of external attacks," Annals of Operations Research, Springer, vol. 212(1), pages 79-91, January.
    20. Montoro-Cazorla, Delia & Pérez-Ocón, Rafael, 2011. "Two shock and wear systems under repair standing a finite number of shocks," European Journal of Operational Research, Elsevier, vol. 214(2), pages 298-307, October.
    21. Ji Hwan Cha & Maxim Finkelstein, 2020. "On optimal life extension for degrading systems," Journal of Risk and Reliability, , vol. 234(3), pages 487-495, June.
    22. Maxim Finkelstein & Gregory Levitin, 2018. "Optimal Mission Duration for Partially Repairable Systems Operating in a Random Environment," Methodology and Computing in Applied Probability, Springer, vol. 20(2), pages 505-516, June.
    23. Ji Hwan Cha & Maxim Finkelstein & Gregory Levitin, 2017. "Bivariate preventive maintenance for repairable systems subject to random shocks," Journal of Risk and Reliability, , vol. 231(6), pages 643-653, December.
    24. Zhou, Wenhui & Zheng, Zhibin & Xie, Wei, 2017. "A control-chart-based queueing approach for service facility maintenance with energy-delay tradeoff," European Journal of Operational Research, Elsevier, vol. 261(2), pages 613-625.

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