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Optimizing a Multi-State Cold-Standby System with Multiple Vacations in the Repair and Loss of Units

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  • Juan Eloy Ruiz-Castro

    (Department of Statistics and O.R., Math Institute (IMAG), University of Granada, 18071 Granada, Spain)

Abstract

A complex multi-state redundant system with preventive maintenance subject to multiple events is considered. The online unit can undergo several types of failure: both internal and those provoked by external shocks. Multiple degradation levels are assumed as both internal and external. Degradation levels are observed by random inspections and, if they are major, the unit goes to a repair facility where preventive maintenance is carried out. This repair facility is composed of a single repairperson governed by a multiple vacation policy. This policy is set up according to the operational number of units. Two types of task can be performed by the repairperson, corrective repair and preventive maintenance. The times embedded in the system are phase type distributed and the model is built by using Markovian Arrival Processes with marked arrivals. Multiple performance measures besides the transient and stationary distribution are worked out through matrix-analytic methods. This methodology enables us to express the main results and the global development in a matrix-algorithmic form. To optimize the model, costs and rewards are included. A numerical example shows the versatility of the model.

Suggested Citation

  • Juan Eloy Ruiz-Castro, 2021. "Optimizing a Multi-State Cold-Standby System with Multiple Vacations in the Repair and Loss of Units," Mathematics, MDPI, vol. 9(8), pages 1-29, April.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:8:p:913-:d:539769
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    References listed on IDEAS

    as
    1. Ruiz-Castro, Juan Eloy, 2016. "Markov counting and reward processes for analysing the performance of a complex system subject to random inspections," Reliability Engineering and System Safety, Elsevier, vol. 145(C), pages 155-168.
    2. Ruiz-Castro, Juan Eloy, 2020. "A complex multi-state k-out-of-n: G system with preventive maintenance and loss of units," Reliability Engineering and System Safety, Elsevier, vol. 197(C).
    3. Juan Eloy Ruiz-Castro, 2018. "A D-MMAP to Model a Complex Multi-state System with Loss of Units," Springer Series in Reliability Engineering, in: Anatoly Lisnianski & Ilia Frenkel & Alex Karagrigoriou (ed.), Recent Advances in Multi-state Systems Reliability, pages 39-58, Springer.
    4. Levitin, Gregory & Xing, Liudong & Xiang, Yanping, 2021. "Optimizing preventive replacement schedule in standby systems with time consuming task transfers," Reliability Engineering and System Safety, Elsevier, vol. 205(C).
    5. 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.
    6. Levitin, Gregory & Finkelstein, Maxim & Dai, Yuanshun, 2017. "Redundancy optimization for series-parallel phased mission systems exposed to random shocks," Reliability Engineering and System Safety, Elsevier, vol. 167(C), pages 554-560.
    7. Q. Zhai & R. Peng & L. Xing & J. Yang, 2015. "Reliability of demand‐based warm standby systems subject to fault level coverage," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 31(3), pages 380-393, May.
    8. Yang, Li & Ma, Xiaobing & Peng, Rui & Zhai, Qingqing & Zhao, Yu, 2017. "A preventive maintenance policy based on dependent two-stage deterioration and external shocks," Reliability Engineering and System Safety, Elsevier, vol. 160(C), pages 201-211.
    9. Peng, Rui & Xiao, Hui & Liu, Hanlin, 2017. "Reliability of multi-state systems with a performance sharing group of limited size," Reliability Engineering and System Safety, Elsevier, vol. 166(C), pages 164-170.
    10. Ruiz-Castro, Juan Eloy & Dawabsha, Mohammed & Alonso, Francisco Javier, 2018. "Discrete-time Markovian arrival processes to model multi-state complex systems with loss of units and an indeterminate variable number of repairpersons," Reliability Engineering and System Safety, Elsevier, vol. 174(C), pages 114-127.
    11. Toshio Nakagawa, 2005. "Maintenance Theory of Reliability," Springer Series in Reliability Engineering, Springer, number 978-1-84628-221-8, March.
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