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Joint signature of two or more systems with applications to multistate systems made up of two-state components

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  • Marichal, Jean-Luc
  • Mathonet, Pierre
  • Navarro, Jorge
  • Paroissin, Christian

Abstract

The structure signature of a system made up of n components having continuous and i.i.d. lifetimes was defined in the eighties by Samaniego as the n-tuple whose kth coordinate is the probability that the kth component failure causes the system to fail. More recently, a bivariate version of this concept was considered as follows. The joint structure signature of a pair of systems built on a common set of components having continuous and i.i.d. lifetimes is a square matrix of order n whose (k, l)-entry is the probability that the kth failure causes the first system to fail and the lth failure causes the second system to fail. This concept was successfully used to derive a signature-based decomposition of the joint reliability of the two systems. In the first part of this paper we provide an explicit formula to compute the joint structure signature of two or more systems and extend this formula to the general non-i.i.d. case, assuming only that the distribution of the component lifetimes has no ties. We also provide and discuss a necessary and sufficient condition on this distribution for the joint reliability of the systems to have a signature-based decomposition. In the second part of this paper we show how our results can be efficiently applied to the investigation of the reliability and signature of multistate systems made up of two-state components. The key observation is that the structure function of such a multistate system can always be additively decomposed into a sum of classical structure functions. Considering a multistate system then reduces to considering simultaneously several two-state systems.

Suggested Citation

  • Marichal, Jean-Luc & Mathonet, Pierre & Navarro, Jorge & Paroissin, Christian, 2017. "Joint signature of two or more systems with applications to multistate systems made up of two-state components," European Journal of Operational Research, Elsevier, vol. 263(2), pages 559-570.
    • Handle: RePEc:eee:ejores:v:263:y:2017:i:2:p:559-570
    DOI: 10.1016/j.ejor.2017.06.022
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    References listed on IDEAS

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    1. Marichal, Jean-Luc & Mathonet, Pierre & Waldhauser, Tamás, 2011. "On signature-based expressions of system reliability," Journal of Multivariate Analysis, Elsevier, vol. 102(10), pages 1410-1416, November.
    2. Marichal, Jean-Luc & Mathonet, Pierre & Spizzichino, Fabio, 2015. "On modular decompositions of system signatures," Journal of Multivariate Analysis, Elsevier, vol. 134(C), pages 19-32.
    3. Marichal, Jean-Luc & Mathonet, Pierre, 2011. "Extensions of system signatures to dependent lifetimes: Explicit expressions and interpretations," Journal of Multivariate Analysis, Elsevier, vol. 102(5), pages 931-936, May.
    4. Francisco J. Samaniego, 2007. "System Signatures and their Applications in Engineering Reliability," International Series in Operations Research and Management Science, Springer, number 978-0-387-71797-5, December.
    5. Navarro, Jorge & Spizzichino, Fabio & Balakrishnan, N., 2010. "Applications of average and projected systems to the study of coherent systems," Journal of Multivariate Analysis, Elsevier, vol. 101(6), pages 1471-1482, July.
    6. Navarro, Jorge & Rychlik, Tomasz, 2007. "Reliability and expectation bounds for coherent systems with exchangeable components," Journal of Multivariate Analysis, Elsevier, vol. 98(1), pages 102-113, January.
    7. Richard E. Barlow & Alexander S. Wu, 1978. "Coherent Systems with Multi-State Components," Mathematics of Operations Research, INFORMS, vol. 3(4), pages 275-281, November.
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    Cited by:

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    3. 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.
    4. Eryilmaz, Serkan, 2020. "Age-based preventive maintenance for coherent systems with applications to consecutive-k-out-of-n and related systems," Reliability Engineering and System Safety, Elsevier, vol. 204(C).
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    6. Yi, He & Cui, Lirong & Balakrishnan, Narayanaswamy, 2021. "Computation of survival signatures for multi-state consecutive-k systems," Reliability Engineering and System Safety, Elsevier, vol. 208(C).
    7. Da, Gaofeng & Chan, Ping Shing & Xu, Maochao, 2018. "On the signature of complex system: A decomposed approach," European Journal of Operational Research, Elsevier, vol. 265(3), pages 1115-1123.
    8. Zhu, Xiaojun & Balakrishnan, N., 2023. "Non-parametric inference based on reliability life-test of non-identical coherent systems with application to warranty time," Reliability Engineering and System Safety, Elsevier, vol. 232(C).
    9. Ji Hwan Cha & Maxim Finkelstein, 2019. "Optimal preventive maintenance for systems having a continuous output and operating in a random environment," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(2), pages 327-350, July.
    10. He Yi & Narayanaswamy Balakrishnan & Lirong Cui, 2022. "On Dependent Multi-State Semi-Coherent Systems Based on Multi-State Joint Signature," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1717-1734, September.
    11. Somayeh Ashrafi & Majid Asadi & Jorge Navarro, 2022. "Joint Reliability Function of Coherent Systems with Shared Heterogeneous Components," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1485-1502, September.
    12. Yi, He & Balakrishnan, Narayanaswamy & Li, Xiang, 2024. "Joint reliability of linear consecutive k-type systems with shared components in a zigzag structure," Reliability Engineering and System Safety, Elsevier, vol. 241(C).
    13. Ji Hwan Cha & Maxim Finkelstein, 2020. "Stochastic modelling of operational quality of k-out-of-n systems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(2), pages 424-441, July.
    14. Jia, Xujie & Shen, Jingyuan & Xu, Fanqi & Ma, Ruihong & Song, Xueying, 2019. "Modular decomposition signature for systems with sequential failure effect," Reliability Engineering and System Safety, Elsevier, vol. 189(C), pages 435-444.

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