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RELATIONSHIPS BETWEEN IMPORTANCE MEASURES AND REDUNDANCY IN SYSTEMS WITH DEPENDENT COMPONENTS

Published online by Cambridge University Press:  17 May 2019

Jorge Navarro Open the ORCID record for Jorge Navarro [Opens in a new window] ,
Pedro Fernández-Martínez ,
Juan Fernández-Sánchez  and
Antonio Arriaza
Jorge Navarro
Affiliation:
Universidad de Murcia, Murcia, Spain E-mail: jorgenav@um.es
Pedro Fernández-Martínez
Affiliation:
Universidad de Murcia, Murcia, Spain E-mail: jorgenav@um.es
Juan Fernández-Sánchez
Affiliation:
Universidad de Almería, Almería, Spain
Antonio Arriaza
Affiliation:
Universidad de Cádiz, Cádiz, Spain
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Abstract

The paper shows the connections between some importance indices for the components in an engineering coherent system and the performance of the system obtained when a redundancy mechanism is applied to a specific component. A copula approach is used to model the dependency among the components. This approach includes the popular case of independent components. Under some assumptions, it is proved that if component i is more important than component j, then the system obtained by applying a redundancy procedure to the ith component is better, under different stochastic criteria, than that obtained with the jth component. These results can be applied to several redundancy mechanisms. A new importance index is defined to study active redundancies. Some illustrative examples are provided.

Keywords

coherent systemscopuladistorted distributionsimportance measuresredundancy
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 34 , Issue 4 , October 2020 , pp. 583 - 604
Copyright
Copyright © Cambridge University Press 2019

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References

1.Arriaza, A., Navarro, J., & Suárez-Llorens, A. (2018). Stochastic comparisons of replacement policies in coherent systems under minimal repair. Naval Research Logistics 65: 550–565.CrossRefGoogle Scholar
2.Barlow, R.E. & Proschan, F. (1975). Statistical theory of reliability and life testing. International Series in Decision Processes. New York: Holt, Rinehart and Winston, Inc.Google Scholar
3.Belzunce, F., Martínez-Riquelme, C., & Ruiz, J.M. (2018). Allocation of a relevation in redundancy problems. To appear in Applied Stochastic Models in Business and Industry. Published online first April 6, 2018. DOI: 10.1002/asmb.2328.CrossRefGoogle Scholar
4.Burkschat, M. & Navarro, J. (2018). Stochastic comparisons of systems based on sequential order statistics via properties of distorted distributions. Probability in the Engineering and Informational Sciences 32: 246–274.CrossRefGoogle Scholar
5.Calì, C., Longobardi, M., & Navarro, J. (2018). Properties for generalized cumulative past measures of information. To appear in Probability in the Engineering and Informational Sciences. Published online first October 29, 2018. DOI: 10.1017/S0269964818000360.Google Scholar
6.Durante, F. & Sempi, C. (2016). Principles of Copula Theory. London: CRC/Chapman & Hall.Google Scholar
7.Durante, F., Saminger-Platz, S., & Sarkoci, P. (2009). Rectangular patchwork for bivariate copulas and tail dependence. Communications in Statistics Theory and Methods 38: 2515–2527.CrossRefGoogle Scholar
8.Federer, H. (1969). Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften 153. Berlin-Heidelberg-New York: Springer-Verlag.Google Scholar
9.Gertsbakh, I.B. & Shpungin, Y. (2012). Combinatorial approach to computing component importance indexes in coherent systems. Probability in the Engineering and Informational Sciences 26: 117–128.CrossRefGoogle Scholar
10.Hazra, N.K. & Nanda, A.K. (2014). Component redundancy versus system redundancy in different stochastic orderings. IEEE Transactions on Reliability 63: 567–582.CrossRefGoogle Scholar
11.Iyer, S. (1992). The Barlow–Proschan importance and its generalizations with dependent components. Stochastic Processes and their Applications 42: 353–359.CrossRefGoogle Scholar
12.Krakowski, M. (1973). The relevation transform and a generalization of the Gamma distribution function. Revue française d'automatique, informatique, recherche opérationnelle, tome 7 (no. V-2): 107–120.CrossRefGoogle Scholar
13.Kuo, W. & Zhu, X. (2012). Importance measures in reliability, risk, and optimization. Principles and applications. West Sussex, UK: John Wiley & Sons Ltd.CrossRefGoogle Scholar
14.Marichal, J.-L. & Mathonet, P. (2013). On the extensions of Barlow–Proschan importance index and system signature to dependent lifetimes. Journal of Multivariate Analysis 115: 48–56.CrossRefGoogle Scholar
15.Miziuła, P. & Navarro, J. (2017). Sharp bounds for the reliability of systems and mixtures with ordered components. Naval Research Logistics 64: 108–116.CrossRefGoogle Scholar
16.Miziuła, P. & Navarro, J. (2019). A new importance measure for reliability systems with dependent components. To appear in IEEE Transactions on Reliability, Published online first February 2019. DOI: 10.1109/TR.2019.2895400.Google Scholar
17.Navarro, J. & Gómis, M.C. (2016). Comparisons in the mean residual life order of coherent systems with identically distributed components. Applied Stochastic Models in Business and Industry 32: 33–47.CrossRefGoogle Scholar
18.Navarro, J., del Águila, Y., Sordo, M.A., & Suárez-Llorens, A. (2013). Stochastic ordering properties for systems with dependent identically distributed components. Applied Stochastic Models in Business and Industry 29: 264–278.CrossRefGoogle Scholar
19.Navarro, J., del Águila, Y., Sordo, M.A., & Suárez-Llorens, A. (2016). Preservation of stochastic orders under the formation of generalized distorted distributions. Applications to coherent systems. Methodology and Computing in Applied Probability 18: 529–545.CrossRefGoogle Scholar
20.Nelsen, R.B. (2006). An introduction to copulas, 2nd ed., New York: Springer.Google Scholar
21.Shaked, M. & Shanthikumar, J.G. (2007). Stochastic orders. Springer series in statistics. New York: Springer.CrossRefGoogle Scholar
22.Zhang, X. & Wilson, A. (2017). System reliability and component importance under dependence. Technometrics 59: 215–224.CrossRefGoogle Scholar