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Load-Sharing Reliability Models with Different Component Sensitivities to Other Components’ Working States

Part of: Nonparametric inference Survival analysis and censored data

Published online by Cambridge University Press:  17 March 2021

Tomasz Rychlik  and
Fabio Spizzichino
Tomasz Rychlik*
Affiliation:
Polish Academy of Sciences
Fabio Spizzichino*
Affiliation:
Sapienza University of Rome
*
*Postal address: Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00 656 Warsaw, Poland. Email address: trychlik@impan.pl
**Postal address: Sapienza University of Rome, Rome, Italy. Email address: fabio.spizzichino@fondazione.uniroma1.it
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Abstract

We study the distributions of component and system lifetimes under the time-homogeneous load-sharing model, where the multivariate conditional hazard rates of working components depend only on the set of failed components, and not on their failure moments or the time elapsed from the start of system operation. Then we analyze its time-heterogeneous extension, in which the distributions of consecutive failure times, single component lifetimes, and system lifetimes coincide with mixtures of distributions of generalized order statistics. Finally we focus on some specific forms of the time-nonhomogeneous load-sharing model.

Keywords

Multivariate conditional hazard ratelinear breakdown modelMarkov property of failure processdecomposition of reliability functiongeneralized order statisticsexchangeabilityuniform supporting abilityuniform frailty

MSC classification

Primary: 62N05: Reliability and life testing
Secondary: 62G30: Order statistics; empirical distribution functions
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 1 , March 2021 , pp. 107 - 132
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Aki, S. and Hirano, K. (1997). Lifetime distributions of consecutive-k-out-of-n:F systems. Nonlinear Anal. 30, 555–562.CrossRefGoogle Scholar
Arjas, E. (1981). The failure and hazard processes in multivariate reliability systems. Math. Operat. Res. 6, 551–562.CrossRefGoogle Scholar
Arjas, E. and Norros, I. (1984). Life lengths and association: a dynamic approach. Math. Operat. Res. 9, 151–158.CrossRefGoogle Scholar
Barlow, R. E. and Proschan, F. (1975). Importance of system components and fault tree events. Stoch. Process. Appl. 3, 153–173.CrossRefGoogle Scholar
Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life-Testing. Holt, Rinehart, and Winston, New York.Google Scholar
Boland, P. J. (2001). Signatures of indirect majority systems. J. Appl. Prob. 38, 597–603.CrossRefGoogle Scholar
Bremaud, P. (1981). Point Processes and Queues: Martingale Dynamics. Springer, New York.CrossRefGoogle Scholar
Burkschat, M. (2009). Systems with failure-dependent lifetimes of components. J. Appl. Prob. 46, 1052–1072.CrossRefGoogle Scholar
Burkschat, M. and Rychlik, T. (2018). Sharp inequalities for quantiles of system lifetime distributions from failure dependent proportional hazard model. TEST 27, 618–638.CrossRefGoogle Scholar
Cramer, E. and Kamps, U. (2003). Marginal distributions of sequential and generalized order statistics. Metrika 58, 293–310.CrossRefGoogle Scholar
Dempsey, W. and McCullagh, P. (2017). Exchangeable Markov survival process and weak continuity of predictive distributions. Electron. J. Statist. 11, 5406–5451.CrossRefGoogle ScholarPubMed
De Santis, E., Malinovsky, Y. and Spizzichino, F. (2020). Stochastic precedence and minima among dependent variables. Methodology Comput. Appl. Prob. Available online, DOI: 10.1007/s11009-020-09772-3.Google Scholar
Deshpande, J. V., Dewan, I. and Naik-Nimbalkar, U. V. (2010). A family of distributions to model load sharing systems. J. Statist. Planning Infer. 140, 1441–1451.CrossRefGoogle Scholar
Finkelstein, M. and Hazra, N. K. (2017). On stochastic comparisons for load-sharing series and parallel systems. Prob. Eng. Inf. Sci. 31, 311–329.CrossRefGoogle Scholar
Freund, J. E. (1961). A bivariate extension of the exponential distribution. J. Amer. Statist. Assoc. 56, 971–977.CrossRefGoogle Scholar
Grabski, F. (2003). The reliability of an object with semi-Markov failure rate. Appl. Math. Comput. 135, 1–16.Google Scholar
Hollander, M. and Peña, E. A. (1995). Dynamic reliability models with conditional proportional hazard rates. Lifetime Data Anal. 1, 377–401.CrossRefGoogle Scholar
Kamps, U. (1995). A Concept of Generalized Order Statistics. Teubner, Stuttgart.CrossRefGoogle Scholar
Kopocińska, I. and Kopociński, B. (1980). On system reliability under random load of elements. Appl. Math. (Warsaw) 17, 5–14.CrossRefGoogle Scholar
Li, S., Gleaton, J. and Lynch, J. (2019). What is the shape of a bundle? An analysis of Rosen’s fibrous composites experiments using the chain-of-bundles model. Scand. J. Statist. 46, 59–86.CrossRefGoogle Scholar
Marichal, J.-L. and Mathonet, P. (2011). Extensions of system signatures to dependent lifetimes: explicit expressions and interpretations. J. Multivariate Anal. 102, 931–936.CrossRefGoogle Scholar
Marichal, J.-L., Mathonet, P. and Waldhauser, T. (2011). On signature-based expressions of system reliability. J. Multivariate Anal. 102, 1410–1416.CrossRefGoogle Scholar
Navarro, J. and Rychlik, T. (2007). Reliability and expectation bounds for coherent systems with exchangeable components. J. Multivariate Anal. 98, 102–113.CrossRefGoogle Scholar
Navarro, J., Samaniego, F. J., Balakrishnan, N. and Bhattacharya, D. (2008). On the application and extension of system signatures in engineering reliability. Naval Res. Logistics 55, 313–327.CrossRefGoogle Scholar
Navarro, J., Spizzichino, F. and Balakrishnan, N. (2010). The role of average and projected systems in the study of coherent systems. J. Multivariate Anal. 101, 1471–1482.CrossRefGoogle Scholar
Norros, I. (1985). System weakened by failures. Stoch. Process. Appl. 20, 181–196.CrossRefGoogle Scholar
Rajan, V. P. and Curtin, W. A. (2016). Micromechanical design of hierarchical composites using global load sharing theory. J. Mech. Phys. Solids 90, 1–17.CrossRefGoogle Scholar
Ross, S. M., Shahshahani, M. and Weiss, G. (1980). On the number of component failures in systems whose component lives are exchangeable. Math. Operat. Res. 5, 358–365.CrossRefGoogle Scholar
Ross, S. M. (1984). A model in which component failure rates depend on the working set. Naval Res. Logistics 31, 297–300.CrossRefGoogle Scholar
Samaniego, F. J. (1985). On closure of the IFR class under formation of coherent systems. IEEE Trans. Reliab. 34, 69–72.CrossRefGoogle Scholar
Samaniego, F. J. (2007). System Signatures and Their Applications in Engineering Reliability. Springer, New York.CrossRefGoogle Scholar
Scala, A. and De Sanctis Lucentini, P. G. (2016). The equal load-sharing model of cascade failures in power grids. Physica A 462, 737–742.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (1990). Multivariate stochastic orderings and positive dependence in reliability theory. Math. Operat. Res. 15, 545–552.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (2006). Stochastic Orders. Springer, New York.Google Scholar
Shaked, M. and Shanthikumar, J. G. (2015). Multivariate conditional hazard rate functions: an overview. Appl. Stoch. Models Business Industry 31, 285–296.CrossRefGoogle Scholar
Schechner, Z. (1984). A load-sharing model: the linear breakdown rule. Naval Res. Logistics 31, 137–144.CrossRefGoogle Scholar
Smaili, K., Kadri, T. and Kadry, S. (2016). Finding the PDF of the hypoexponential random variable using the Kad matrix similar to the general Vandermonde matrix. Commun. Statist. Theory Meth. 45, 1542–1549.CrossRefGoogle Scholar
Spizzichino, F. (2001). Subjective Probability Models for Lifetimes. Chapman and Hall/CRC, Boca Raton, FL.CrossRefGoogle Scholar
Spizzichino, F. (2008). The role of signature and symmetrization for systems with non-exchangeable components. In Advances in Mathematical Modeling for Reliability, eds. T. Bedford, J. Quigley, L Walls, B. Alkali, A. Daneshkhah, and G. Hardman, IOS, Amsterdam, pp. 138–148.Google Scholar
Spizzichino, F. (2019). Reliability, signature, and relative quality functions for systems under time-homogeneous load-sharing models. Appl. Stoch. Models Business Industry 35, 158–176.CrossRefGoogle Scholar
Sutar, S. and Naik-Nimbalkar, U. V. (2019). A load share model for non-identical components of a k-out-of-m system. Appl. Math. Modelling 72, 486–498.CrossRefGoogle Scholar
Swolfs, Y., McMeeking, R. M., Rajan, V. P., Zok, F. W., Verpoest, I. and Gorbatikh, L. (2015). Global load-sharing model for unidirectional hybrid fibre-reinforced composites. J. Mech. Phys. Solids 84, 380–394.CrossRefGoogle Scholar
Zhang, Z. and Balakrishnan, N. (2017). Stochastic properties and parameter estimation for a general load-sharing system. Commun. Statist. Theory Meth. 46, 747–760.CrossRefGoogle Scholar