Hostname: page-component-848d4c4894-xm8r8 Total loading time: 0 Render date: 2024-06-23T06:06:14.407Z Has data issue: false hasContentIssue false
  • English
  • Français

Systems with Failure-Dependent Lifetimes of Components

Part of: Distribution theory Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

M. Burkschat
M. Burkschat*
Affiliation:
Otto-von-Guericke University Magdeburg
*
∗Postal address: Institute of Mathematical Stochastics, Otto-von-Guericke University Magdeburg, D-39016 Magdeburg, Germany. Email address: marco.burkschat@ovgu.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A model for describing the lifetimes of coherent systems, where the failures of components may have an impact on the lifetimes of the remaining components, is proposed. The model is motivated by the definition of sequential order statistics (cf. Kamps (1995)). Sequential order statistics describe the successive failure times in a sequential k-out-of-n system, where the distribution of the remaining components' lifetimes is allowed to change after every failure of a component. In the present paper, general component lifetimes which can be influenced by failures are considered. The ordered failure times of these components can be used to extend the concept of sequential order statistics. In particular, a definition of sequential order statistics based on exchangeable components is proposed. By utilizing the system signature (cf. Samaniego (2007)), the distribution of the lifetime of a coherent system with failure-dependent exchangeable component lifetimes is shown to be given by a mixture of the distributions of sequential order statistics. Furthermore, some results on the joint distribution of sequential order statistics based on exchangeable components are given.

Keywords

Coherent systemk-out-of-n systemsequential order statisticsexchangeable lifetimessignature

MSC classification

Secondary: 60E05: Distributions 62E10: Characterization and structure theory
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 4 , December 2009 , pp. 1052 - 1072
Copyright
Copyright © Applied Probability Trust 2009 

References

Balakrishnan, N., Beutner, E. and Kamps, U. (2008). “Order restricted inference for sequential k-out-of-n systems.” J. Multivariate Anal. 99, 1489–1502.Google Scholar
Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
Beutner, E. (2008). Nonparametric inference for sequential k-out-of-n systems. Ann. Inst. Statist. Math. 60, 605–626.CrossRefGoogle Scholar
Beutner, E., Burkschat, M. and Kamps, U. (2007). Sequential k-out-of-n systems: model and estimation. In Proc. 5th Internat. Math. Meth. Reliab. Conf., Glasgow.Google Scholar
Boland, P. J. and Samaniego, F. J. (2004{a}). Stochastic ordering results for consecutive k-out-of-n: F systems. IEEE Trans. Reliab. 53, 7–10.Google Scholar
Boland, P. J. and Samaniego, F. (2004{b}). The signature of a coherent system and its applications in reliability. In Mathematical Reliability: An Expository Perspective, eds Soyer, R., Mazzuchi, T. and Singpurwalla, N. D., Kluwer, Boston, MA, pp. 1–29.Google Scholar
Boland, P. J., Samaniego, F. and Vestrup, E. M. (2003). Linking dominations and signatures in network reliability theory. In Mathematical and Statistical Methods in Reliability, eds Lindquist, B. and Doksum, K. A., World Scientific, Singapore, pp. 89–103.Google Scholar
Cramer, E. (2001). Inference for stress-strength models based on Weinman multivariate exponential samples. Commun. Statist. Theory Methods 30, 331–346.CrossRefGoogle Scholar
Cramer, E. (2006{a}). Dependence structure of generalized order statistics. Statistics 40, 409–413.Google Scholar
Cramer, E. (2006{b}). Sequential order statistics. In Encyclopedia of Statistical Sciences, Vol. 12, 2nd edn. John Wiley, Hoboken, NJ, pp. 7629–7634.Google Scholar
Cramer, E. and Kamps, U. (1996). Sequential order statistics and k-out-of-n systems with sequentially adjusted failure rates. Ann. Inst. Statist. Math. 48, 535–549.CrossRefGoogle Scholar
Cramer, E. and Kamps, U. (1998). Sequential k-out-of-n systems with Weibull components. Econom. Quality Control 13, 227–239.Google Scholar
Cramer, E. and Kamps, U. (2001{a}). Estimation with sequential order statistics from exponential distributions. Ann. Inst. Statist. Math. 53, 307–324.Google Scholar
Cramer, E. and Kamps, U. (2001{b}). Sequential k-out-of-n systems. In Advances in Reliability (Handbook Statist. 20), eds Balakrishnan, N. and Rao, C. R., North-Holland, Amsterdam, pp. 301–372.Google Scholar
Cramer, E. and Kamps, U. (2003). Marginal distributions of sequential and generalized order statistics. Metrika 58, 293–310.Google Scholar
David, H. A. and Nagaraja, H. N. (2003). Order Statistics, 3rd edn. John Wiley, Hoboken, NJ.CrossRefGoogle Scholar
Esary, J. D. and Marshall, A. W. (1970). Coherent life functions. SIAM J. Appl. Math. 18, 810–814.Google Scholar
Garren, S. T. and Richards, D. S. P. (1998). General conditions for comparing the reliability functions of systems of components sharing a common environment. J. Appl. Prob. 35, 124–135.Google Scholar
Gupta, R. C. (2002). Reliability of a k-out-of-n system of components sharing a common environment. Appl. Math. Lett. 15, 837–844.CrossRefGoogle Scholar
Hájek, J., Šidák, Z. and Sen, P. K. (1999). Theory of Rank Tests, 2nd edn. Academic Press, San Diego, CA.CrossRefGoogle Scholar
Kamps, U. (1995). A Concept of Generalized Order Statistics. Teubner, Stuttgart.Google Scholar
Kamps, U. and Cramer, E. (2001). On distributions of generalized order statistics. Statistics 35, 269–280.Google Scholar
Khaledi, B.-E. and Shaked, M. (2007). Ordering conditional lifetimes of coherent systems. J. Statist. Planning Infer. 137, 1173–1184.CrossRefGoogle Scholar
Kochar, S., Mukerjee, H. and Samaniego, F. J. (1999). The ‘signature’ of a coherent system and its application to comparisons among systems. Naval Res. Logistics 46, 507–523.Google Scholar
Kotz, S., Balakrishnan, N. and Johnson, N. L. (2000). Continuous Multivariate Distributions, Vol. 1, 2nd edn. Wiley-Interscience, New York.Google Scholar
Lindley, D. V. and Singpurwalla, N. D. (1986). Multivariate distributions for the life lengths of components of a system sharing a comment environment. J. Appl. Prob. 23, 418–431.CrossRefGoogle Scholar
Navarro, J. and Eryilmaz, S. (2007). Mean residual lifetimes of consecutive k-out-of-n systems. J. Appl. Prob. 44, 82–98.Google Scholar
Navarro, J. and Hernandez, P. J. (2008). Mean residual life functions of finite mixtures, order statistics and coherent systems. Metrika 67, 277–298.Google Scholar
Navarro, J. and Rychlik, T. (2007). Reliability and expectation bounds for coherent systems with exchangeable components. J. Multivariate Anal. 98, 102–113.Google Scholar
Navarro, J. and Shaked, M. (2006). Hazard rate ordering of order statistics and systems. J. Appl. Prob. 43, 391–408.Google Scholar
Navarro, J., Balakrishnan, N. and Samaniego, F. J. (2008{a}). Mixture representations of residual lifetimes of used systems. J. Appl. Prob. 45, 1097–1112.Google Scholar
Navarro, J., Ruiz, J. M. and Sandoval, C. J. (2005). A note on comparisons among coherent systems with dependent components using signatures. Statist. Prob. Lett. 72, 179–185.Google Scholar
Navarro, J., Ruiz, J. M. and Sandoval, C. J. (2007{a}). Properties of coherent systems with dependent components. Commun. Statist. Theory Meth. 36, 175–191.CrossRefGoogle Scholar
Navarro, J., Ruiz, J. M. and Sandoval, C. J. (2008{b}). Properties of systems with two exchangeable Pareto components. Statist. Papers 49, 177–190.Google Scholar
Navarro, J., Rychlik, T. and Shaked, M. (2007{b}). Are the order statistics ordered? A survey of recent results. Commun. Statist. Theory Meth. 36, 1273–1290.Google Scholar
Navarro, J., Samaniego, F. J., Balakrishnan, N. and Bhattacharya, D. (2008{c}). On the application and extension of system signatures in engineering reliability. Naval Res. Logistics 55, 313–327.Google Scholar
Revathy, S. A. and Chandrasekar, B. (2007). Equivariant estimation of parameters based on sequential order statistics from (1,3) and (2,3) systems. Commun. Statist. Theory Meth. 36, 541–548.CrossRefGoogle Scholar
Samaniego, F. J. (1985). On closure of the IFR class under formation of coherent systems. IEEE Trans. Reliab. 34, 69–72.Google Scholar
Samaniego, F. J. (2007). System Signatures and Their Applications in Engineering Reliability. Springer, New York.Google Scholar
Triantafyllou, I. S. and Koutras, M. V. (2008). On the signature of coherent systems and applications. Prob. Eng. Inf. Sci. 22, 19–35.Google Scholar
Zhuang, W. and Hu, T. (2007). Multivariate stochastic comparisons of sequential order statistics. Prob. Eng. Inf. Sci. 21, 47–66.CrossRefGoogle Scholar