Hostname: page-component-745bb68f8f-5r2nc Total loading time: 0 Render date: 2025-01-28T22:48:16.172Z Has data issue: false hasContentIssue false
  • English
  • Français

Mixture Representations of Inactivity Times of Conditional Coherent Systems and their Applications

Part of: Special processes Distribution theory

Published online by Cambridge University Press:  14 July 2016

Zhengcheng Zhang
Zhengcheng Zhang*
Affiliation:
Lanzhou Jiaotong University
*
∗Postal address: School of Mathematics, Physics and Software Engineering, Lanzhou Jiaotong University, 730070 Lanzhou, P. R. China. Email address: [email protected]
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.

In this paper we obtain several mixture representations of the reliability function of the inactivity time of a coherent system under the condition that the system has failed at time t (> 0) in terms of the reliability functions of inactivity times of order statistics. Some ordering properties of the inactivity times of coherent systems with independent and identically distributed components are obtained, based on the stochastically ordered coefficient vectors between systems.

Keywords

Coherent systeminactivity timemixture representationsignaturestochastic order

MSC classification

Secondary: 62E15: Exact distribution theory 60K10: Applications (reliability, demand theory, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 3 , September 2010 , pp. 876 - 885
Copyright
Copyright © Applied Probability Trust 2010 

Footnotes

Research supported by the Science and Technology Program of Gansu Province, China (project no. 1010RJZA076).

References

[1] Asadi, M. (2006). On the mean past lifetime of components of a parallel system. J. Statist. Planning Infer. 136, 1197–1206.Google Scholar
[2] Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
[3] Bhattacharya, D. and Samaniego, F. J. (2010). On estimating component characteristics from system failure-time data. Naval Res. Logistics 57, 380–389.Google Scholar
[4] Boland, P. J. and Samaniego, F. J. (2004). The signature of a coherent system and its applications in reliability. In Mathematical Reliability: An Expository Perspective, (Internat. Ser. Operat. Res. Management Sci.), eds Soyer, R., Mazzuchi, T. and Singpurwalla, N., Kluwer, Boston, MA, pp. 3–30.Google Scholar
[5] Esary, J. D. and Marshall, A. W. (1970). Coherent life functions. SIAM J. Appl. Math. 18, 810–814.Google Scholar
[6] Gåsemyr, J. and Natvig, B. (1998). The posterior distribution of the parameters of component lifetimes based on autopsy data in a shock model. Scand. J. Statist. 25, 271–292.Google Scholar
[7] Gåsemyr, J. and Natvig, B. (2001). Bayesian inference based on partial monitoring of components with applications to preventive system maintenance. Naval Res. Logistics 48, 551–577.Google Scholar
[8] Hu, T., Li, X., Xu, M. and Zhuang, W. (2007). Some new results on ordering conditional distributions of generalized order statistics. Statistics 21, 401–417.Google Scholar
[9] Jasinski, K., Navarro, J. and Rychlik, T. (2009). Bounds on variances of lifetimes of coherent and mixed systems. J. Appl. Prob. 46, 894–908.CrossRefGoogle Scholar
[10] Khaledi, B. E. and Shaked, M. (2007). Ordering conditional lifetimes of coherent systems. J. Statist. Planning Infer. 137, 1173–1184.CrossRefGoogle Scholar
[11] Kochar, S. C., 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
[12] Li, X. and Zhang, Z. (2008). Some stochastic comparisons of conditional coherent systems. Appl. Stoch. Models Business Industry 24, 541–549.Google Scholar
[13] Li, X. and Zhao, P. (2008). Stochastic comparison on general inactivity time and general residual life of k-out-of-n systems. Commun. Statist. Simul. Comput. 37, 1005–1019.Google Scholar
[14] Meilijson, I. (1981). Estimation of the lifetime distribution of the parts from the autopsy statistics of the machine. J. Appl. Prob. 18, 829–838.CrossRefGoogle Scholar
[15] Navarro, J. and Balakrishnan, N. (2010). Study of some measures of dependence between order statistics and systems. J. Multivariate Anal. 101, 52–67.Google Scholar
[16] Navarro, J. and Rychlik, T. (2007). Reliability and expectation bounds for coherent systems with exchangeable components. J. Multivariate Anal. 98, 102–113.Google Scholar
[17] Navarro, J., Balakrishnan, N. and Samaniego, F. J. (2008). Mixture representations of residual lifetimes of used systems. J. Appl. Prob. 45, 1097–1112.CrossRefGoogle Scholar
[18] 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.CrossRefGoogle Scholar
[19] Navarro, J., Ruiz, J. M. and Sandoval, C. J. (2007). Properties of coherent systems with dependent components. Commun. Statist. Theory Meth. 36, 175–191.Google Scholar
[20] 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.Google Scholar
[21] Poursaeed, M. H. (2010). A note on the mean past and the mean residual life of a (n-k+1)-out-of-n system under multi monitoring. Statist. Papers 51, 409–419.Google Scholar
[22] Poursaeed, M. H. and Nematollahi, A. R. (2008). On the mean past and the mean residual life under double monitoring. Commun. Statist. Theory Meth. 37, 1119–1133.Google Scholar
[23] Samaniego, F. J. (1985). On closure of the IFR class under formation of coherent systems. IEEE Trans. Reliab. 34, 69–72.Google Scholar
[24] Samaniego, F. J., Balakrishnan, N. and Navarro, J. (2009). Dynamic signatures and their use in comparing the reliability of new and used systems. Naval Res. Logistics 56, 577–591.Google Scholar
[25] Shaked, M. and Shanthikumar, J. G.} (2007). Stochastic Orders. Springer, New York.Google Scholar
[26] Tavangar, M. and Asadi, M. (2010). A study on the mean past lifetime of the components of (n-k+1)-out-of-n system at the system level. Metrika 72, 59–73.Google Scholar
[27] Wang, Y., Zhuang, W. and Hu, T. (2010). Conditionally stochastic domination of generalized order statistics from two samples. Statist. Papers 51, 369–373.Google Scholar
[28] Zhang, Z. (2010). Ordering conditional general coherent systems with exchangeable components. J. Statist. Planning Infer. 140, 454–460.Google Scholar