Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-27T20:45:04.358Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamic Signatures of Coherent Systems Based on Sequential Order Statistics

Part of: Special processes Distribution theory

Published online by Cambridge University Press:  30 January 2018

M. Burkschat  and
J. Navarro
M. Burkschat*
Affiliation:
Otto-von-Guericke University Magdeburg
J. Navarro*
Affiliation:
Universidad de Murcia
*
∗ Postal address: Institute of Mathematical Stochastics, Otto-von-Guericke University Magdeburg, D-39016 Magdeburg, Germany. Email address: [email protected]
∗∗ Postal address: Facultad de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain. 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.

Sequential order statistics can be used to describe the ordered lifetimes of components in a system, where the failure of a component may affect the performance of remaining components. In this paper mixture representations of the residual lifetime and the inactivity time of systems with such failure-dependent components are considered. Stochastic comparisons of differently structured systems are obtained and properties of the weights in the mixture representations are examined. Furthermore, corresponding representations of the residual lifetime and the inactivity time of a system given the additional information about a previous failure time are derived.

Keywords

signaturecoherent systemsequential order statisticsmixturemixed system

MSC classification

Primary: 62E15: Exact distribution theory
Secondary: 60K10: Applications (reliability, demand theory, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 1 , March 2013 , pp. 272 - 287
Copyright
Copyright © Applied Probability Trust 2013 

References

Balakrishnan, N., Beutner, E. and Kamps, U. (2011). Modeling parameters of a load-sharing system through link functions in sequential order statistics models and associated inference. IEEE Trans. Reliab. 60, 605–611.Google Scholar
Balakrishnan, N., Cramer, E. and Kamps, U. (2001). Bounds for means and variances of progressive type II censored order statistics. Statist. Prob. Lett. 54, 301–315.Google Scholar
Balakrishnan, N., Kamps, U. and Kateri, M. (2009). Minimal repair under a step-stress test. Statist. Prob. Lett. 79, 1548–1558.Google Scholar
Barlow, R. E. and Proschan, F. (1981). Statistical Theory of Reliability and Life Testing. To Begin With, Silver Spring, MD.Google Scholar
Bieniek, M. (2007). Variation diminishing property of densities of uniform generalized order statistics. Metrika 65, 297–309.Google Scholar
Burkschat, M. (2009). Systems with failure-dependent lifetimes of components. J. Appl. Prob. 46, 1052–1072.CrossRefGoogle Scholar
Burkschat, M. and Lenz, B. (2009). Marginal distributions of the counting process associated with generalized order statistics. Commun. Statist. Theory Methods 38, 2089–2106.Google Scholar
Burkschat, M. and Navarro, J. (2011). Aging properties of sequential order statistics. Prob. Eng. Inf. Sci. 25, 449–467.Google Scholar
Cramer, E. (2006). Sequential order statistics. In Encyclopedia of Statistical Sciences, Vol. 12, 2nd edn, eds Kotz, S. et al., John Wiley, Hoboken, pp. 7629–7634.Google Scholar
Cramer, E. and Kamps, U. (2001). 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
Cramer, E., Kamps, U. and Raqab, M. Z. (2003). Characterizations of exponential distributions by spacings of generalized order statistics. Appl. Math. 30, 257–265.Google Scholar
Cramer, E., Kamps, U. and Rychlik, T. (2004). Unimodality of uniform generalized order statistics, with applications to mean bounds. Ann. Inst. Statist. Math. 56, 183–192.Google Scholar
Hashemi, M., Tavangar, M. and Asadi, M. (2010). Some properties of the residual lifetime of progressively type-II right censored order statistics. Statist. Prob. Lett. 80, 848–859.Google Scholar
Hu, T. and Zhuang, W. (2005). A note on stochastic comparisons of generalized order statistics. Statist. Prob. Lett. 72, 163–170.CrossRefGoogle Scholar
Kamps, U. (1995{a}). A Concept of Generalized Order Statistics. Teubner, Stuttgart.Google Scholar
Kamps, U. (1995{b}). A concept of generalized order statistics. J. Statist. Planning Inf. 48, 1–23.Google Scholar
Kamps, U. and Cramer, E. (2001). On distributions of generalized order statistics. Statistics 35, 269–280.Google Scholar
Luke, Y. L. (1969). The Special Functions and Their Approximations, Vol. I. Academic Press, New York.Google Scholar
Mathai, A. M. (1993). A Handbook of Generalized Special Functions for Statistical and Physical Sciences. Clarendon Press, Oxford.Google Scholar
Navarro, J. and Burkschat, M. (2011). Coherent systems based on sequential order statistics. Naval Res. Logistics 58, 123–135.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.CrossRefGoogle Scholar
Navarro, J., Balakrishnan, N. and Samaniego, F. J. (2008a). Mixture representations of residual lifetimes of used systems. J. Appl. Prob. 45, 1097–1112.CrossRefGoogle Scholar
Navarro, J., Samaniego, F. J., Balakrishnan, N. and Bhattacharya, D. (2008{b}). On the application and extension of system signatures in engineering reliability. Naval Res. Logistics 55, 313–327.Google 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.CrossRefGoogle Scholar
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
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.Google Scholar
Xie, H. and Hu, T. (2008). Conditional ordering of generalized order statistics revisited. Prob. Eng. Inf. Sci. 22, 333–346.CrossRefGoogle Scholar
Zhang, Z. (2010{a}). Mixture representations of inactivity times of conditional coherent systems and their applications. J. Appl. Prob. 47, 876–885.Google Scholar
Zhang, Z. (2010{b}). Ordering conditional general coherent systems with exchangeable components. J. Statist. Planning Inf. 140, 454–460.Google Scholar