Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-06-10T22:18:50.329Z Has data issue: false hasContentIssue false
  • English
  • Français

Mixture Representations for the Joint Distribution of Lifetimes of two Coherent Systems with Shared Components

Part of: Distribution theory - Probability Special processes

Published online by Cambridge University Press:  04 January 2016

Jorge Navarro ,
Francisco J. Samaniego  and
N. Balakrishnan
Jorge Navarro*
Affiliation:
Universidad de Murcia
Francisco J. Samaniego*
Affiliation:
University of California, Davis
N. Balakrishnan*
Affiliation:
McMaster University and King Abdulaziz University
*
∗ Postal address: Facultad de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain. Email address: jorgenav@um.es
∗∗ Postal address: Department of Statistics, University of California, Davis, 1 Shields Avenue, Davis, CA 95616, USA. Email address: fjsamaniego@ucdavis.edu
∗∗∗ Postal address: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario L8S 4K1, Canada. Email address: bala@mcmaster.ca
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.

The signature of a system is defined as the vector whose ith element is the probability that the system fails concurrently with the ith component failure. The signature vector is known to be a distribution-free measure and a representation of the system's survival function has been developed in terms of the system's signature. The present work is devoted to the study of the joint distribution of lifetimes of pairs of systems with shared components. Here, a new distribution-free measure, the ‘joint bivariate signature’, of a pair of systems with shared components is defined, and a new representation theorem for the joint survival function of the system lifetimes is established. The theorem is shown to facilitate the study of the dependence between systems and the comparative performance of two pairs of such systems.

Keywords

Coherent systemk-out-of-n systemorder statisticssignaturemixturestochastic order

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60K10: Applications (reliability, demand theory, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 4 , December 2013 , pp. 1011 - 1027
Copyright
© Applied Probability Trust 

References

Arcones, M. A., Kvam, P. H. and Samaniego, F. J. (2002). Nonparametric estimation of a distribution subject to a stochastic precedence constraint. J. Amer. Statist. Assoc. 97, 170–182.Google Scholar
Asadi, M. and Goliforushani, S. (2008). On the mean residual life function of coherent systems. IEEE Trans. Reliab. 57, 574–580.Google Scholar
Bairamov, I. and Arnold, B. C. (2008). On the residual lifelengths of the remaining components in an n-k+1-out-of-n system. Statist. Prob. Lett. 78, 945–952.Google Scholar
Balakrishnan, N. (2007). Permanents, order statistics, outliers, and robustness. Rev. Mat. Complut. 20, 7–107.Google Scholar
Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
Burkschat, M. and Navarro, J. (2013). Dynamic signatures of coherent systems based on sequential order statistics. J. Appl. Prob. 50, 272–287.Google Scholar
Da, G., Zheng, B. and Hu, T. (2012). On computing signatures of coherent systems. J. Multivariate Anal. 103, 142–150.CrossRefGoogle Scholar
David, H. A. and Nagaraja, H. N. (2003). Order Statistics, 3rd edn. John Wiley, Hoboken, NJ.CrossRefGoogle Scholar
Epstein, B. and Sobel, M. (1953). Life testing. J. Amer. Statist. Assoc. 48, 486–502.Google Scholar
Gertsbakh, I. and Shpungin, Y. (2011). Network Reliability and Resilience. Springer, New York.CrossRefGoogle Scholar
Gertsbakh, I. and Shpungin, Y. (2012). Multidimensional spectra of multistate systems with binary components. In Recent Advances in System Reliability, eds Lisniansky, A. and Frenkel, I., Springer, London, pp. 49–61.CrossRefGoogle Scholar
Gertsbakh, I. and Shpungin, Y. (2012). Stochastic models of network survivability. Quality Tech. Quantitative Manag. 9, 45–58.Google Scholar
Gertsbakh, I., Shpungin, Y. and Spizzichino, F. (2012). Two-dimensional signatures. J. Appl. Prob. 49, 416–429.Google Scholar
Goliforushani, S., Asadi, M. and Balakrishnan, N. (2012). On the residual and inactivity times of the components of used coherent systems. J. Appl. Prob. 49, 385–404.Google Scholar
Hollander, M. and Samaniego, F. J. (2008). On comparing the reliability of arbitrary systems via stochastic precedence. In Advances in Mathematical Modeling for Reliability, eds Bedford, T. et al., IOS Press, Amsterdam, pp. 129–137.Google Scholar
Khaledi, B.-E. and Shaked, M. (2007). Ordering conditional lifetimes of coherent systems. J. Statist. Planning Infer. 137, 1173–1184.Google Scholar
Kochar, S., Mukerjee, H. and Samaniego, F. J. (1999). The “signature” of a coherent system and its application to comparison among systems. Naval Res. Logistics 46, 507–523.Google Scholar
Li, X. and Zhang, Z. (2008). Some stochastic comparisons of conditional coherent systems. Appl. Stoch. Models Business Industry 24, 541–549.Google Scholar
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
Marichal, J.-L. and Mathonet, P. (2011). Extensions of system signatures to dependent lifetimes: explicit expressions and interpretations. J. Multivariate Anal. 102, 931–936.Google Scholar
Marichal, J.-L., Mathonet, P. and Waldhauser, T. (2011). On signature-based expressions of system reliability. J. Multivariate Anal. 102, 1410–1416.Google Scholar
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
Navarro, J. and Rubio, R. (2010). Comparisons of coherent systems using stochastic precedence. TEST 19, 469–486.Google Scholar
Navarro, J. and Rubio, R. (2011). A note on necessary and sufficient conditions for ordering properties of coherent systems with exchangeable components. Naval Res. Logistics 58, 478–489.Google Scholar
Navarro, J., Balakrishnan, N. and Samaniego, F. J. (2008). Mixture representations of residual lifetimes of used systems. J. Appl. Prob. 45, 1097–1112.Google Scholar
Navarro, J., Samaniego, F. J. and Balakrishnan, N. (2010). The Joint signature of coherent systems with shared components. J. Appl. Prob. 47, 235–253.Google Scholar
Navarro, J., Samaniego, F. J. and Balakrishnan, N. (2011). Signature-based representations for the reliability of systems with heterogeneous components. J. Appl. Prob. 48, 856–867.Google Scholar
Navarro, J., Samaniego, F. J. and Balakrishnan, N. (2013). Mixture representations for the Joint distribution of two coherent systems with shared components. Tech. Rep. 1-2013, Department of Statitstics and Operational Research, Universidad de Murcia.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.Google Scholar
Samaniego, F. J. (1985). On closure of the IFR class under formation of coherent systems. IEEE Trans. Reliab. R-34, 69–72.CrossRefGoogle Scholar
Samaniego, F. J. (2007). System Signatures and Their Applications in Engineering Reliability (Internat. Ser. Operat. Res. Manag. Sci. 110). Springer, New York.Google 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
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
Zhang, Z. (2010). Ordering conditional general coherent systems with exchangeable components. J. Statist. Planning Infer. 140, 454–460.Google Scholar