Hostname: page-component-74656f9884-54qdv Total loading time: 0 Render date: 2026-11-10T05:52:38.033Z Has data issue: false hasContentIssue false
  • English
  • Français

Reliability modeling of coherent systems with shared components based on sequential order statistics

Part of: Special processes Operations research and management science

Published online by Cambridge University Press:  16 November 2018

S. Ashrafi ,
S. Zarezadeh  and
M. Asadi
S. Ashrafi*
Affiliation:
University of Isfahan
S. Zarezadeh*
Affiliation:
Shiraz University
M. Asadi*
Affiliation:
University of Isfahan
*
* Postal address: Department of Statistics, University of Isfahan, Isfahan, 81744, Iran.
*** Postal address: Department of Statistics, Shiraz University, Shiraz, 71454, Iran. Email address: s.zarezadeh@shirazu.ac.ir
* Postal address: Department of Statistics, University of Isfahan, Isfahan, 81744, Iran.
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 are concerned with the reliability properties of two coherent systems having shared components. We assume that the components of the systems are two overlapping subsets of a set of n components with lifetimes X1,...,Xn. Further, we assume that the components of the systems fail according to the model of sequential order statistics (which is equivalent, under some mild conditions, to the failure model corresponding to a nonhomogeneous pure-birth process). The joint reliability function of the system lifetimes is expressed as a mixture of the joint reliability functions of the sequential order statistics, where the mixing probabilities are the bivariate signature matrix associated to the structures of systems.We investigate some stochastic orderings and dependency properties of the system lifetimes. We also study conditions under which the joint reliability function of systems with shared components of order m can be equivalently written as the joint reliability function of systems of order n (n>m). In order to illustrate the results, we provide several examples.

Keywords

Bivariate signature matrixpositive quadrant dependentnonhomogeneous pure-birth processstochastic orderingequivalent systems

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Secondary: 90B25: Reliability, availability, maintenance, inspection

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 3 , September 2018 , pp. 845 - 861
Copyright
Copyright © Applied Probability Trust 2018 

References

[1]Ashrafi, S. and Asadi, M. (2015). On the stochastic and dependence properties of the three-state systems. Metrika 78, 261–281.Google Scholar
[2]Belzunce, F., Lillo, R. E., Ruiz, J.-M. and Shaked, M. (2001). Stochastic comparisons of nonhomogeneous processes. Prob. Eng. Inf. Sci. 15, 199–224.Google Scholar
[3]Belzunce, F., Mercader, J.-A., Ruiz, J.-M. and Spizzichino, F. (2009). Stochastic comparisons of multivariate mixture models. J. Multivariate Anal. 100, 1657–1669.Google Scholar
[4]Burkschat, M. and Navarro, J. (2011). Aging properties of sequential order statistics. Prob. Eng. Inf. Sci. 25, 449–467.Google Scholar
[5]Burkschat, M. and Navarro, J. (2013). Dynamic signatures of coherent systems based on sequential order statistics. J. Appl. Prob. 50, 272–287.Google Scholar
[6]Burkschat, M. and Navarro, J. (2014). Asymptotic behavior of the hazard rate in systems based on sequential order statistics. Metrika 77, 965–994.Google Scholar
[7]Cramer, E. (2006). Sequential order statistics. In Encyclopedia of Statistical Sciences, John Wiley, Hoboken, NJ, pp. 7629–7634.Google Scholar
[8]Cramer, E. and Kamps, U. (2003). Marginal distributions of sequential and generalized order statistics. Metrika 58, 293–310.Google Scholar
[9]David, H. A. and Joshi, P. C. (1968). Recurrence relations between moments of order statistics for exchangeable variates. Ann. Math. Statist. 39, 272–274.Google Scholar
[10]Gertsbakh, I. and Shpungin, Y. (2011). Network Reliability and Resilience. Springer, Berlin.Google Scholar
[11]Gertsbakh, I. B. and Shpungin, Y. (2012). Stochastic models of network survivability. Quality Tech. Quant. Manag. 9, 45–58.Google Scholar
[12]Kamps, U. (1995). A Concept of Generalized Order Statistics. Teubner, Stuttgart.Google Scholar
[13]Kamps, U. and Cramer, E. (2001). On distributions of generalized order statistics. Statistics 35, 269–280.Google Scholar
[14]Karlin, S. and Rinott, Y. (1980). Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions. J. Multivariate Anal. 10, 467–498.Google Scholar
[15]Navarro, J. and Burkschat, M. (2011). Coherent systems based on sequential order statistics. Naval Res. Logistics 58, 123–135.Google Scholar
[16]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
[17]Navarro, J., Samaniego, F. J. and Balakrishnan, N. (2013). Mixture representations for the joint distribution of lifetimes of two coherent systems with shared components. Adv. Appl. Prob. 45, 1011–1027.Google Scholar
[18]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
[19]Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.Google Scholar
[20]Torrado, N., Lillo, R. E. and Wiper, M. P. (2012). Sequential order statistics: ageing and stochastic orderings. Methodol. Comput. Appl. Prob. 14, 579–596.Google Scholar
[21]Zarezadeh, S., Asadi, M. and Hesampour, A. D. (2018). On Jensen-Shannon divergence of liftime states of three-state networks. Submitted.Google Scholar
[22]Zarezadeh, S., Mohammadi, L. and Balakrishnan, N. (2018). On the joint signature of several coherent systems with some shared components. Europ. J. Operat. Res. 264, 1092–1100.Google Scholar
[23]Zhuang, W. and Hu, T. (2007). Multivariate stochastic comparisons of sequential order statistics. Prob. Eng. Inf. Sci. 21, 47–66.Google Scholar