Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-27T18:44:15.643Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamic Reliability Modeling of Three-State Networks

Part of: Operations research and management science Special processes

Published online by Cambridge University Press:  30 January 2018

S. Ashrafi  and
M. Asadi
S. Ashrafi*
Affiliation:
University of Isfahan
M. Asadi*
Affiliation:
University of Isfahan and IPM
*
∗ Postal address: Department of Statistics, University of Isfahan, Isfahan, 81744, Iran. Email address: [email protected]
∗∗ Postal address: School of Mathematics, Institute of Research in Fundamental Sciences (IPM), PO Box 19395-5764, Tehran, Iran. 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.

This paper is an investigation into the reliability and stochastic properties of three-state networks. We consider a single-step network consisting of n links and we assume that the links are subject to failure. We assume that the network can be in three states, up (K = 2), partial performance (K = 1), and down (K = 0). Using the concept of the two-dimensional signature, we study the residual lifetimes of the networks under different scenarios on the states and the number of failed links of the network. In the process of doing so, we define variants of the concept of the dynamic signature in a bivariate setting. Then, we obtain signature based mixture representations of the reliability of the residual lifetimes of the network states under the condition that the network is in state K = 2 (or K = 1) and exactly k links in the network have failed. We prove preservation theorems showing that stochastic orderings and dependence between the elements of the dynamic signatures (which relies on the network structure) are preserved by the residual lifetimes of the states of the network (which relies on the network ageing). Various illustrative examples are also provided.

Keywords

Dynamic signature matrixequivalent three-state networkstochastic orderingpositively quadrant dependentassociationNBU

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Secondary: 90B25: Reliability, availability, maintenance, inspection
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 4 , December 2014 , pp. 999 - 1020
Copyright
© Applied Probability Trust 

References

Amari, S. V., Zuo, M. J. and Dill, G. (2009). A fast and robust reliability evaluation algorithm for generalized multi-state k-out-of-n systems. IEEE Trans. Reliab. 58, 88–97.Google Scholar
Asadi, M. and Berred, A. (2012). On the number of failed components in a coherent operating system. Statist. Prob. Lett. 82, 2156–2163.CrossRefGoogle Scholar
Asadi, M. and Goliforushani, S. (2008). On the mean residual life function of coherent systems. IEEE Trans. Reliab. 57, 574–580.Google Scholar
Balakrishnan, N. (2007). Permanents, order statistics, outliers, and robustness. Rev. Mat. Complut. 20, 7–107.Google Scholar
Balakrishnan, N. and Asadi, M. (2012). A proposed measure of residual life of live components of a coherent system. IEEE Trans. Reliab. 61, 41–49.Google Scholar
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
Boland, P. J., Samaniego, F. J. and Vestrup, E. M. (2003). Linking dominations and signatures in network reliability theory. In Mathematical and Statistical Methods in Reliability, World Scientific, River Edge, NJ, pp. 89–103.CrossRefGoogle Scholar
Da, G. and Hu, T. (2013). On bivariate signatures for systems with independent modules. In Stochastic Orders in Reliability and Risk, Springer, New York, pp. 143–166.Google Scholar
Eryilmaz, S. (2010). Conditional lifetimes of consecutive k-out-of-n systems. IEEE Trans. Reliab. 59, 178–182.Google Scholar
Eryilmaz, S. (2010). Mean residual and mean past lifetime of multi-state systems with identical components. IEEE Trans. Reliab. 59, 644–649.CrossRefGoogle Scholar
Eryilmaz, S. (2010). Mixture representations for the reliability of consecutive-k systems. Math. Comput. Modelling 51, 405–412.CrossRefGoogle Scholar
Eryilmaz, S. (2010). Number of working components in consecutive k-out-of-n system while it is working. Commun. Statist. Simul. Comput. 39, 683–692.CrossRefGoogle Scholar
Eryilmaz, S. (2011). Dynamic reliability and performance evaluation of multi-state systems with two components. Hacet. J. Math. Statist. 40, 125–133.Google Scholar
Eryilmaz, S. and İşçioğlu, F. (2011). Reliability evaluation for a multi-state system under stress-strength setup. Commun. Statist. Theory Meth. 40, 547–558.Google Scholar
Eryilmaz, S. and Zuo, M. J. (2010). Computing and applying the signature of a system with two common failure criteria. IEEE Trans. Reliab. 59, 576–580.Google Scholar
Feng, X., Zhang, S. and Li, X. (2013). A note on the mixture representation of the conditional residual lifetime of a coherent system. J. Appl. Prob. 50, 475–485.CrossRefGoogle Scholar
Gertsbakh, I. B. and Shpungin, Y. (2012). Stochastic models of network survivability. QTQM 9, 45–58.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
Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman and Hall, London.Google Scholar
Khaledi, B.-E. and Shaked, M. (2010). Stochastic comparisons of multivariate mixtures. J. Multivariate Anal. 101, 2486–2498.Google Scholar
Lai, C.-D. and Xie, M. (2006). Stochastic Ageing and Dependence for Reliability. Springer, New York.Google Scholar
Li, W. and Zuo, M. J. (2008). Reliability evaluation of multi-state weighted k-out-of-n systems. Reliab. Eng. Syst. Saf. 93, 160–167.CrossRefGoogle Scholar
Li, X. and Pellerey, F. (2011). Generalized Marshall–Olkin distributions and related bivariate aging properties. J. Multivariate Anal. 102, 1399–1409.Google Scholar
Lisnianski, A. and Levitin, G. (2003). Multi-State System Reliability: Assessment, Optimization and Applications. World Scientific, River Edge, NJ.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., 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.CrossRefGoogle 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., 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
Ramirez-Marquez, J. E. and Coit, D. W. (2005). A Monte-Carlo simulation approach for approximating multi-state two-terminal reliability. Reliab. Eng. Syst. Saf. 87, 253–264.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., 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.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.Google Scholar
Sillitto, G. P. (1964). Some relations between expectations of order statistics in samples of different sizes. Biometrika 51, 259–262.CrossRefGoogle Scholar
Zarezadeh, S. and Asadi, M. (2013). Network reliability modeling under stochastic process of component failures. IEEE Trans. Reliab. 62, 917–929.CrossRefGoogle Scholar
Zarezadeh, S., Asadi, M. and Balakrishnan, N. (2014). Dynamic network reliability modeling under nonhomogeneous Poisson processes. Europ. J. Operat. Res. 232, 561–571.CrossRefGoogle Scholar
Zhang, Z. (2010). Mixture representations of inactivity times of conditional coherent systems and their applications. J. Appl. Prob. 47, 876–885.Google Scholar
Zuo, M. J. and Tian, Z. (2006). Performance evaluation of generalized multi-state k-out-of-n systems. IEEE Trans. Reliab. 55, 319–327.Google Scholar