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Published online by Cambridge University Press:  14 June 2018

S. Zarezadeh
Department of Statistics, Shiraz University, Shiraz 71454, Iran E-mail:
M. Asadi
Department of Statistics, University of Isfahan, Isfahan 81744, Iranand School of Mathematics, Institute of Research in Fundamental Sciences (IPM), P.O Box 19395-5746, Tehran, Iran E-mail:
S. Eftekhar
Department of Statistics, Shiraz University, Shiraz 71454, Iran E-mail:


The signature matrix of an n-component three-state network (system), which depends only on the network structure, is a useful tool for comparing the reliability and stochastic properties of networks. In this paper, we consider a three-state network with states up, partial performance, and down. We assume that the network remains in state up, for a random time T1 and then moves to state partial performance until it fails at time T>T1. The signature-based expressions for the conditional entropy of T given T1, the joint entropy, Kullback-Leibler (K-L) information, and mutual information of the lifetimes T and T1 are presented. It is shown that the K-L information, and mutual information between T1 and T depend only on the network structure (i.e., depend only to the signature matrix of the network). Some signature-based stochastic comparisons are also made to compare the K-L of the state lifetimes in two different three-state networks. Upper and lower bounds for the K-L divergence and mutual information between T1 and T are investigated. Finally the results are extended to n-component multi-state networks. Several examples are examined graphically and numerically.

Research Article
Copyright © Cambridge University Press 2018 

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1.Ashrafi, S. & Asadi, M. (2014). Dynamic reliability modeling of three-state networks. Journal of Applied Probability 51(4): 9991020.CrossRefGoogle Scholar
2.Ashrafi, S. & Asadi, M. (2015). On the stochastic and dependence properties of the three-state systems. Metrika 78(3): 261281.CrossRefGoogle Scholar
3.Asadi, M., Ebrahimi, N., Soofi, E.S., & Zohrevand, Y. (2016). Jensen-Shannon information of the coherent system lifetime. Reliability Engineering & System Safety 156: 244255.CrossRefGoogle Scholar
4.Ebrahimi, N., Soofi, E.S., & Soyer, R. (2013). When are observed failures more informative than observed survivals? Naval Research Logistics 60(2): 102110.CrossRefGoogle Scholar
5.Ebrahimi, N., Soofi, E.S., & Zahedi, H. (2004). Information properties of order statistics and spacings. IEEE Transactions on Information Theory 50: 177183.CrossRefGoogle Scholar
6.Gertsbakh, I. & Shpungin, Y. (2011). Network Reliability and Resilience. Springer Briefs in Electrical and Computer Engineering. Berlin, Germany: Springer-Verlag.CrossRefGoogle Scholar
7.Gertsbakh, I. & Shpungin, Y. (2012). Stochastic models of network survivability. Quality Technology & Quantitative Management 9(1): 4558.CrossRefGoogle Scholar
8.Gertsbakh, I., Shpungin, Y., & Spizzichino, F. (2012). Two-dimensional signatures. Journal of Applied Probability 49(2): 416429.CrossRefGoogle Scholar
9.Kochar, S., Mukerjee, H., & Samaniego, F.J. (1999). The ‘signature’ of a coherent system and its application to comparison among systems. Naval Research Logistics 46: 507523.3.0.CO;2-D>CrossRefGoogle Scholar
10.Kullback, S. & Leibler, R.A. (1951). On information and sufficiency. Annals of Mathematical Statistics 22(1): 7986.CrossRefGoogle Scholar
11.Navarro, J., Balakrishnan, N., & Samaniego, F.J. (2008). Mixture reptesentations of residual lifetimes of used systems. Journal of Applied Probability 45: 10971112.CrossRefGoogle Scholar
12.Navarro, J., Samaniego, F.J., & Balakrishnan, N. (2013). Mixture representations for the joint distribution of two coherent systems with shared components. Advances in Applied Probability 45(4): 10111027.CrossRefGoogle Scholar
13.Parzen, E. (1979). Nonparametric statistical data modelling. Journal of American Statistical Association 74: 105122.CrossRefGoogle Scholar
14.Samaniego, F.J. (1985). On closure of the IFR class under formation of coherent systems. IEEE Transactions on Reliability 34: 6972.CrossRefGoogle Scholar
15.Samaniego, F.J. (2007). System signatures & their applications in reliability engineering. New York, Berlin: Springer.CrossRefGoogle Scholar
16.Shaked, M. & Shanthikumar, J.G. (2007). Stochastic orders. New York: Springer.CrossRefGoogle Scholar
17.Shannon, C.E. (1948). A mathematical theory of communication. Bell System Technical Journal 27: 379423.CrossRefGoogle Scholar
18.Toomaj, A. & Doostparast, M. (2014). A note on signature-based expressions for the entropy of mixed r-out-of-n systems. Naval Research Logistics 61(3): 202206.CrossRefGoogle Scholar
19.Toomaj, A. & Doostparast, M. (2016). On the Kullback Leibler information for mixed systems. International Journal of Systems Science 47(10): 24582465.CrossRefGoogle Scholar
20.Toomaj, A., Sunoj, S.M., & Navarro, J. (2017). Some properties of the cumulative residual entropy of coherent and mixed systems. Journal of Applied Probability 54(2): 379393.CrossRefGoogle Scholar
21.Mahmoudi, M. & Asadi, M. (2011). The dynamic signature of coherent systems. IEEE Transactions on Reliability 60(4): 817822.CrossRefGoogle Scholar
22.Marichal, J.L., Mathonet, P., Navarro, J., & Paroissin, C. (2017). Joint signature of two or more systems with applications to multistate systems made up of two-state components. European Journal of Operational Research 263: 559570.CrossRefGoogle Scholar
23.Wilks, S.S. (1962). Mathematical statistics. New York: Wiley.Google Scholar
24.Zarezadeh, S., Ashrafi, S., & Asadi, M. (2016). A shock model based approach to network reliability. IEEE Transactions on Reliability 65(2): 9921000.CrossRefGoogle Scholar
25.Zarezadeh, S. & Asadi, M. (2010). Results on residual Rényi entropy of order statistics and record values. Information Sciences 180(21): 41954206.CrossRefGoogle Scholar
26.Zarezadeh, S., Mohammadi, L., & Balakrishnan, N. (2018). On the joint signature of several coherent systems with some shared components. European Journal of Operational Research 264(3): 10921100.CrossRefGoogle Scholar