Hostname: page-component-7c8c6479df-p566r Total loading time: 0 Render date: 2024-03-28T21:16:42.046Z Has data issue: false hasContentIssue false

SIGNATURE-BASED INFORMATION MEASURES OF MULTI-STATE NETWORKS

Published online by Cambridge University Press:  14 June 2018

S. Zarezadeh
Affiliation:
Department of Statistics, Shiraz University, Shiraz 71454, Iran E-mail: s.zarezadeh@shirazu.ac.ir
M. Asadi
Affiliation:
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: m.assadi@sci.ui.ac.ir
S. Eftekhar
Affiliation:
Department of Statistics, Shiraz University, Shiraz 71454, Iran E-mail: sana.eftekhar@shirazu.ac.ir

Abstract

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.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Ashrafi, S. & Asadi, M. (2014). Dynamic reliability modeling of three-state networks. Journal of Applied Probability 51(4): 9991020.Google Scholar
2.Ashrafi, S. & Asadi, M. (2015). On the stochastic and dependence properties of the three-state systems. Metrika 78(3): 261281.Google 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.Google 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.Google Scholar
5.Ebrahimi, N., Soofi, E.S., & Zahedi, H. (2004). Information properties of order statistics and spacings. IEEE Transactions on Information Theory 50: 177183.Google Scholar
6.Gertsbakh, I. & Shpungin, Y. (2011). Network Reliability and Resilience. Springer Briefs in Electrical and Computer Engineering. Berlin, Germany: Springer-Verlag.Google Scholar
7.Gertsbakh, I. & Shpungin, Y. (2012). Stochastic models of network survivability. Quality Technology & Quantitative Management 9(1): 4558.Google Scholar
8.Gertsbakh, I., Shpungin, Y., & Spizzichino, F. (2012). Two-dimensional signatures. Journal of Applied Probability 49(2): 416429.Google 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.Google Scholar
10.Kullback, S. & Leibler, R.A. (1951). On information and sufficiency. Annals of Mathematical Statistics 22(1): 7986.Google Scholar
11.Navarro, J., Balakrishnan, N., & Samaniego, F.J. (2008). Mixture reptesentations of residual lifetimes of used systems. Journal of Applied Probability 45: 10971112.Google 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.Google Scholar
13.Parzen, E. (1979). Nonparametric statistical data modelling. Journal of American Statistical Association 74: 105122.Google Scholar
14.Samaniego, F.J. (1985). On closure of the IFR class under formation of coherent systems. IEEE Transactions on Reliability 34: 6972.Google Scholar
15.Samaniego, F.J. (2007). System signatures & their applications in reliability engineering. New York, Berlin: Springer.Google Scholar
16.Shaked, M. & Shanthikumar, J.G. (2007). Stochastic orders. New York: Springer.Google Scholar
17.Shannon, C.E. (1948). A mathematical theory of communication. Bell System Technical Journal 27: 379423.Google 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.Google Scholar
19.Toomaj, A. & Doostparast, M. (2016). On the Kullback Leibler information for mixed systems. International Journal of Systems Science 47(10): 24582465.Google 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.Google Scholar
21.Mahmoudi, M. & Asadi, M. (2011). The dynamic signature of coherent systems. IEEE Transactions on Reliability 60(4): 817822.Google 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.Google 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.Google Scholar
25.Zarezadeh, S. & Asadi, M. (2010). Results on residual Rényi entropy of order statistics and record values. Information Sciences 180(21): 41954206.Google 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.Google Scholar