On the Stochastic Properties of Unfailed Components in Used Networks

Document Type : Original Paper


Department of Statistics, Shiraz University, Shiraz, Iran


We consider a two-state network consists of n components and assume that the failure of components occur according to a nonhomogeneous Poisson process. Some networks have the property that after the failure, some of the components remain unfailed. The remaining unfailed components might be resumed from the network and be used again in a new network. In this paper, we explore some aging properties and stochastic comparisons of the residual lifetime of remaining unfailed components of the failed network.


Main Subjects

 [1] Boland, P.J., Samaniego, F.J. and Vestrup, E.M. (2003). Linking dominations and signatures in network reliability theory, In: Lindquist, B.H., Doksum, K.A. (Eds) Mathematical and statistical methods in reliability. World Scientific, Singapore, 89-103.
[2] Samaniego, F.J. (1985). On closure of the IFR class under formation of coherent systems, IEEE Transactions on Reliability, 34, 69-72.
[3] Zarezadeh, S., Mohammadi, L. and Balakrishnan, N. (2018). On the joint signature of several coherent systems with some shared components, European Journal of Operational Research, 264(3), 1092-1100.
[4] Navarro, J., Balakrishnan, N. and Samaniego. F.J. (2008). Mixture representations of residual lifetimes of used systems. Journal of Applied Probability, 45, 1097-1112.
[5] Eryilmaz, S. (2014). A study on reliability of coherent systems equipped with a cold standby component, Metrika, 77, 349-359.
[6] Gertsbakh, I., Rubinstein, R., Shpungin, Y. and Vaisman, R. (2014). Permutational methods for performance analysis of stochastic flow networks, Probability in the Engineering and Informational Sciences, 28(1), 21-38.
[7] Patelli, E., Feng, G., Coolen, F.P. and Coolen-Maturi, T. (2017). Simulation methods for system reliability using the survival signature, Reliability Engineering & System Safety, 167, 327-337.
[8] Eryilmaz, S. and Bayramoglu, I. (2012). On extreme residual lives after the failure of the system, Mathematical Problems in Engineering, 1-11.
[9] Zarezadeh, S. and Asadi, M. (2013). Network reliability modeling under stochastic process of component failures, IEEE Transactions on Reliability, 62(4), 917-929.
[10] Nakagawa, T. (2011). Stochastic Processes: With Applications to Reliability Theory, New York: Springer-Verlag.
[11] Arnold, B.C., Balakrishnan, N. and Nagaraja, H.N. (2011). Records, Wiley.
[12] Gupta, R. C. and Kirmani, S.N.U.A. (1988). Closure and monotonicity properties of nonhomogeneous Poisson processes and record values, Probability in the Engineering and Informational Sciences, 2, 475-484.
 [13] Balakrishnan, N. and Asadi, M. (2012). A proposed measure of residual life of live components of a coherent system, IEEE Transactions on Reliability, 61(1), 41-49.
[14] Bairamov, I. and Arnold, B.C. (2008). On the residual life lengths of the remaining components in a (n - k + 1)-out-of-n system, Statistics & Probability Letters, 78, 945-952.
[15] Gurler, S. (2012). On residual lifetimes in sequential (n-k+1)-out-of-n systems. Statistical Papers, 53(1), 23-31.
[16] Balakrishnan, N., Barmalzan, G. and Haidari, A. (2014). Stochastic orderings and ageing properties of residual life lengths of live components in (n-k+ 1)-out-of-n systems, Journal of Applied Probability, 51(1), 58-68.
[17] Balakrishnan, N., Barmalzan, G. and Haidari, A. (2016). Multivariate stochastic comparisons of multivariate mixture models and their applications, Journal of Multivariate Analysis, 145, 37-43.
[18] Kelkinnama, M. and Asadi, M. (2014). Stochastic properties of components in a used coherent system, Methodology and Computing in Applied Probability, 16(3), 675-691.
[19] Kelkin Nama, M., Asadi, M. and Zhang, Z. (2013). On the residual life lengths of the remaining components in a coherent system, Metrika, 76(7), 979-996.
[20] Shaked, M. and Shanthikumar, J.G. (2007). Stochastic Orders, New York: Springer-Verlag.
[21] Rao, M., Chen, Y., Vemuri, B.C. and Wang, F. (2004). Cumulative residual entropy: A new measure of information, IEEE Transactions on Information Theory, 50(6), 1220-1228.
[22] Elperin, T., Gertsbakh, I. and Lomonosov, M. (1991). Estimation of network reliability using graph evolution models, IEEE Transactions on Reliability, 40(5), 572-581.
[23] Khaledi, B.E. and Shaked, M. (2010). Stochastic comparisons of multivariate mixtures. Journal of Multivariate Analysis, 101(10), 2486-2498.
[24] Belzunce, F., Mercader, J.A., Ruiz, J.M. and Spizzichino, F. (2009). Stochastic comparisons of multivariate mixture models. Journal of Multivariate Analysis, 100(8), 1657-1669.
[25] Ahmadi, J. and Arghami, N.R. (2001). Some univariate stochastic orders on record values. Communications in Statistics - Theory and Methods, 30, 69-74.