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The failure probability of components in three-state networks with applications to age replacement policy

  • S. Ashrafi (a1) and M. Asadi (a2)


In this paper we investigate the stochastic properties of the number of failed components of a three-state network. We consider a network made up of n components which is designed for a specific purpose according to the performance of its components. The network starts operating at time t = 0 and it is assumed that, at any time t > 0, it can be in one of states up, partial performance, or down. We further suppose that the state of the network is inspected at two time instants t 1 and t 2 (t 1 < t 2). Using the notion of the two-dimensional signature, the probability of the number of failed components of the network is calculated, at t 1 and t 2, under several scenarios about the states of the network. Stochastic and ageing properties of the proposed failure probabilities are studied under different conditions. We present some optimal age replacement policies to show applications of the proposed criteria. Several illustrative examples are also provided.


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* Postal address: Department of Statistics, University of Isfahan, Isfahan, 81744, Iran.
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[1] Asadi, M. and Berred, A. (2012). On the number of failed components in a coherent operating system. Statist. Prob. Lett. 82, 21562163.
[2] Ashrafi, S. and Asadi, M. (2014). Dynamic reliability modeling of three-state networks. J. Appl. Prob. 51, 9991020.
[3] Ashrafi, S. and Asadi, M. (2015). On the stochastic and dependence properties of the three-state systems. Metrika 78, 261281.
[4] 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. 143166.
[5] Da, G., Zheng, B. and Hu, T. (2012). On computing signatures of coherent systems. J. Multivariate Anal. 103, 142150.
[6] Eryilmaz, S. (2010). Mean residual and mean past lifetime of multi-state systems with identical components. IEEE Trans. Reliab. 59, 644649.
[7] Eryilmaz, S. (2010). Number of working components in consecutive k-out-of-n system while it is working. Commun. Statist. Simul. Comput. 39, 683692.
[8] Eryilmaz, S. (2011). Dynamic behavior of k-out-of-n : G systems. Operat. Res. Lett. 39, 155159.
[9] Eryilmaz, S. (2012). The number of failed components in a coherent system with exchangeable components. IEEE Trans. Reliab. 61, 203207.
[10] Eryilmaz, S. (2015). On the mean number of remaining components in three-state k-out-of-n system. Operat. Res. Lett. 43, 616621.
[11] Eryilmaz, S. and Xie, M. (2014). Dynamic modeling of general three-state k-out-of-n : G systems: permanent-based computational results. J. Comput. Appl. Math. 272, 97106.
[12] Gertsbakh, I. B. and Shpungin, Y. (2010). Models of Network Reliability: Analysis, Combinatorics, and Monte Carlo. CRC, Boca Raton, FL.
[13] Gertsbakh, I. B. and Shpungin, Y. (2012). Stochastic models of network survivability. Quality Tech. Quant. Manag. 9, 4558.
[14] Gertsbakh, I., Shpungin, Y. and Spizzichino, F. (2011). Signatures of coherent systems built with separate modules. J. Appl. Prob. 48, 843855.
[15] Gertsbakh, I., Shpungin, Y. and Spizzichino, F. (2012). Two-dimensional signatures. J. Appl. Prob. 49, 416429.
[16] Harris, R. (1970). A multivariate definition for increasing hazard rate distribution functions. Ann. Math. Statist. 41, 713717.
[17] Huang, J., Zuo, M. J. and Wu, Y. (2000). Generalized multi-state k-out-of-n : G systems. IEEE Trans. Reliab. 49, 105111.
[18] Karlin, S. and Rinott, Y. (1980). Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions. J. Multivariate Anal. 10, 467498.
[19] Kelkinnama, M., Tavangar, M. and Asadi, M. (2015). New developments on stochastic properties of coherent systems. IEEE Trans. Reliab. 64, 12761286.
[20] Levitin, G., Gertsbakh, I. and Shpungin, Y. (2011). Evaluating the damage associated with intentional network disintegration. Reliab. Eng. System Safety 96, 433439.
[21] Lindqvist, B. H., Samaniego, F. J. and Huseby, A. B. (2016). On the equivalence of systems of different sizes, with applications to system comparisons. Adv. Appl. Prob. 48, 332348.
[22] Ling, X. and Li, P. (2013). Stochastic comparisons for the number of working components of a system in random environment. Metrika 76, 10171030.
[23] Lisnianski, A. and Levitin, G. (2003). Multi-State System Reliability: Assessment, Optimization and Applications. World Scientific, River Edge, NJ.
[24] Marichal, J.-L. (2015). Algorithms and formulae for conversion between system signatures and reliability functions. J. Appl. Prob. 52, 490507.
[25] Navarro, J., Samaniego, F. J. and Balakrishnan, N. (2011). Signature-based representations for the reliability of systems with heterogeneous components. J. Appl. Prob. 48, 856867.
[26] 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, 10111027.
[27] Samaniego, F. J. (2007). System Signatures and their Applications in Engineering Reliability. Springer, New York.
[28] Samaniego, F. J. and Navarro, J. (2016). On comparing coherent systems with heterogeneous components. Adv. Appl. Prob. 48, 88111.
[29] Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.
[30] Tian, Z., Yam, R. C. M., Zuo, M. J. and Huang, H.-Z. (2008). Reliability bounds for multi-state k-out-of-n systems. IEEE Trans. Reliab. 57, 5358.
[31] Zarezadeh, S. and Asadi, M. (2013). Network reliability modeling under stochastic process of component failures. IEEE Trans. Reliab. 62, 917929.
[32] Zarezadeh, S., Ashrafi, S. and Asadi, M. (2016). A shock model based approach to network reliability. IEEE Trans. Reliab. 65, 9921000.
[33] Zhao, X. and Cui, L. (2010). Reliability evaluation of generalized multi-state k-out-of-n systems based on FMCI approach. Internat. J. System Sci. 41, 14371443.
[34] Zuo, M. J. and Tian, Z. (2006). Performance evaluation for generalized multi-state k-out-of-n systems. IEEE Trans. Reliab. 55, 319327.
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