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

Part of: Special processes Operations research and management science

Published online by Cambridge University Press:  30 November 2017

S. Ashrafi  and
M. Asadi
S. Ashrafi*
Affiliation:
University of Isfahan
M. Asadi*
Affiliation:
University of Isfahan and Institute of Research in Fundamental Sciences
*
* Postal address: Department of Statistics, University of Isfahan, Isfahan, 81744, Iran.
* Postal address: Department of Statistics, University of Isfahan, Isfahan, 81744, Iran.
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Abstract

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 t1 and t2 (t1 < t2). Using the notion of the two-dimensional signature, the probability of the number of failed components of the network is calculated, at t1 and t2, 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.

Keywords

Signature matrixmulti-state systembivariate increasing failure ratetotally positive of order twostochastic order

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Secondary: 90B25: Reliability, availability, maintenance, inspection
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 4 , December 2017 , pp. 1051 - 1070
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Asadi, M. and Berred, A. (2012). On the number of failed components in a coherent operating system. Statist. Prob. Lett. 82, 2156–2163. Google Scholar
[2] Ashrafi, S. and Asadi, M. (2014). Dynamic reliability modeling of three-state networks. J. Appl. Prob. 51, 999–1020. Google Scholar
[3] Ashrafi, S. and Asadi, M. (2015). On the stochastic and dependence properties of the three-state systems. Metrika 78, 261–281. Google Scholar
[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. 143–166. Google Scholar
[5] Da, G., Zheng, B. and Hu, T. (2012). On computing signatures of coherent systems. J. Multivariate Anal. 103, 142–150. CrossRefGoogle Scholar
[6] Eryilmaz, S. (2010). Mean residual and mean past lifetime of multi-state systems with identical components. IEEE Trans. Reliab. 59, 644–649. Google Scholar
[7] 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. Google Scholar
[8] Eryilmaz, S. (2011). Dynamic behavior of k-out-of-n : G systems. Operat. Res. Lett. 39, 155–159. Google Scholar
[9] Eryilmaz, S. (2012). The number of failed components in a coherent system with exchangeable components. IEEE Trans. Reliab. 61, 203–207. CrossRefGoogle Scholar
[10] Eryilmaz, S. (2015). On the mean number of remaining components in three-state k-out-of-n system. Operat. Res. Lett. 43, 616–621. Google Scholar
[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, 97–106. CrossRefGoogle Scholar
[12] Gertsbakh, I. B. and Shpungin, Y. (2010). Models of Network Reliability: Analysis, Combinatorics, and Monte Carlo. CRC, Boca Raton, FL. Google Scholar
[13] Gertsbakh, I. B. and Shpungin, Y. (2012). Stochastic models of network survivability. Quality Tech. Quant. Manag. 9, 45–58. CrossRefGoogle Scholar
[14] Gertsbakh, I., Shpungin, Y. and Spizzichino, F. (2011). Signatures of coherent systems built with separate modules. J. Appl. Prob. 48, 843–855. Google Scholar
[15] Gertsbakh, I., Shpungin, Y. and Spizzichino, F. (2012). Two-dimensional signatures. J. Appl. Prob. 49, 416–429. Google Scholar
[16] Harris, R. (1970). A multivariate definition for increasing hazard rate distribution functions. Ann. Math. Statist. 41, 713–717. Google Scholar
[17] Huang, J., Zuo, M. J. and Wu, Y. (2000). Generalized multi-state k-out-of-n : G systems. IEEE Trans. Reliab. 49, 105–111. CrossRefGoogle Scholar
[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, 467–498. Google Scholar
[19] Kelkinnama, M., Tavangar, M. and Asadi, M. (2015). New developments on stochastic properties of coherent systems. IEEE Trans. Reliab. 64, 1276–1286. Google Scholar
[20] Levitin, G., Gertsbakh, I. and Shpungin, Y. (2011). Evaluating the damage associated with intentional network disintegration. Reliab. Eng. System Safety 96, 433–439. Google Scholar
[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, 332–348. Google Scholar
[22] Ling, X. and Li, P. (2013). Stochastic comparisons for the number of working components of a system in random environment. Metrika 76, 1017–1030. Google Scholar
[23] Lisnianski, A. and Levitin, G. (2003). Multi-State System Reliability: Assessment, Optimization and Applications. World Scientific, River Edge, NJ. Google Scholar
[24] Marichal, J.-L. (2015). Algorithms and formulae for conversion between system signatures and reliability functions. J. Appl. Prob. 52, 490–507. Google Scholar
[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, 856–867. Google Scholar
[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, 1011–1027. Google Scholar
[27] Samaniego, F. J. (2007). System Signatures and their Applications in Engineering Reliability. Springer, New York. Google Scholar
[28] Samaniego, F. J. and Navarro, J. (2016). On comparing coherent systems with heterogeneous components. Adv. Appl. Prob. 48, 88–111. Google Scholar
[29] Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York. CrossRefGoogle Scholar
[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, 53–58. Google Scholar
[31] Zarezadeh, S. and Asadi, M. (2013). Network reliability modeling under stochastic process of component failures. IEEE Trans. Reliab. 62, 917–929. Google Scholar
[32] Zarezadeh, S., Ashrafi, S. and Asadi, M. (2016). A shock model based approach to network reliability. IEEE Trans. Reliab. 65, 992–1000. Google Scholar
[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, 1437–1443. Google Scholar
[34] Zuo, M. J. and Tian, Z. (2006). Performance evaluation for generalized multi-state k-out-of-n systems. IEEE Trans. Reliab. 55, 319–327. Google Scholar