Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-24T03:23:40.079Z Has data issue: false hasContentIssue false
  • English
  • Français

Computing minimal signature of coherent systems through matrix-geometric distributions

Part of: Survival analysis and censored data Special processes

Published online by Cambridge University Press:  16 September 2021

Serkan Eryilmaz Open the ORCID record for Serkan Eryilmaz [Opens in a new window]  and
Fatih Tank
Serkan Eryilmaz*
Affiliation:
Atilim University
Fatih Tank*
Affiliation:
Ankara University
*
*Postal address: Atilim University, Department of Industrial Engineering, Ankara, Turkey. Email address: [email protected]
**Postal address: Ankara University, Department of Actuarial Sciences, Ankara, Turkey. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Signatures are useful in analyzing and evaluating coherent systems. However, their computation is a challenging problem, especially for complex coherent structures. In most cases the reliability of a binary coherent system can be linked to a tail probability associated with a properly defined waiting time random variable in a sequence of binary trials. In this paper we present a method for computing the minimal signature of a binary coherent system. Our method is based on matrix-geometric distributions. First, a proper matrix-geometric random variable corresponding to the system structure is found. Second, its probability generating function is obtained. Finally, the companion representation for the distribution of matrix-geometric distribution is used to obtain a matrix-based expression for the minimal signature of the coherent system. The results are also extended to a system with two types of components.

Keywords

Matrix-geometric distributionminimal signatureprobability generating functionreliabilitysignature

MSC classification

Primary: 62N05: Reliability and life testing
Secondary: 60K10: Applications (reliability, demand theory, etc.)
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 3 , September 2021 , pp. 621 - 636
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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

Balakrishnan, N. and Koutras, M. V. (2002). Runs and Scans with Applications. John Wiley, New York.Google Scholar
Bladt, M. and Nielsen, B. F. (2017). Matrix-Exponential Distributions in Applied Probability. Springer, New York.10.1007/978-1-4939-7049-0CrossRefGoogle Scholar
Da, G., Chan, P. S. and Xu, M. (2018). On the signature of complex system: a decomposed approach. Europ. J. Operat. Res. 265, 1115–1123.10.1016/j.ejor.2017.08.052CrossRefGoogle Scholar
Da, G., Xia, L. and Hu, T. (2014). On computing signatures of k-out-of-n systems consisting of modules. Methodology Comput. Appl. Prob. 16, 223–233.10.1007/s11009-012-9308-5CrossRefGoogle Scholar
Da, G., Xu, M. and Chan, P. S. (2018). An efficient algorithm for computing the signatures of systems with exchangeable components and applications. IISE Trans. 50, 584–595.10.1080/24725854.2018.1429694CrossRefGoogle Scholar
Eryilmaz, S. (2014). On signatures of series and parallel systems consisting of modules with arbitrary structures. Commun. Statist. Simul. Comput. 43, 1202–1211.10.1080/03610918.2012.732174CrossRefGoogle Scholar
Eryilmaz, S. and Tuncel, A. (2015). Computing the signature of a generalized k-out-of-n system. IEEE Trans. Reliab. 64, 766–771.10.1109/TR.2015.2405594CrossRefGoogle Scholar
Eryilmaz, S. and Zuo, M. J. (2010). Computing and applying the signature of a system with two common failure criteria. IEEE Trans. Reliab. 59, 576–580.10.1109/TR.2010.2054170CrossRefGoogle Scholar
Eryilmaz, S., Coolen, F. P. A. and Coolen-Maturi, T. (2018). Mean residual life of coherent systems consisting of multiple types of components. Naval Res. Logistics 65, 86–97.10.1002/nav.21782CrossRefGoogle Scholar
Franko, C. and Yalcin, F. (2017). Signatures of series and parallel systems consisting of non disjoint modules. Commun. Statist. Theory Meth. 46, 11425–11439.10.1080/03610926.2015.1080839CrossRefGoogle Scholar
Gertsbakh, I., Shpungin, Y. and Spizzichino, F. (2011). Signatures of coherent systems built with separate modules. J. Appl. Prob. 48, 843–855.10.1017/S0021900200008366CrossRefGoogle Scholar
Marichal, J. L. (2015). Algorithms and formulae for conversion between system signatures and reliability functions. J. Appl. Prob. 52, 490–507.10.1239/jap/1437658611CrossRefGoogle Scholar
Marichal, J. L., Mathonet, P. and Spizzichino, F. (2015). On modular decompositions of system signature. J. Multivariate Anal. 134, 19–32.10.1016/j.jmva.2014.10.002CrossRefGoogle Scholar
Mohan, P., Agarwal, M. and Sen, K. (2009). Reliability analysis of sparsely connected consecutive-k systems: GERT approach. In 8th International Conference on Reliability, Maintainability and Safety, Chengdu, pp. 213–218. Available at doi: 10.1109/ICRMS.2009.5270207.CrossRefGoogle Scholar
Navarro, J. and Rubio, R. (2009). Computations of signatures of coherent systems with five components. Commun. Statist. Simul. Comput. 39, 68–84.10.1080/03610910903312185CrossRefGoogle Scholar
Navarro, J. and Spizzichino, F. (2020). Aggregation and signature based comparisons of multi-state systems via decompositions of fuzzy measures. Fuzzy Sets and Systems 396, 115–137.10.1016/j.fss.2019.10.011CrossRefGoogle Scholar
Navarro, J., Ruiz, J. M. and Sandoval, C. J. (2007). Properties of coherent systems with dependent components. Commun. Statist. Theory Meth. 36, 175–191.10.1080/03610920600966316CrossRefGoogle Scholar
Samaniego, F. J. (2007). System Signatures and Their Applications in Engineering Reliability. Springer, New York.10.1007/978-0-387-71797-5CrossRefGoogle Scholar
Triantafyllou, I. S. and Koutras, M. V. (2008). On the signature of coherent systems and applications. Prob. Eng. Inf. Sci. 22, 19–35.10.1017/S0269964808000028CrossRefGoogle Scholar
Triantafyllou, I. S. and Koutras, M. V. (2014). Reliability properties of (n,f,k) systems. IEEE Trans. Reliab. 63, 357–366.10.1109/TR.2014.2299495CrossRefGoogle Scholar
Yi, H. and Cui, L. (2018). A new computation method for signature: Markov process method. Naval Res. Logistics 65, 410–426.10.1002/nav.21811CrossRefGoogle Scholar
Zhao, X., Cui, L. and Kuo, W. (2007). Reliability for sparsely connected consecutive-k systems. IEEE Trans. Reliab. 56, 516–524.10.1109/TR.2007.903202CrossRefGoogle Scholar