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Signatures of Coherent Systems Built with Separate Modules

Part of: Operations research and management science Special processes

Published online by Cambridge University Press:  14 July 2016

Ilya Gertsbakh ,
Yoseph Shpungin  and
Fabio Spizzichino
Ilya Gertsbakh*
Affiliation:
Ben Gurion University
Yoseph Shpungin*
Affiliation:
Shamoon College of Engineering
Fabio Spizzichino*
Affiliation:
Sapienza University of Rome
*
∗ Postal address: Shai Agnon 30/25, 69362 Tel-Aviv, Israel. Email address: elyager@bezeqint.net
∗∗ Postal address: Department of Software Engineering, Shamoon College of Engineering, Beer Sheva, Bialik/Basel Streets, Beer Sheva, Israel. Email address: yosefs@sce.ac.il
∗∗∗ Postal address: Department of Mathematics, Sapienza University of Rome, Piazzale Aldo Moro 2, 00185 Rome, Italy. Email address: fabio.spizzichino@uniroma1.it
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Abstract

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The signature is an important structural characteristic of a coherent system. Its computation, however, is often rather involved and complex. We analyze several cases where this complexity can be considerably reduced. These are the cases when a ‘large’ coherent system is obtained as a series, parallel, or recurrent structure built from ‘small’ modules with known signature. Corresponding formulae can be obtained in terms of cumulative notions of signatures. An algebraic closure property of families of homogeneous polynomials plays a substantial role in our derivations.

Keywords

Cumulative and tail signaturesrecurrentseriesparallel systemanchorhomogeneous polynomial

MSC classification

Secondary: 60K10: Applications (reliability, demand theory, etc.) 90B25: Reliability, availability, maintenance, inspection
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 3 , September 2011 , pp. 843 - 855
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability, and Life Testing. Holt, Riehart and Wiston, New York.Google Scholar
[2] Boland, P. J. (2001). Signatures of indirect majority systems. J. Appl. Prob. 38, 597–603.Google Scholar
[3] Block, H., Dugas, M. R. and Samaniego, F. J. (2007). Characterizations of the relative behavior of two systems via properties of their signature vectors. In Advances in Distribution Theory, Order Statistics, and Inference, Birkhäuser, Boston, MA, pp. 279–289.Google Scholar
[4] Block, H. W., Dugas, M. R. and Samaniego, F. J. (2007). Signature-related results on system lifetimes. In Advances in Statistical Modeling and Inference, World Scientific, Hackensack, NJ, pp. 115–129.Google Scholar
[5] Boland, P. J. and Samaniego, F. J. (2004). Stochastic ordering results for consecutive k-out-of-n: F systems. IEEE Trans. Reliab. 53, 7–10.Google Scholar
[6] Boland, P. J. and Samaniego, F. J. (2004). The signature of a coherent system and its applications in reliability. In Mathematical Reliability: an Expository Perspective, Kluwer, Boston, MA, pp. 3–30.Google Scholar
[7] Boland, P. J., Samaniego, F. J. and Vestrup, E. M. (2003). Linking dominations and signatures in network reliability theory. In Mathematical and Statistical Methods in Reliability (Trondheim, 2002), World Scientific, River Edge, NJ, pp. 89–103.Google Scholar
[8] Elperin, T., Gertsbakh, I. B. and Lomonosov, M. (1991). Estimation of network reliability using graph evolution models. IEEE Trans. Reliab. 40, 572–581.Google Scholar
[9] Gertsbakh, I. B. and Shpungin, Y. (2010). Models of Network Reliability. CRC Press, Boca Raton, FL.Google Scholar
[10] Harms, D. D., Kraetzl, M., Colbourn, C. J. and Devitt, S. J. (1995). Network Reliability: Experiments with a Symbolic Algebra Environment. CRC Press, Boca Raton, FL.Google Scholar
[11] Kochar, S., Mukerjee, H. and Samaniego, F. J. (1999). The “signature” of a coherent system and its application to comparisons among systems. Naval Res. Logistics 46, 507–523.Google Scholar
[12] Marichal, J.-L. and Mathonet, P. (2011). System signatures for dependent lifetimes: explicit expressions and interpretations. J. Multivariate Anal. 102, 931–936.Google Scholar
[13] Navarro, J. and Rubio, R. (2010). Computation of signatures of coherent systems with five components. Commun. Statist. Simul. Comput. 39, 68–84.Google Scholar
[14] Navarro, J. and Rychlik, T. (2007). Reliability and expectation bounds for coherent systems with exchangeable components. J. Multivariate Anal. 98, 102–113.Google Scholar
[15] Navarro, J. and Shaked, M. (2006). Hazard rate ordering of order statistics and systems. J. Appl. Prob. 43, 391–408.Google Scholar
[16] Navarro, J., Samaniego, F. J., Balakrishnan, N. and Bhattacharaya, D. (2008). On the application and extension of system signatures in engineering reliability. Naval Res. Logistics 55, 313–327.Google Scholar
[17] Navarro, J., Spizzichino, F. and Balakrishnan, N. (2010). Applications of average and projected systems to the study of coherent systems. J. Multivariate Anal. 101, 1471–1482.Google Scholar
[18] Samaniego, F. J. (1985). On closure of the IFR class under formation of coherent systems. IEEE Trans Reliab. 34, 69–72.Google Scholar
[19] Samaniego, F. J. (2007). System Signatures and Their Application in Engineering Reliability. Springer, New York.Google Scholar
[20] Satyanarayana, A. and Prabhakar, A. (1978). New topological formula and rapid algorithm for reliability analysis of complex networks. IEEE Trans. Reliab. 27, 82–100.Google Scholar
[21] Spizzichino, F. (2008). The role of signature and symmetrization for systems with non-exchangeable components. In Mathematical Modeling for Reliability, IOS, Amsterdam, pp. 138–148.Google Scholar
[22] Triantafyllou, I. S. and Koutras, M. V. (2008). On the signature of coherent systems and applications. Prob. Eng. Inf. Sci. 22, 19–35.Google Scholar
[23] Wolfram, S. (1991). MATHEMATICA: A System for doing Mathematics by Computer, 2nd edn. Addison-Wesley, Redwood, CA.Google Scholar