Journal of Applied Probability

Signatures of coherent systems built with separate modules

Ilya Gertsbakh, Yoseph Shpungin, and Fabio Spizzichino

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

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.

Article information

Source
J. Appl. Probab., Volume 48, Number 3 (2011), 843-855.

Dates
First available in Project Euclid: 23 September 2011

Permanent link to this document
https://projecteuclid.org/euclid.jap/1316796919

Digital Object Identifier
doi:10.1239/jap/1316796919

Mathematical Reviews number (MathSciNet)
MR2884820

Zentralblatt MATH identifier
1230.60096

Subjects
Primary: 60K10: Applications (reliability, demand theory, etc.)
Secondary: 90B25: Reliability, availability, maintenance, inspection [See also 60K10, 62N05]

Keywords
Cumulative and tail signatures recurrent series parallel system anchor homogeneous polynomial

Citation

Gertsbakh, Ilya; Shpungin, Yoseph; Spizzichino, Fabio. Signatures of coherent systems built with separate modules. J. Appl. Probab. 48 (2011), no. 3, 843--855. doi:10.1239/jap/1316796919. https://projecteuclid.org/euclid.jap/1316796919


Export citation

References

  • Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability, and Life Testing. Holt, Riehart and Wiston, New York.
  • Boland, P. J. (2001). Signatures of indirect majority systems. J. Appl. Prob. 38, 597–603.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • Elperin, T., Gertsbakh, I. B. and Lomonosov, M. (1991). Estimation of network reliability using graph evolution models. IEEE Trans. Reliab. 40, 572–581.
  • Gertsbakh, I. B. and Shpungin, Y. (2010). Models of Network Reliability. CRC Press, Boca Raton, FL.
  • 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.
  • 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.
  • Marichal, J.-L. and Mathonet, P. (2011). System signatures for dependent lifetimes: explicit expressions and interpretations. J. Multivariate Anal. 102, 931–936.
  • Navarro, J. and Rubio, R. (2010). Computation of signatures of coherent systems with five components. Commun. Statist. Simul. Comput. 39, 68–84.
  • Navarro, J. and Rychlik, T. (2007). Reliability and expectation bounds for coherent systems with exchangeable components. J. Multivariate Anal. 98, 102–113.
  • Navarro, J. and Shaked, M. (2006). Hazard rate ordering of order statistics and systems. J. Appl. Prob. 43, 391–408.
  • 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.
  • 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.
  • Samaniego, F. J. (1985). On closure of the IFR class under formation of coherent systems. IEEE Trans Reliab. 34, 69–72.
  • Samaniego, F. J. (2007). System Signatures and Their Application in Engineering Reliability. Springer, New York.
  • Satyanarayana, A. and Prabhakar, A. (1978). New topological formula and rapid algorithm for reliability analysis of complex networks. IEEE Trans. Reliab. 27, 82–100.
  • 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.
  • Triantafyllou, I. S. and Koutras, M. V. (2008). On the signature of coherent systems and applications. Prob. Eng. Inf. Sci. 22, 19–35.
  • Wolfram, S. (1991). MATHEMATICA: A System for doing Mathematics by Computer, 2nd edn. Addison-Wesley, Redwood, CA.