Journal of Applied Probability

Two-dimensional signatures

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


The notion of the signature is a basic concept and a powerful tool in the analysis of networks and reliability systems of binary type. An appropriate definition of this concept has recently been introduced for systems that have ν possible states (with ν ≥ 3). In this paper we analyze in detail several properties and the most relevant aspects of such a general definition. For simplicity's sake, we focus our attention on the case ν = 3. Our analysis will however provide a number of hints for understanding the basic aspects of the general case.

Article information

J. Appl. Probab., Volume 49, Number 2 (2012), 416-429.

First available in Project Euclid: 16 June 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Two-dimensional anchor two-dimensional signature G-series and G-parallel systems multistate recurrent system homogeneous polynomial


Gertsbakh, Ilya; Shpungin, Yoseph; Spizzichino, Fabio. Two-dimensional signatures. J. Appl. Probab. 49 (2012), no. 2, 416--429. doi:10.1239/jap/1339878795.

Export citation


  • Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.
  • Boland, P. J. (2001). Signatures of indirect majority systems. J. Appl. Prob. 38, 597–603.
  • 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.
  • Da, G., Zheng, B. and Hu, T. (2012). On computing signatures of coherent systems. J. Multivariate Anal. 103, 142–150.
  • Elperin, T., Gertsbakh, I. and Lomonosov, M. (1991). Estimation of network reliability using graph evolution models. IEEE Trans. Reliab. 40, 572–581.
  • Gertsbakh, I. and Shpungin, Y. (2004). Combinatorial approaches to Monte Carlo estimation of network lifetime distribution. Appl. Stoch. Models Business Industry 20, 49–57.
  • Gertsbakh, I. B. and Shpungin, Y. (2009). Models of Network Reliability: Analysis, Combinatorics, and Monte Carlo. CRC Press, Boca Raton, FL.
  • Gertsbakh, I. and Shpungin, Y. (2011). Network Reliability and Resilience. Springer.
  • Gertsbakh, I. and Shpungin, Y. (2012). Multidimensional spectra of multistate systems with binary components. In Recent Advances in System Reliability, eds A. Lisniansky and I. Frenkel, Springer, London, pp. 49–61.
  • Gertsbakh, I., Shpungin, Y. and Spizzichino, F. (2011). Signatures of coherent systems built with separate modules. J. Appl. Prob. 48, 843–855.
  • 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.
  • Levitin, G., Gertsbakh, I. and Shpungin, Y. (2011). Evaluating the damage associated with intentional network disintegration. Reliab. Eng. System Safety 96, 433–439.
  • Lisnianski, A. and Levitin, G. (2003). Multi-State System Reliability. World Scientific, River Edge, NJ.
  • Navarro, J., Samaniego, F. J. and Balakrishnan, N. (2010). The joint signature of coherent systems with shared components. J. Appl. Prob. 47, 235–253.
  • 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, 314–326.
  • Samaniego, F. J. (1985). On the closure of the IFR class under formation of coherent systems. IEEE Trans. Reliab. R-34, 69–72.
  • Samaniego, F. J. (2007). System Signatures and Their Application in Engineering Reliability. Springer, New York.
  • Spizzichino, F. (2008). The role of signature and symmetrization for systems with non-exchangeable components. In Advances in Mathematical Modeling for Reliability, eds T. Bedford \et, IOS Press, Amsterdam, pp. 138–148.
  • Spizzichino, F. and Navarro, J. (2012). Signatures and symmetry properties of coherent systems. In Recent Advances in System Reliability, eds A. Lisniansky and I. Frenkel, Springer, London, pp. 33–48.