Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-25T01:39:04.059Z Has data issue: false hasContentIssue false
  • English
  • Français

D-SPECTRUM AND RELIABILITY OF A BINARY SYSTEM WITH TERNARY COMPONENTS

Published online by Cambridge University Press:  14 October 2015

Ilya B. Gertsbakh ,
Yoseph Shpungin  and
Radislav Vaisman
Ilya B. Gertsbakh
Affiliation:
Department of Mathematics, Ben-Gurion University, P. O. Box 653, Beer-Sheva 84105, Israel E-mail: elyager@bezeqint.net
Yoseph Shpungin
Affiliation:
Software Engineering Department, Sami Shamoon College of Engineering, Beer Sheva 84100, Israel E-mail: yosefs@sce.ac.il
Radislav Vaisman
Affiliation:
School of Mathematics and Physics, The University of Queensland, Brisbane 4072, Australia E-mail: r.vaisman@uq.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a monotone binary system with ternary components. “Ternary” means that each component can be in one of three states: up, middle (mid) and down. Handling such systems is a hard task, even if a part of the components have no mid state. Nevertheless, the permutation Monte Carlo methods, that proved very useful for dealing with binary components, can be efficiently used also for ternary monotone systems. It turns out that for “ternary” system there also exists a combinatorial invariant by means of which it becomes possible to count the number C(r;x) of system failure sets which have a given number r and x of components in up and down states, respectively. This invariant is called ternary D-spectrum and it is an analogue of the D-spectrum (or signature) of a system with binary components. Its value is the knowledge of system failure or path set properties which do not depend on stochastic mechanism governing component failures. In case of independent and identical components, knowing D-spectrum makes it easy to calculate system UP or DOWN probability for a variety of UP/DOWN definitions suitable for systems of many types, like communication networks, flow and supply networks, etc.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 30 , Issue 1 , January 2016 , pp. 25 - 39
Copyright
Copyright © Cambridge University Press 2015 

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

1.Barlow, R.E. & Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, Inc, New York.Google Scholar
2.Elperin, T., Gertsbakh, I.B., & Lomonosov, M. (1991). Estimation of network reliability using graph evolution models. IEEE Transactions on Reliability, 40(5): 572–581.CrossRefGoogle Scholar
3.Gertsbakh, I., Rubinstein, R., Shpungin, Y., & Vaisman, R. (2014). Permutational methods for performance analysis of stochastic flow networks. Probability in Engineering and Informational Sciences, 28(1): 21–38.CrossRefGoogle Scholar
4.Gertsbakh, I. & Shpungin, Y. (2009). Models of Network Reliability: Analysis, Combinatorics and Monte Carlo. CRC Press, Boca Raton, New York.Google Scholar
5.Gertsbakh, I. & Shpungin, Y. (2011). Network Reliability and Resilience, Springer Briefs in Electrical and Computer Engineering. Springer, New York.CrossRefGoogle Scholar
6.Gertsbakh, I. & Shpungin, Y. (2012). Spectral approach to reliability evaluation of flow networks. In Proceedings of the European Modeling and Simulation Symposium, 68–73.Google Scholar
7.Karger, D.R. (1996). A randomized fully polynomial scheme for all terminal network reliability problem. SIAM Journal on Computing, 25, 11–17.Google Scholar
8.Newman, M.E.J. (2010). Networks. An Introduction. Cambridge University Press, New York.CrossRefGoogle Scholar
9.Ramirez-Marquez, J. & David, W. Coit. (2005). A Monte Carlo simulation approach for approximating multi-state two-terminal reliability. Reliability Engineering and System Safety 87(2): 253–264.CrossRefGoogle Scholar
10.Samaniego, F.J. (1985). On closure under IFR formation of coherent systems. IEEE Transaction on Reliability, 34: 69–72.CrossRefGoogle Scholar
11.Samaniego, F.J. (2007). System Signatures and Their Application in Engineering Reliability. Springer, New York.CrossRefGoogle Scholar