Ternary Networks pp 1-23 | Cite as

# Networks with Ternary Components: Ternary Spectrum

## Abstract

In this chapter we consider a monotone binary system with ternary components. “Ternary” or (“trinary”) means that each component can be in one of three states: *up, middle (mid)* and *down*. It turns out that for this system exists a combinatorial invariant by means of which it is possible to count the number \(C(r;x)\) of system failure sets with a given number of \(r\) components in *up*, \(x\) components in \(down\) and the remaining components in state \(mid\). This invariant is called ternary D-spectrum and it is an analogue of signature or D-spectrum for a binary system with binary components. Contrary to D-spectrum, it is not a single set of probabilities, but a collection of such sets. The \(r\)-th member of this collection resembles a D-spectrum computed for a special case for which \(r\) components are permanently turned into state *up*. If system (network) components are statistically independent and identical, and have probabilities \(p_2, p_1 \) and \(p_0 \), to be in *up, mid* and *down*, respectively, then the ternary D-spectrum allows obtaining a simple formula for calculating system *DOWN* probability. We consider also so-called ternary importance spectrum by means of which it becomes possible to rank system components by their importance measures. These importance measures are similar to Birnbaum importance measures that are well-known in Reliability Theory. The chapter is concluded by a description of Monte Carlo procedures used for approximating
the ternary spectra.

### Keywords

Ternary components Ternary network Signature Ternary D-spectrum Failure sets Ternary importance measure### References

- 1.Barlow RE, Proschan F (1975) Statistical theory of reliability and life testing. Rinehart and Winston Inc., HoltGoogle Scholar
- 2.Birnbaum ZW (1969) On the importance of different components in multicomponent system. In: Krishnaiah PR (ed) Multivarite analysis-II, Academic Press, New York, pp 581–592Google Scholar
- 3.Elperin T, Gertsbakh IB, Lomonosov M (1991) Estimation of network reliability using graph evolution models. IEEE Trans Reliab 40(5):572–581Google Scholar
- 4.Gertsbakh I, Shpungin Y (2009) Models of network reliability: analysis, Combinatorics and Monte Carlo. CRC Press, Boca RatonGoogle Scholar
- 5.Gertsbakh I, Shpungin Y (2011) Network reliability and resilience. Springer Briefs in Electrical and Computer Engineering, Springer, New YorkGoogle Scholar
- 6.Gertsbakh I, Shpungin Y (2012) Combinatorial approach to computing importance indices of coherent systems. Probab Eng Inf Sci 26:117–128CrossRefMATHMathSciNetGoogle Scholar
- 7.Gertsbakh I, Rubinstein R, Shpungin Y (2014) Permutational methods for performance analysis of stochastic flow networks. Probab Eng Inf Sci 28:21–38CrossRefGoogle Scholar
- 8.Gertsbakh I, Shpungin Y (2012) Failure development in a system of two connected networks. Transp Commun 13(4):255–260Google Scholar
- 9.Gertsbakh I, Shpungin Y (2012) Stochastic models of network survivability. Qual Technol Quant Manag 9(1):45–58MathSciNetGoogle Scholar
- 10.Samaniego FJ (1985) On closure under ifr formation of coherent systems. IEEE Trans Reliab 34:69–72CrossRefMATHGoogle Scholar
- 11.Samaniego FJ (2007) System signatures and their application in engineering reliability. Springer, New YorkGoogle Scholar