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.