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Vaisman Ternary Networks Reliability and Monte Carlo 123

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 systemwith 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 p2, p1 and p0, 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.