Publisher’s description: Ternary means “based on three”. This book deals with reliability investigations of networks whose components subject to failures can be in three states – up, down and middle (mid), contrary to traditionally considered networks having only binary (up/down) components. Extending binary case to ternary allows to consider more realistic and flexible models for communication, flow and supply networks.
From the text: In the first chapter (Networks with ternary components: ternary spectrum) 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.
Sections 2.1–2.4 of chapter 2 (Applications) present numerical illustrations and applications of the theory developed in Chap. 1. Section 2.5 deals with networks having independent and nonidentical components.
Chapter 3 (Interaction of Networks). The simplest form of two interacting networks is sharing the same set of nodes by two independent networks. For example, the power supply and water supply networks in the same geographic area share the same set of nodes (houses or residencies). Section 3.1 presents several simple results concerning the size of the set of nodes which receive “full” supply, i.e. are adjacent to edges of both types. Here we use some basic facts from the theory of large random Erdős-Renyi or Poisson networks. Section 3.2 considers a system of two or more finite interacting networks. Here the interaction means that a node \(v_a\) of network \(A\) delivers “infection” to a randomly chosen node \(v_B\) in \(B\) which in turn, bounces back and infects another randomly chosen node \(w_a\) in network \(A\), and so on. As a result, random number \(Y\) of nodes in \(B\) gets “infected” and fails. We compute, using \(D\)-spectra technique, the DOWN probability for network \(B\). This model is generalized to the case of several peripheral networks attacking one “central” network \(B\). In this “attack”, some of nodes in \(B\) will receive more than one hit. The use of DeMoivre combinatorial formula together with \(D\)-spectra technique allows obtaining an expression for network \(B\) DOWN probability in a close form. Finally, Sect. 3.3 extends the results of Sect. 3.2 to the case when the “central” network is ternary. In that case, we must take into account different node behavior that are hit once or more. It is assumed that a node hit only once changes its state from up to mid. When this node receives another hit, it turns into down and remains in it forever. Network DOWN probability for this case can be estimated by an efficient Monte Carlo algorithm.