Applications

Abstract

Sections 2.1–2.4 present numerical illustrations and applications of the theory developed in Chap. 1. Section 2.1 presents reliability calculations for $H_4$ network. The network has 16 nodes, 32 edges and two sets of terminals, $T_1$ and $T_2$. An edge $e=(a,b)$ in state up provides high communication speed between $a$ and $b$. If this edge is in state mid, the $a\leftrightarrow b$ communication goes with reduced speed; down state for an edge means that this edge is erased. Edge state is chosen randomly and independently according to probabilities $p_2,p_1,p_0$ for up, mid and down state, respectively. System $UP$ state is defined as the existence of high speed communication between nodes of $T_1$ and existence of a path of operational edges between any pair of nodes of $T_2$. We present data on network reliability and characterize the ternary D-spectrum numerically and graphically. Section 2.2 considers a stochastic source-terminal flow network problem for a dodecahedron network. In this network, an edge $e=(e,b)$ is a pair of directed links for $a\rightarrow b$ and $b\rightarrow a$ flows. Each link has capacity 6, 3, or 0 for up, mid and down state, respectively. The network has two DOWN states, $DOWN1$ and $DOWN2$, for the flow less than $L_1$ or $L_2$, respectively ($L_1 > L_2$). We present data on network reliability for various values of edge state probabilities $\mathbf{p }=(p_2,p_1,p_0)$. Section 2.3 is an example of a rectangular grid network with 100 nodes and 180 edges. Components subject to failure are the nodes. If a node is down all edges adjacent to it are erased and the node gets isolated. If a node is in mid state, it has only horizontal or vertical edges, depending on the position of the node. For this network we calculate the probability that the largest connected node set (an analogue to a “giant” component) has less than $L$ nodes. Section 2.4 analyzes edges importance data for $H_4$ network. Section 2.5 deals with networks having independent and nonidentical components. Component $i$ is in state up, mid and down, with probability $p_2^{(i)},p_1^{(i)}$ and $p_0^{(i)}$, respectively. In this situation, different failure sets with the same number of components in up, mid, down have different probabilistic weights and it is not possible to use the ternary spectrum technique. For calculating network reliability, we present an efficient and accurate Monte Carlo method based on a modification of M.V. Lomonosov’s evolution algorithm [3, 4]. Its action is illustrated by examples of a flow network and a grid network.