Ternary Networks pp 25-45 | Cite as

# 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.

### Keywords

Dodecahedron network Hypercube network Flow-network Grid network Giant component Component importance Nonidentical components Evolution algorithm### References

- 1.Barlow RE, Proschan F (1975) Statistical theory of reliability and life testing. Holt Rinehart and Winston Inc, New YorkMATHGoogle 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, Lomonsov MV (1991) Estimation of network reliability using graph evolution models. IEEE Trans Reliab 40(5):572–581CrossRefMATHGoogle 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, HeidelbergGoogle Scholar
- 6.Gertsbakh I, Shpungin Y (2012) Combinatorial approach to computing importance indices of coherent systems. Probab Eng Inf Sci 26:117–128Google Scholar
- 7.Gertsbakh I, Shpungin Y (2012) Stochastic models of network survivability. Qual Technol Quant Manag 9(1):45–58MathSciNetGoogle Scholar
- 8.Gertsbakh I, Rubinstein R, Shpungin Y, Vaisman R (2014) Permutational methods for performance analysis of stochastic flow networks. Probab Eng Inf Sci 28:21–38CrossRefGoogle Scholar
- 9.Gertsbakh I, Shpungin Y, Vaisman R (2014) Network reliability Monte Carlo with nodes subject to failure. Int J Performability Eng 10(2):161–170Google Scholar
- 10.Kroese D, Taimre T, Botev IZ (2011) Handbook of Monte Carlo methods. Wiley, New YorkCrossRefMATHGoogle Scholar
- 11.Lewis TG (2009) Network science: theory and applications. Wiley, HobokenGoogle Scholar
- 12.Ross S (2007) Introduction to probability models, 9th edn. Academic Press, New YorkGoogle Scholar