Design optimization for reliable networking is necessary for Ethernet ring mesh networks with ITU-T G.8032 Ethernet ring protection recommendation or with the IEC 62439-3 high-availability seamless redundancy protocol. To achieve this goal, the precise analysis of reliability/availability in Ethernet ring mesh consisting of single or multiple rings is one of the basic needs. Calculation of network reliability in these networks is an NP-hard problem. Therefore, most of the reliability and availability analyses were based on approximation and inaccurate methods. In this paper, at first, using decomposition method, a careful reliability analysis of Ethernet single-ring mesh networks will be presented. In addition, an accurate analysis of the reliability of Ethernet multiple-ring mesh networks will be performed by a novel approach named the spanning tree set method. Then, the availability of the ring networks with imperfect coverage and online repair assumptions will be analyzed using a hierarchical model in which each subsystem is modeled as a series system of independent Markov components.

Reliability Analysis of Ethernet Ring Mesh Networks

In the continuing flurry of graph theory research fields, we propose a combinatorial approach based on the deletion and the contraction operations in order to evaluate the number of spanning trees in a very large network. This latter can be represented by three families of book graphs: The simple book graph Bm,p, the chain of book graph Cm,p,n and the closed chain of book graph Gm,p,n with n number of book graph where m number of p-cycles.. Finally, we give the numerical results for each book graph to show the simplicity and efficiency of our proposed methods when compared with the matrix tree theorem.

Combinatorial methods evaluating the complexity of book networks

The spanning trees number of a network, also known as the complexity of a network is an essential measure characterizing the network reliability and its availability. In this paper, we enumerate the spanning trees of a book network with a common path. We propose two combinatorial methods to calculate the spanning trees number of this network. We present two types of the book network: A book network having the same number of vertices in each cycle and a book network with different number of vertices in each cycle. For the first type of book network, we propose the geometrical transformation approaches which are the bipartition and the reduction that make the enumeration of spanning trees of a large network (containing a large number of vertices and edges) easy to calculate. For the second type of a book network, we use the generalization of Fussner Theorem to find its complexity whatever its size. Finally, we give the spanning trees number of some networks derived from our two book networks, such as a chain and a closed chain. As an application in computer networking, we give a formula to estimate the reliability of our network.

Enumeration and Applications of Spanning Trees in Book Networks with Common Path

Enumeration of trees is a new line of research in graph theory; many researchers worked on this area, starting with the Matrix Tree Theorem given by Kirchho who dened the number of spanning trees in graph G as the determinant of its Laplacian matrix, since this later is easy to compute but it cannot give the recurrences of spanning trees. In this paper, we present two dierent methods to derive recursive functions, the deletion and contraction method and the splitting method that calculate the number of spanning trees in some families of graph, many recursions are given for the wheel graph, fan graph and family of corn graphs. The purpose of this paper is to propose an algorithm that calculates the number of spanning trees in a chained graph. Finally, we give the complexity of the chained wheel graph and the chained corn graph as applications of the spanning trees algorithm. Mathematics Subject Classication: 05C85, 05C30

Spanning Tree Algorithm for Families of Chained Graphs

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Spanning Tree Algorithm for Families of Chained Graphs