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Graph reliability evaluation via random $K$-out-of-$N$ systems

Hiroaki Mohri, Jun-ichi Takeshita

Abstract

The present study was concerned with network failure problems for simple connected undirected graphs. A connected graph becomes unconnected through edge failure, under the assumptions that only edges can fail and each edge has an identical failure distribution. The main purpose of the present study was to show recurrent relations with respect to the number of edges in graph generation procedures. To this end, simple connected undirected graphs were correlated to random $K$-out-of-$N$ systems, and key features of such systems were applied. In addition, some simple graph cases and examples were analyzed.

Graph reliability evaluation via random $K$-out-of-$N$ systems

Abstract

The present study was concerned with network failure problems for simple connected undirected graphs. A connected graph becomes unconnected through edge failure, under the assumptions that only edges can fail and each edge has an identical failure distribution. The main purpose of the present study was to show recurrent relations with respect to the number of edges in graph generation procedures. To this end, simple connected undirected graphs were correlated to random -out-of- systems, and key features of such systems were applied. In addition, some simple graph cases and examples were analyzed.

Paper Structure

This paper contains 8 sections, 10 theorems, 8 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic graph definitions
  4. Random $K$-out-of-$N$ systems
  5. Formulation of the graph reliability
  6. Main theorems
  7. Simple cases
  8. Examples

Key Result

Lemma 2.4

Suppose that, for a given constant integer $N$, $K$ is a random variable with the probability function $p_k := P(K = k) \ (k=1, \ldots, N)$, and each unit has the identical failure distribution function $F(t) \ (t \geq 0)$. Then the failure distribution function of the system at time $t$, say $\math

Figures (3)

  • Figure 1: The edge set $\{e_1, e_2, e_3, e_4\}$ is a $uv$-disconnected set, but not a $uv$-cut set. However, the set includes two $uv$-cut sets, $\{e_1\}$ and $\{e_2, e_3\}$.
  • Figure 2: Graph $G_1$ is an example of a tree, graph $G_2$ that of a cycle.
  • Figure 3: Graph generation, from a graph with one edge ($G_1$) to a graph with five edges ($G_5$).

Theorems & Definitions (20)

  • Definition 2.1: walk, path, closed walk
  • Definition 2.2: undirected, simple, connected
  • Definition 2.3: Conventional and random $K$-out-of-$N$ system
  • Lemma 2.4
  • Remark 2.5
  • Definition 3.1: Failure of graphs
  • Definition 4.1: $G$-disconnected set, $G$-cut set
  • Definition 4.2: $uv$-disconnected set, $uv$-cut set
  • Theorem 4.3
  • proof
  • ...and 10 more