A path-based algorithm for evaluating the k-out-of-n flow network reliability

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Abstract

Evaluating the network reliability is an important topic in the planning, designing and control of systems. The minimal path (MP, an edge set) set is one of the major, fundamental tools for evaluating network reliability. A k-out-of-n MP is a union of some MPs in a k-out-of-n flow network in which some nodes must receive flows from their k input distinctive edges (each input edge has one flow) to generate one flow, where k is an integer number between 2 and n. In this study, a very simple a-lgorithm based on some intuitive theorems that characterize the k-out-of-n MP structure and the relationship between k-out-of-n MPs and k-out-of-n MP candidates is developed to solve the k-out-of-n flow network reliability by finding the k-out-of-n MPs between two special nodes. The proposed algorithm is easier to understand and implement. The correctness of the proposed algorithm will be analyzed and proven. One example is illustrated to show how all k-out-of-n MPs are generated and verified in a k-out-of-n flow network using the proposed algorithm. The reliability of one example is then computing using the same example.

Introduction

The reliability evaluation problem has been a popular research area that has received significant attention during the past four decades [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37] because of reliability's critical importance in various kinds of systems. In recent years, network reliability theory has been applied extensively in many real-world systems such as computers and communication systems [1], [2], power transmission and distribution systems [3], transportation systems [4], oil/gas production system [5], etc. Thus, network reliability plays important role in modern society [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37].

The reliability of a network represents its ability to continuing to perform a specified operation despite the effects of malfunctions or damage [6]. For example, in the st network, which are graphs with two special terminals: one source node s and one sink node t as a reliability model is widely used for systems reliability analysis. The reliability of a st network is given by the probability that there exists a flow from the source node to the sink node.

The traditional network always assumes that each node can generate one flow output to one of its output edges if and only if any flow from any one of its input edges exits. However, in real applications, some nodes (called k-out-of-n nodes here) with n input edges must receive flows from its k input edges to generate output one flow. Different nodes may have different k values and k is defined on a node with k∈{2,3,…,n}. For example, the voter output the result in an n-version programming system only when there are at least k modules that have the same results, where k is a defined value [7], [8]. Similar applications are observed in other redundant systems [7], [8]. Hence, it is necessary to extend traditional networks without k-out-of-n nodes to k-out-of-n networks.

Both of the evaluations of the reliability of the traditional networks or the k-out-of-n networks are NP-hard problems [6]. The traditional network reliability evaluation approaches exploit a variety of tools for system modeling and reliability index calculation. Among the most popular tools are network-based algorithms founded in terms of either minimal cuts (MCs) or minimal paths (MPs) [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37]. Tan [7] first investigated the search for enumerating the reliability of the k-out-of-n binary network in which the capacity of each edge is 1 only. In the k-out-of-n flow network reliability evaluation, Yeh [8] is the first researcher who discussed how to search for all of the k-out-of-n MCs to find the reliability of the k-out-of-n flow network in which the capacity of each edge is not limited. Unfortunately, to the author's best knowledge, no published papers have investigated an algorithm to evaluate the k-out-of-n flow network reliability based on MPs. The need for an efficient algorithm based on MPs to enumerate the k-out-of-n flow network reliability thus arises.

The purpose of this paper is to develop a first one path-based algorithm to enumerate the k-out-of-n general flow network reliability in terms of all MPs. The proposed algorithm is based on some simple concepts that were found in this study. The computational complexity of the proposed algorithm is analyzed and compared with the existing methods.

This paper is organized as follows. Section 2 describes the notations, nomenclature and assumptions required. In Section 3, some important properties, and theorems are also discussed included the relationships among k-out-of-n nodes, MPs, k-out-of-n MPs, and k-out-of-n MP candidates. Section 4 presents the proposed method for finding k-out-of-n MPs in the networks. One example is illustrated to show how to generate all k-out-of-n MPs using the proposed algorithm in the k-out-of-n network in Section 5. The reliability of one example is also calculated in terms of k-out-of-n MPs by further applying the inclusion–exclusion method. Concluding remarks are given in Section 6.

Section snippets

Acronym

  • MC/MP

    minimal cut/path.

Nomenclature

  • Reliability

    the probability that a signal can be transmitted from the source node to the sink node.

  • k-out-of-n node

    a node with n input edges must receive flows from k distinctive input edges to generate exactly one flow, where k is an integer assigned to the node and its value can be any number from 2 to n.

  • k-out-of-n binary network

    a connected network with k-out-of-n nodes s.t. the capacity of each edge is 1 only.

  • k-out-of-n flow network

    a connected network with k-out-of-n nodes s.t. the capacity of each edge is not limited.

  • Cut

    a cut is an edge set such that

Preliminaries

Before introducing the proposed algorithm, some useful properties and results will be described in this section. The following properties discuss the relationships among the MPs, k-out-of-n MP candidates, and k-out-of-n MPs.

Property 1

If a MP is not via any k-out-of-n nodes, then it is a k-out-of-n MP.

Based on the definition of k-out-of-n MP, a k-out-of-n MP is also a k-out-of-n MP candidate.

Property 2

Each k-out-of-n MP is a k-out-of-n MP candidate, i.e. a union of some MPs.

Using Property 2, all of the k-out-of-n

The proposed algorithm

This section outlines the proposed algorithm, which is very simple for finding and verifying all k-out-of-n MP candidates between the source and sink nodes in G(V, E, k). There are two main steps in the proposed algorithm. First, Corollary 1 and Property 5, Property 6, Property 7 are implemented to enumerate the k-out-of-n MP candidates. Next, these k-out-of-n MP candidates are verified using Theorem 3. The details are described in the following steps:

Algorithm

Find the k-out-of-n MPs in a flow network G(

Examples

To enumerate all of the MPs in a flow network is a NP-hard problem [6], [22], [23], [24], [25], [26], [27], [28]. It possesses a computational difficulty that, in the worse case, grows exponentially with network size. Owing to this inherent problem, instead of presenting practically large network systems [8], one moderate size network was selected to demonstrate this methodology for the k-out-of-n MP problem.

Example 1

Consider the k-out-of-n flow network in Fig. 1 of Section 2, where k(1)=k(2)=k(t)=1, k

Conclusions

A very simple algorithm for finding all k-out-of-n MPs between two special nodes was developed in this study. This is the first algorithm to find the k-out-of-n flow network reliability based on MPs, and achieves this in the following four ways: (1) it proposes a simplified method way to verify the k-out-of-n MPs candidates; (2) it can be used to predict the exact number of MPs in a k-out-of-n MP; (3) the number of k-out-of-n MP candidates is reduced; and (4) it is simple to understand and

Acknowledgements

This research was supported in part by the National Science Council of Taiwan, ROC under grant NSC 92-2213-E-035-041.

References (37)

  • W.C. Yeh

    Search for all MCs in networks with unreliable nodes and arcs

    Reliab Eng Syst Safety

    (2003/1)
  • W.C. Yeh

    A simple algorithm to search for all d-MPs with unreliable nodes

    Reliab Eng Syst Safety

    (2001)
  • W.C. Yeh

    Search for minimal paths in modified networks

    Reliab Eng Syst Safety

    (2002/3)
  • W.C. Yeh

    Multistate-node acyclic network reliability evaluation

    Reliab Eng Syst Safety

    (2002/11)
  • Y. Shen

    A new simple algorithm for enumerating all minimal paths and cuts of a graph

    Microelectron Reliab

    (1995)
  • J. Malinowski et al.

    Reliability evaluation for tree-structured systems with multistate components

    Microelectron Reliab

    (1996)
  • D.W. Lee et al.

    Determination of minimal upper paths for reliability analysis of planar flow networks

    Reliab Eng Syst Safety

    (1993)
  • F.S. Roberts et al.

    Reliability of computer and communication networks

    (1991)
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