Complexity

Volume 2019, Article ID 3971597, 11 pages

https://doi.org/10.1155/2019/3971597

## Measuring Component Importance for Network System Using Cellular Automata

College of Computer Science, Chongqing University of Posts and Telecommunication, Chongqing 400065, China

Correspondence should be addressed to Li He; moc.qq@646925273

Received 13 December 2018; Revised 26 March 2019; Accepted 2 April 2019; Published 2 May 2019

Guest Editor: Md Sarder

Copyright © 2019 Li He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper concentrates on the component importance measure of a network whose arc failure rates are not deterministic and imprecise ones. Conventionally, a computing method of component importance and a measure method of reliability stability are proposed. Three metrics are analyzed first: Birnbaum measurement, component importance, and component risk growth factor. Based on them, the latter can measure the impact of the component importance on the reliability stability of a system. Examples in some typical structures illustrate how to calculate component importance and reliability stability, including uncertain random series, parallel, parallel-series, series-parallel, and bridge systems. The comprehensive numerical experiments demonstrate that both of these methods can efficiently and accurately evaluate the impact of an arc failure on the reliability of a network system.

#### 1. Introduction

As a quantitative measure, reliability can be broadly interpreted as the ability of a system to perform its intended function. During the past ten years, a significant amount of research has been conducted to address reliability evaluation. Network reliability can be estimated using Bayesian approach [1], Monte Carlo simulation [2, 3], genetic algorithm [4], fault-tree analysis [5], etc. Obviously, all those methods apply numerical reliability or boundary value to indicate the reliability of network systems. However, two main questions must be answered for designing a network system. Question 1: Which component is the most important? Question 2: How does the importance of component impact the system reliability stability? For answering such questions, component importance measures must show the effect and rank in system design and preventive maintenance.

Determining the importance of components in complexity networks is crucial. Several importance measure methods have been introduced in [6, 7], including Birnbaum measure, criticality importance, improvement potential, risk achievement worth, and risk reduction worth. Based on the fundamental component importance theory initially proposed by Birnbaum [6], there have been a number of approaches used to show component importance. Generally speaking, the traditional component importance evaluation methods are classified into two kinds, one is on the components failure rates, and the other is without taking components characteristic into account. The first category method is mainly based on the graph theory, including reliability Boolean polynomial [7], minimum trees and their number [8], and minimal cut set. In [9], criticality importance measures for components with respect to system failure intensity and the total system failure count are presented. To evaluate reliability importance of components in a network system, Zio et al. [10] present generalized importance measures based on Monte Carlo simulation. Meanwhile, Wang et al. [11] introduce the failure critically index, restoring critical index and operational index. In [12], importance measures with respect to system failure intensity are developed and it also points out that the Barlow [11] importance only measures the contribution of a component as the last failure in a minimal cut set, not the total contribution. Contini et al. [13] also evaluate network system importance with respect to the system failure. However, an obvious shortcoming is that the impact of the component characteristic on the failure rate of network is not considered. Some examples, including the reliability of Boolean polynomial [7], minimum spanning tree [14, 15], minimum cut set and minimum path set [16], and fault-tree analysis [9], attempting to incorporate more features of network topology consisting of multiple terminals and dependency between topology are researched. Meanwhile, simulation based on Monte Carlo method [10] often depends more on the convergence of probability than the number of network components; statistical error during reliability analysis may result in slow convergence for achieving acceptable accuracy in low probability estimations. Therefore, these methods depend on the model to decompose the network topology and calculate the reliability of network. And the complexity of calculation will increase by index level as the size of network grows. Although these methods have adequately considered the characteristic of component in network system, how to improve the efficiency of calculation to strength the practical of importance measurements is still a focus.

Recently, importance measure to estimate the effect of a component residing at certain states on the performance is proposed in [17]. Importance measure of components when the system may be reconfigured is designed [18]. Liu [19] presents a chance theory, which contains some basic concepts including chance measure, uncertain random variable, and chance distribution. Then, Gao and Yao [20] research the importance index of components in uncertain random systems; a concept of importance index on a component in uncertain random variable and Boolean system is proposed. At the same time, link component importance is analyzed in [21]. Component maintenance priority is used to select component for preventive maintenance. And a Monte Carlo-based method to generate probability distributions of the two metrics for all of the components of the network is proposed and a stochastic ranking approach based on the Copeland’s pairwise aggregation is used to rank components importance in [22]. In addition, a strategy for solving the component placement problem by maximizing the information gain in terms of users’ choices in [23] is proposed. At last, Zhu et al. [24] present a nonlinear binary programming model, which focuses on embedding the Birnbaum importance in heuristics and the method of dealing with more than one type of components.

However, to the best of our knowledge, none of the exiting classical importance methods based on Cellular Automata are directly applied to measure the impact of the component importance on the reliability of system. The Birnbaum importance, risk growth factor, and reliability stability to measure the importance of a component or a group of components are defined. A computing method of component importance (NEA) based on Cellular Automata is designed; in addition, a new measure method of reliability stability (NSA) is proposed in this paper. At last, the validities of NSA and NEA are proved by experiments. And it is also proved that the proposed algorithm NSA is more accurate in calculating the importance of the system components compared to the classical algorithm in [25].

The main contributions are as follows:(i)defining the Birnbaum importance, risk growth factor, and reliability stability;(ii)designing a model for measuring the network system component importance;(iii)designing a method for measuring the reliability stability of network system.

The rest of this paper is organized as follows. Three metrics of component importance are introduced in Section 2, and the validity of these measurements is verified in this section. In Section 3, we put forward our system models. In Section 4, we evaluate the component importance and describe our algorithms. In Section 5, the parameters of experiment are given and the performance of the algorithms is analyzed. The conclusion is drawn in Section 6.

#### 2. Preliminaries

Let be a network system, where is the set of nodes, . is the set of arcs, . is a directed, connected, acyclic graph which contains an initial node and a terminal node . In order to study the issue of this paper, there are some assumptions.(i)The state of each node and arc is statistically independent.(ii)The evaluation of network reliability is considered so that the probability of the initial node successfully connects to the terminal node .

##### 2.1. Birnbaum Measure

The significance of network system component importance is the influence degree of network system components (nodes or links) on network system connectivity, which can be expressed by the Birnbaum measure [9], denoting the importance of network system decided by the reliability of network system and component. The nature of this definition mathematically denotes the partial derivative on the reliability of component to the reliability of network system; that is, when the component reliability is changed, the network system reliability will be accordingly changed. For a network system with components, its Birnbaum measure can be defined as Here is the function of network system failure, is the failure function of network system component , and the relation of failure function and reliability function is .

Theorem 1. *For a network system with components, the mathematical expression of Birnbaum measure is Here, is the Birnbaum measure of component , is the function of network system reliability, and is the reliability function of component . If is larger, the impact on network system reliability is greater when the reliability of component is changed.**Equation (2) shows the changes of network system reliability in the case of component from normal state to failure state. Thus, Birnbaum measure can be defined asHere is the network system reliability when component is normal and is the network system reliability when component is failure.*

##### 2.2. Critical Importance

In network system, the failure rate of each component is different, so Lambert [9] proposes a critical importance to describe the probability of network system failure caused by network system component . It can also be functioned asFrom the perspective of the whole system, (4) can be further converted to the following one:Equation (5) shows that the reliability of network system is the product of the Birnbaum measure of component and the ratio of system failure rate, when the state of component is from the normal to failure.

##### 2.3. Network System Reliability Stability

Assuming that a network system contains components, the risk growth factor [26] of component can be defined asHere is the network system failure rate when component is in malfunction. This formula describes the impact of the failure of component on system reliability. In addition, based on (6), the other two reliability metrics, average risk and reliability stability, can be concluded, which measure the impact of single component malfunction on network reliability. The average risk growth factor can be expressed as follows by its own definition:where is the average impact of all components failure individually on the network system reliability. On the basis of (7), the network reliability stability [26] can be formulated asIt can be known, by the definition of network system reliability stability, that the network system reliability stability and network system reliability are greatly related to the network system average risk growth factor. When , the network system component failure has little impact on the network system reliability, and vice versa.

##### 2.4. Experimental Analysis

For any network system, it is noted that the complexity of network topology can make network components decomposed as combination of series and parallel system, and the complexity needed by optimal solution grows exponentially with the network size [27]. Next we will verify the validity of the previous measurements for different network structures using typical data recommended by [5, 14].

*(1) Series System. *Assuming that a system has components connected in series configuration, the system will operate as long as all components are working. For Figure 1(a), the failure rates of components 1, 2, 3 in the network are . When , their reliabilities are, respectively, , and . By [27], the reliability of network system is , so the Birnbaum measurements of three components are as follows.Obviously, . Component 1 has more impact on the system. Increasing or decreasing the failure of component 3 will be the biggest change to the reliability of the system, so component 3 is the most important component of the system. In addition, the critical importance of components can be computed based on the Birnbaum measurements and (5).By and the definition of critical importance, the probability of component 3 leads to the malfunction when the system is failure.