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A Study on the Reliability of Consecutive k-Out-of-n: GSystems Based on CopulaXujie Jia a , Lirong Cui b & Jinqiu Yan ca School of Science , Minzu University of China , Beijing, Chinab School of Management and Economics , Beijing Institute of Technology , Beijing, Chinac Bauer Business School , University of Houston , Houston, Texas, USAPublished online: 23 Jun 2010.

To cite this article: Xujie Jia , Lirong Cui & Jinqiu Yan (2010) A Study on the Reliability of Consecutive k-Out-of-n: G SystemsBased on Copula, Communications in Statistics - Theory and Methods, 39:13, 2455-2472, DOI: 10.1080/03610921003778134

To link to this article: http://dx.doi.org/10.1080/03610921003778134

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Communications in Statistics—Theory and Methods, 39: 2455–2472, 2010Copyright © Taylor & Francis Group, LLCISSN: 0361-0926 print/1532-415X onlineDOI: 10.1080/03610921003778134

A Study on the Reliability of Consecutivek-Out-of-n: G Systems Based on Copula

XUJIE JIA1, LIRONG CUI2, AND JINQIU YAN3

1School of Science, Minzu University of China, Beijing, China2School of Management and Economics,Beijing Institute of Technology, Beijing, China3Bauer Business School, University of Houston, Houston, Texas, USA

The computation of reliability characteristics of a system that consists of dependentcomponents is sometimes difficult especially when the type of dependence is notknown. This article introduces the copula method to calculate the reliability ofdependent consecutive k-out-of-n: G systems. The components in these systems aredependent on each other and the dependency may be either linear or nonlinear. Thecopula is a popular tool for modeling the dependence structure of data. It containsthe information about the dependency structure of a vector of random variables andcan capture nonlinear dependence. Based on the copula theory, the article analyzesthe consecutive k-out-of-n: G systems and gets the reliability indexes. Finally, somenumerical examples are presented to illustrate the results obtained in this article.

Keywords Copula; Consecutive k-out-of-n: Dependent failure; G system;Reliability.

Mathematics Subject Classification 62xx; 62N05.

1. Introduction

A consecutive k-out-of-n: G system consists of an ordered sequence of ncomponents such that the system works if and only if at least k consecutivecomponents work. Communication systems, street parking systems, and oil pipelinesystems are examples where such consecutive k-out-of-n: G systems are used. Thereis an extensive literature on this system. Kontoleon (1980) reported the first study ofthe consecutive k-out-of-n: G system in 1980. Since Chiang and Niu (1981) formallyintroduced these systems, there have been a lot of follow-up studies (Chang et al.,2000; Cui and Hawkes, 2008; Cui et al., 2006; Zhao et al., 2007). For calculatingthe reliability for consecutive k-out-of-n: G systems, a basic assumption of previousmethods is that the components are mutually statistically independent (Chiang andNiu, 1981; Iyer, 1990). There are some researches on dependent components and

Received February 6, 2009; Accepted March 15, 2010Address correspondence to Xujie Jia, School of Science, Minzu University of China,

Beijing 100081, China; E-mail: jiaxujie@126.com

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then the Markov dependence was studied (Ge and Wang, 1990; Papastavridis andLambiris, 1987). Markov dependence is that the failure probability of workingcomponent depends on the number of consecutive failures immediately next to thecomponent. When this type of dependence goes forwards for k− 1 steps it is k− 1-step Markov dependence.

Some researches considered the dependence situation of common cause failures(CCF). Common cause failures are multiple dependent component failures within asystem that are a direct result of a common cause (CC) (Papastavridis and Lambiris,1987). Such as design deficiency, operating & maintenance errors, operatingenvironment, external catastrophe, and common power source, they can result ina common cause failure. It has been shown by many studies that the presence ofCCF tends to increase a system’s joint failure probabilities, and thus contributessignificantly to the overall unreliability of the system subject to CCF (Vaurio,1998). Therefore, it is crucial that CCF be modeled and analyzed appropriately.Considerable research efforts have been expended in the study of CCF for thesystem reliability analysis. There are some methods to deal with the CCF systems,such as � factor model (Flemining and Hunnaman, 1976), � factor model (Moslehand Siu, 1987), BFR model (Atwood, 1986), MCBFR model (Mankamo andKosonen, 1992), and environmental factor method (Eryilmaz, 2008). There are someother papers studied the dependent systems. Kotz et al. (2003) investigated how thedegree of correlation affects the increase in the mean lifetime for parallel redundancywhen the two components are positively quadrant dependent. Gera (2000) studieddependency using matrix formulation and Navarro et al. (2005) used signaturesmethod to solve dependent problem. But all the dependency they researched was fora specific type.

This article, referenced on their results, introduces the copula method tocalculate the reliability of consecutive k-out-of-n: G systems. The componentsin system are dependent on each other and the dependency relation may beeither linear or nonlinear. Copulas have become a popular tool for modelingthe dependence structure of data. Nelsen defined copulas as “functions that joinor couple multivariate distribution functions to their one-dimensional marginaldistribution functions” (Nelsen, 1999). Copulas contain the information about thedependence structure of a vector of random variables and can capture nonlineardependence, while correlation is only a linear measure of dependence.

On the foundation of the past’s work, this article makes a further study onreliability based on the copula theory. Section 2 introduces copula and calculatesthe reliability indexes of consecutive k-out-of-n: G systems. Section 3 studies howto choose the right copulas. Then, in Sec. 4 we show several special cases andgive numerical examples to illustrate the results obtained in this article. Section 5compares the dependent systems with the independent systems and some concludingremarks and extensions for further research are provided in Sec. 6.

2. Copula and Dependent Consecutive k-out-of-n G System

A d-dimensional copula is a distribution function on �0� 1�d with standarduniform marginal distributions. Reserve the notation C�U� = C�u1� � � � � ud� for themultivariate distribution functions which are copulas. ui is the distribution functionof variable i. Hence, C is a mapping of the form C �0� 1�d → �0� 1�, i.e., a mapping

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A Study on the Reliability Based on Copula 2457

of the unit hypercube into the unit interval. The following three properties musthold:

(1) C�u1� � � � � ud� is increasing in each component ui;(2) C�1� � � � � 1� ui� 1� � � � � 1� = ui for all i,ui ∈ �0� 1�;(3) For all �a1� � � � � ad�, �b1� � � � � bd� ∈ �0� 1�d with ai < bi:

2∑i1=1

· · ·2∑

id=1

�−1�i1+···+idC�u1i1� � � � � udid

� ≥ 0�

where uj1 = aj , uj2 = bj , for all j ∈ �1� � � � � d�.

Sklar’s theorem states that for all real-valued random variables �X1� � � � � Xd�with joint distribution function H and unvaried margins �F1� � � � � Fd� there exists acopula C such that

H�x1� � � � � xd� = C�F1�x1�� � � � � Fd�xd���

Based on the property of copula, we calculate the reliability of consecutivek-out-of-n: G systems when the component failures are dependent and thecomponent failure probabilities are not equal.

Let X1� X2� � � � � Xn denote the dependent lifetime of components in a consecutivek-out-of-n: G system. Every component is new at time t = 0, and works at this time.The components in the system are dependent on each other and the dependencyrelation may be either linear or nonlinear. Let Xi be the lifetime variable of theith component with the distribution function Fi�t� = P�Xi ≤ t�, and the reliabilityof it is Ri�t� = 1− Fi�t�. The joint distribution of X1� X2� � � � � Xn is H�x1� � � � � xn� =P�X1 ≤ x1� � � � � Xn ≤ xn�.

And there exit a copula function:

H�x1� � � � � xn� = C�F1�x1�� � � � � Fn�xn���

For a consecutive k-out-of-n: G system, let X11� � � � � X

j1 denote the lifetime of

the first j components of the n components. And X12� � � � � X

j2 denotes the lifetime

of the second j components of the n components, it means from X2 to Xj+1.Then X1

m� � � � � Xjm denotes the life of the mth j components in the n components,

where j ∈ �k� n�, m ∈ �1� n− j + 1�. The distributions of their lives are denoted byF 1m� � � � � F

jm, respectively. The joint distribution is denoted by Hj

m �x1m� � � � � xjm� and

their copula is Cjm�F

1m�x

1m�� � � � � F

jm�x

jm��, shorted as Cj

m. The life of remain parts inthe system are Y 1

m� � � � � Yn−jm and the distributions are F ∗1

m � F ∗2m � � � � � F ∗n−j

m . The jointdistribution is H∗n−j

m �Y 1m� � � � � Y

n−jm � and their copula is C∗n−j

m �F ∗1m � F ∗2

m � � � � � F ∗n−jm �,

shorted as C∗n−jm .

For X�min�u = min�X1� X2� � � � � Xu�, the reliability is:

Rmin�min�X1� X2� � � � � Xu�� = P�min�X1� X2� � � � � Xu� > t�

= P�X1 > t� � � � � Xu > t�

= 1−u∑

i=1

P�Xi ≤ t�+ ∑1≤i1<i2≤u

P�Xi1≤ t� Xi2

≤ t�+ · · ·

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+ �−1�k∑

1≤i1<i2<···<ik≤u

P�Xi1≤ t� Xi2

≤ t� � � � � Xik≤ t�

+ · · · + �−1�uP�X1 ≤ t� X2 ≤ t� � � � � Xu ≤ t�

= ∑sgn�F�C�F��

where F is the subset of F1�t�� F2�t�� � � � � Fu�t�� and C�F� is the copula functionof F . The sum is taken over all subset of F1�t�� F2�t�� � � � � Fu�t��, and sgn�F� isdefined to be 1 if there is an even number in F , and −1 otherwise.

For X�max�v = max�X1� X2� � � � � Xv�, the reliability is:

Rmax�X1� X2� � � � � Xv� = P�max�X1� X2� � � � � Xv� > t�

= 1− P�max�X1� X2� � � � � Xv� < t�

= 1− P�X1 < t� � � � � Xv < t�

= 1− C�F1�t�� F2�t�� � � � � Fv�t���

The joint distribution of min�X1� X2� � � � � Xu� and max�X1� X2� � � � � Xv� is:

H�max�min� = P�max�X1� X2� � � � � Xu� ≤ t�min�X1� X2� � � � � Xv� ≤ t��

By Sklar’s theorem, there exists a bivariate copula C�max�min� respectively.For a consecutive k-out-of-n: G system, if the mth j parts in the n components

are in working state, and the others are failed, then the reliability is:

Rjm�t� = Pmax�Y 1

m� � � � � Yn−jm � ≤ t�min�X1

m� � � � � Xjm� > t�

= Pmax�Y 1m� � � � � Y

n−jm � ≤ t�− Pmax�Y 1

m� � � � � Yn−jm � ≤ t�

min�X1m� � � � � X

jm� ≤ t�

= PY n−j�max�m ≤ t�−PY n−j�max�

m ≤ t� Xj�min�m ≤ t�

= C∗n−jm − C�max�min��C∗n−j

m � 1−∑sgn�F�Cj

m�F���

So the reliability of consecutive k-out-of-n: G system is:

Rcons−k/n =n∑

j=k

n−j+1∑m=1

Rjm�t�� =

n∑j=k

n−j+1∑m=1

Y 1m� � � � � Y

n−jm ≤ t < X1

m� � � � � Xjm�

=n∑

j=k

n−j+1∑m=1

max�Y 1m� � � � � Y

n−jm � ≤ t�min�X1

m� � � � � Xjm� > t�

=n∑

j=k

n−j+1∑m=1

{C∗n−j

m − C�max�min�(C∗n−j

m � 1−∑sgn�F�Cj

m�F�)}� (1)

3. Copula Selection and Parameter Estimation

The results in Sec. 2 are useful in reliability analysis as well as for designers whoare required to take into account the possible dependence among the components.

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A Study on the Reliability Based on Copula 2459

When calculating the indexes, the right copula functions is also needed. Selection ofthe best-fit copula and parameter estimation has been a hot topic of research. Withthe concept of copula, several families of distributions have been constructed suchas Gaussian, Clayton, Frank, Gumbel, Joe, etc.

To define the appropriate copula and get the right parameters � of a copulaC� in order to represent the dependence structure in a good manner, the popularmethods are statistics analysis and simulation techniques. The main statisticsmethods are moment method using Rank Correlation and the maximum likelihoodmethod. This problem has been received considerable attention in the literature.The copula estimation procedure based on empirical values is discussed in detailfor bivariate Archimedean copulas by Genest and Rivest (1993). Iman and Conover(1982) showed the method of calibrating the Gaussian copula with Spearman’srank correlation. Moreover, the Monte Carlo simulation of data is widely used incopula selection and parameter estimation (Iman and Conover, 1982). The IFM orinference functions for margins approach to the estimation of copulas are describedby Palmquist et al. (1999). And the pseudo-likelihood approach to copula estimationis described in Genest and Rivest (1993).

The copula selection problem can be split into three stages:

1. Get the life time data of components and the system.2. Use statistical method to estimate the parameters of the candidate copulas.3. Choose the appropriate copula from the candidates.

An Archimedean copula, which has been extensively studied, has been proveduseful for modeling high dimensional variables. It was shown that if �0� 1� →�0��� is a strict Archimedean copula generator, then

C�u1� � � � � ud� = �−1�� �u1�+ · · · + �ud���

gives a copula in any dimension d if and only if the generator inverse �−1� �0��� → �0� 1�

�−1� ={ −1�t�� 0 ≤ t ≤ ��0�

0� ��0� ≤ t ≤ � �

The common copulas are Gauss copula, t copula, Gumbel copula, Clayton copula,and so on.

If Y � Nd���∑� is a Gaussian random vector, then its copula is a Gauss copula:

CGa� �u1� u2� =

∫ �−1�u1�

−�

∫ �−1�u2�

−�1

2��1− �2�1/2exp

{−�s21 − 2�s1s2 + s22�

2�1− �2�

}ds1 ds2�

For the multivariate normal distribution, there is an implicit copula fromdistribution with continuous marginal DFS:

Ct��P�u� = t��P�t

−1� �u1�� � � � � t

−1� �ud���

The Gauss copula and t copula do not have simple closed forms; an examplewhich does have simple closed forms is Gumble copula and Clayton copula:

CGu� �u1� u2� = exp−��− ln u1�

� + �− ln u1���1/��� 1 ≤ � < ��

CCl� �u1� u2� = �u−�

1 + u−�2 − 1�−1/�� 0 < � < ��

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For the problem of estimating the parameters � of a parametric copula C�,the main method is maximum likelihood estimation. Let C� denote a parametriccopula where � is the vector of parameters to be estimated. The maximum likelihoodestimator (MLE) is obtained by maximizing

lnL(�� U1� � � � � Un

) = n∑t=1

ln c��Ut�� (2)

with respect to �, where c� denotes the copula density and Ut denotes a pseudo-observation from the copula.

Take an example to illustrate the method. Consider two variables X1 and X2

generated by a bivariate Gaussian copula with exponential distributions. Then,

c��x1� x2� =1√

1− �2exp

(−12

(�21 + 2�1�2 + �21

1− �2

)+ 1

2��21 + �21�

)

with

�1 = �−1�1− e−�1t�� �2 = �−1�1− e−�2t�

and the parameter is �.From (2), we get

lnL��� =n∑

t=1

ln c��x1� x1�t

Figure 1. The scatter plot of the random datas.

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A Study on the Reliability Based on Copula 2461

Let

�L���

�= 0

and we can get �.To choose the appropriate copula from the candidates, the widely used copula

is Archimedean copula. An Archimedean copula has been extensively studiedand proved useful for modeling dependence in low-dimensional applications. If �0� 1� → �0��� is a strict Archimedean copula generator, then

C�u1� � � � � ud� = �−1�� �u1�+ · · · + �ud��

gives a copula in any dimension d if and only if the generator inverse �−1�

�0��� → �0� 1�

�−1� ={ −1�t�� 0 ≤ t ≤ ��0�

0� ��0� ≤ t ≤ �

Theorem 3.1. Archimedean copula C is generated by , then,

C >∏ ⇔ ln �−1��x + y� ≥ ln �−1��x�+ ln �−1��y��

So, the right copula in calculating reliability indexes should satisfy:

ln �−1��x + y� ≥ ln �−1��x�+ ln �−1��y��

Due to the lack of data, the focus of this study is the calculation method.To verify the effectiveness of the method, a set of random dates is generated bysimulation. For simplicity, the copula expression of dependency about two unitsare considered. For about n units, the method is the same. Firstly, two-dimensionalrandom numbers �x� y� are generated by Monte–Carlo method. The scatter plot isshown in Fig. 1.

Considering the common copulas: Gauss copula, t copula, Gumbel copula.According to formula (8), by maximum likelihood estimation method, the results ofthe estimation parameters are shown in Table 1.

Comparing the several copulas, the Gumbel copula is the best; see Fig. 2.

Table 1Results of the parameter estimations for each copulas

Copula Correlation Gauss copula t copula Gumbel copula

Parameter � �Ga �t �Ga

Results 0.952 0.9663 0.9663 6.0372

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Figure 2. Gumbel copula when �Ga = 6�0372.

4. Numerical Examples and Special Cases

4.1. A Numerical Example for Consecutive k-out-of-n: G Systems

To illustrate the results, a numerical example is given in the following.Suppose the supply chain is a consecutive 2 out of 3: G system. The system

consists of three components whose operating lives are X1, X2, and X3. Thedistributions are F1 = 1− e−t, F2 = 1− e−2t, and F2 = 1− e−3t, respectively.

We use Clayton copula

CCl1 �u1� u2� = �u−1

1 + u−12 − 1�−1

to describe the bivariate dependence, and use

CCl1 �u1� u2� u3� = �u−1

1 + u−12 + u−1

3 − 2�−1

to describe the three-dimensional system, then the reliability is:

RC2/3�t� = P�X1 > t�X2 > t�X3 < t�+ P�X1 < t�X2 > t�X3 > t�

+ P�X1 > t�X2 > t�X3 > t�

= 1− P�X2 < t�− P�X1 < t�X3 < t�+ P�X1 < t�X2 < t�X3 < t�

= e−2t − �1− e−t�−1 + �1− e−3t�−1 − 1�−1

+ �1− e−t�−1 + �1− e−2t�−1 + �1− e−3t�−1 − 2�−1�

Using Maple software, we get the comparing curve for RC2/3�t� (see Fig. 3).

Now consider special cases of consecutive k-out-of-n: G systems, that is,consecutive n-out-of-n: G systems, consecutive 1-out-of-n: G systems and partial

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A Study on the Reliability Based on Copula 2463

Figure 3. The curve for RC2/3�t�.

dependent systems. For the three special cases, the reliability indexes will be muchsimpler.

4.2. Consecutive n-Out-of-n: G Systems

The consecutive n-out-of-n: G systems are series systems indeed. Let X denote thelifetime of a series system, RC

S �t� denote the reliability of the system with dependentcomponents. Then:

RCS �t� = P�X > t�

= P�min�X1� X2� � � � � Xn� > t�

= P�X1 > t� � � � � Xn > t�

= 1−n∑

i=1

P�Xi ≤ t�+ ∑1≤i1<i2≤n

P�Xi1≤ t� Xi2

≤ t�+ · · ·+

�−1�k∑

1≤i1<i2<···<ik≤n

P�Xi1≤ t� Xi2

≤ t� � � � � Xik≤ t�+ · · · + �−1�n

× P�X1 ≤ t� X2 ≤ t� � � � � Xn ≤ t�

= 1−n∑

i=1

ui +∑

1≤i1<i2≤n

Ci1�i2�ui1

� ui2�+ · · · + �−1�k

× ∑1≤i1<i2<···<ik≤n

Ci1�i2�����ik�ui1

� � � � � uik�+ · · ·

+ �−1�nC1�2�����n�ui1� � � � � uin

�� (3)

To illustrate the results, a numerical example is given in the following.Suppose the series system consists of two components whose operating life is

X1 and X2. The two components have distributions F1 = 1− e−t and F2 = 1− e−2t,respectively. With Eq. (3), the reliability of the dependent system can be gotten.

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Figure 4. The curve for RCS �t�.

We use Clayton copula

CCl1 �u1� u2� = �u−1

1 + u−12 − 1�−1

to describe the dependence, then the reliability of the series system is:

RCS �t� = 1− u1 − u2 + CCl

1 �u1� u2�

= 1− u1 − u2 + �u−11 + u−1

2 − 1�−1

= e−t + e−2t − 1+ {�1− e−t�−1 + �1− e−2t�−1 − 1

}−1�

Using Maple software, we get the curve for RCS �t�; see Fig. 4.

4.3. Consecutive 1-Out-of-n: G Systems

The consecutive 1-out-of-n: G systems are parallel systems indeed. Let RP�t� denotethe reliability of parallel systems with dependent components. Then the reliabilityof the system is:

RP�t� = P�X > t�

= P�max�X1� X2� � � � � Xn� > t�

= 1− P�max�X1� X2� � � � � Xn� ≤ t�

= 1− P�X1 ≤ t� � � � � Xn ≤ t�

= 1− C�F1�t�� F2�t�� � � � � Fn�t��� (4)

To illustrate the results, a numerical example is given. Suppose the parallelsystem consists of two components whose lifetime and the dependent relationship

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A Study on the Reliability Based on Copula 2465

Figure 5. The curve for RP�t�.

are as same as case (1). That is, the two components have distributions F1 = 1− e−t

and F2 = 1− e−2t, and their copula function is Clayton copula

CCl1 �u1� u2� = �u−1

1 + u−12 − 1�−1�

Then we get the reliability:

RP�t� = 1− CCl1 �u1� u2�

= 1− �u−11 + u−1

2 − 1�−1

= 1− {�1− e−t�−1 + �1− e−2t�−1 − 1

}−1�

Using Maple software, we get the curve for RP�t� in Fig. 5.

4.4. Partial Dependent Systems

In some systems, not all the components are dependent, that is, some componentsare dependent and some of them are independent. In the following, we give anumerical example to illustrate this situation. Considering a consecutive 2-out-of-3: G system, the operating lifetimes of the three components are X1, X2, and X3,and the distributions are F1 = 1− e−t, F2 = 1− e−2t, and F2 = 1− e−3t, respectively.Suppose the first two components are dependent with each other and they areindependent of the third one. We also use the Clayton copula

CCl1 �u1� u2� = �u−1

1 + u−12 − 1�−1

to describe the dependence, then the reliability R2/3�P��t� is:

R2/3�P��t� = P�X1 > t�X2 > t�X3 < t�+ P�X1 < t�X2 > t�X3 > t�

+ P�X1 > t�X2 > t�X3 > t�

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Figure 6. The comparing curves for RC2/3�t� and R2/3�P��t�.

= 1− P�X2 < t�− P�X1 < t�X3 < t�+ P�X1 < t�X2 < t�X3 < t�

= 1− P�X2 < t�− P�X1 < t�P�X3 < t�+ P�X1 < t�X2 < t�P�X3 < t�

= e−2t − �1− e−t� · �1− e−3t�+ {�1− e−t�−1 + �1− e−2t�−1 − 1

}−1 · �1− e−3t��

Using Maple software, we get the curve for R2/3�P��t� and compared with RC2/3�t� in

Fig. 6.From the comparing curves, we get that for the 2 out of 3: G supply chain

system the reliability curve of the dependent system is in the above of the partialdependent system. So the reliability of dependent system is larger than the partialdependent system.

5. Reliability Comparison of Dependent and Independent Systems

The reliability of a consecutive k-out-of-n: G system is less than a consecutive 1-out-of-n: G system which is a n component parallel system, and more than a consecutiven-out-of-n: G system which is a n component serial system. So, we consider theconsecutive 1-out-of-n: G system and the consecutive n-out-of-n: G system to studyand compare the reliability of dependent systems and independent systems.

5.1. Consecutive n-out-of-n G Systems

As mentioned above, the consecutive n-out-of-n: G system is a series system. LetRC

S �t� and RIS�t� denote the reliability of the dependent series system and the

independent series system. Comparing the two systems, we get the inequality as

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A Study on the Reliability Based on Copula 2467

follows:

RCS �t� > RI

S�t�� (5)

Proof. Let Cn denotes the copula of X1� � � � � Xn, and ui = P�Xi ≤ t�. If Cn = u1 ·u2 · · · · · un; then X1� � � � � Xn are independent. If C

n > u1� � � � � un, then X1� � � � � Xn arepositively dependent. If Cn < u1� � � � � un, then X1� � � � � Xn are negatively dependent.Here, we suppose X1� � � � � Xn are positively dependent.

(1) The reliability of dependent system is shown in Eq. (3):

RCS �t� = 1−

n∑i=1

ui +∑

1≤i1<i2≤n

Ci1�i2�ui1

� ui2�+ · · · + �−1�k

× ∑1≤i1<i2<···<ik≤n

Ci1�i2�����ik�ui1

� � � � � uik�+ · · ·

+ �−1�nC1�2�����n�ui1� � � � � uin

��

(2) The reliability of independent systems is:

RIS�t� =

n∏i=1

Ri�t� =n∏

i=1

�1− Fi�t�� =n∏

i=1

�1− ui�

= 1−n∑

i=1

ui +∑

1≤i1<i2≤n

ui1ui2

+ · · · + �−1�k

× ∑1≤i1<i2<···<ik≤n

�ui1ui2

� � � uik�+ · · · + �−1�nu1u2 � � � un� (6)

When n is odd,

RCS �t�− RI

S�t�

={ ∑1≤i1<i2≤n

Ci1�i2�t�+ · · · + �−1�k

∑1≤i1<i2<···<ik≤n

Ci1�i2����ik�t�+ · · · + �−1�nC1�2�����n�t�

}

−{ ∑

1≤i1<i2≤n

ui1ui2

+ · · · + �−1�k

× ∑1≤i1<i2<···<ik≤n

�ui1ui2

� � � uik�+ · · · + �−1�nui1

ui2� � � uik

}

={[ ∑

1≤i1<i2≤n

Ci1�i2�t�− ∑

1≤i1<i2<i3≤n

Ci1�i2�i3�t�

]

−[ ∑1≤i1<i2≤n

ui1ui2

− ∑1≤i1<i2<i3≤n

ui1ui2

ui3

]}+ · · ·

+{[ ∑

1≤i1<i2<···<ik≤n

Ci1�i2����ik�t�− ∑

1≤i1<i2<···<ik+1≤n

Ci1�i2����ik+1�t�

]

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2468 Jia et al.

−[ ∑

1≤i1<i2<···<ik≤n

�ui1ui2

� � � uik�− ∑

1≤i1<i2<···<ik+1≤n

�ui1ui2

� � � uik+1�

]}+ · · ·

> 0

∵

[Ci1�i2�����ik

�t�− Ci1�i2�����ik+1�t�

]− [�ui1

ui2� � � uik

�− �ui1ui2

� � � uik+1�]

= [Ci1�i2�����ik

�t�− �ui1ui2

� � � uik�]− [

Ci1�i2�����ik+1�t�− �ui1

ui2� � � uik+1

�]

>[Ci1�i2�����ik

�t�− �ui1ui2

� � � uik�]− [

Ci1�i2�����ik�t�uik+1

− �ui1ui2

� � � uik+1�]

= [Ci1�i2�����ik

�t�− �ui1ui2

� � � uik�] · �1− uik+1

�

> 0

When n is even, we can suppose it is a two-component system, and onecomponent is an odd-component system. Then we can get the inequality RC

S �t� >RI

S�t�. To illustrate the results, a numerical example is given in the following.Suppose the series system consists of two components whose operating life is

X1 and X2. The two components have distributions F1 = 1− e−t and F2 = 1− e−2t,respectively. With Eq. (6), we get reliability of independent series systems:

RIS�t� = e−te−2t = e−3t�

The reliability index and the curve of dependent series systems are given inSec. 4.2. Using Maple software, we get the comparing curves for RI

S�t� and RCS �t�

which is consistent with the results we obtained in the previous case (see Fig. 7).

5.2. Consecutive 1-out-of-n: G Systems

As mentioned above, the consecutive 1-out-of-n: G system is a parallel system. LetRC

P �t� and RIP�t� denote the reliabilities of the dependent parallel system and the

independent parallel system. Comparing the two systems, we get the inequality asfollows:

RCP �t� < RI

P�t�� (7)

Proof. The reliability of dependent parallel systems is shown in (4):

RCS �t� = 1− C1�2�����n�t��

The reliability of independent systems:

RIP�t� = 1−

n∏i=1

�1− Ri�t�� = 1−n∏

i=1

ui� (8)

Because C1�2�����n�t� >∏n

i=1 ui, then 1− C1�2�����n�t� < 1−∏ni=1 ui can be obtained.

An example is given to illustrate the results. Suppose the parallel system consistsof two components whose operating lifetime are X1 and X2. The two componentshave distributions F1 = 1− e−t and F2 = 1− e−2t, respectively.

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A Study on the Reliability Based on Copula 2469

Figure 7. The comparing curves for RIS�t� and RC

S �t�.

Figure 8. The comparing curves for RCP �t� and RI

P�t�.

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(1) Reliability of independent systems

RIP�t� = 1− u1u2 = e−te−2t − e−t − e−2t�

(2) Reliability of dependent systems.

The reliability index and the curve of dependent parallel systems are given inSec. 4.3. Using Maple software, we get the comparing curves for RI

P�t� and RCP �t�

which is consistent with the results we obtained in the previous case, see Fig. 8.

5.3. Consecutive k-out-of-n: G Systems

For the consecutive k-out-of-n: G systems, the reliability is depended on thedependence structure, k and n. So the comparison result is not fixed. Using anumerical example, we shall illustrate it.

Suppose the supply chain is a consecutive 2 out of 3: G system. The systemconsists of three components whose operating lives are X1, X2, and X3. Thedistributions are F1 = 1− e−t, F2 = 1− e−2t, and F3 = 1− e−3t, respectively.

(1) Reliability of the independent systemThe reliability of the independent 2 out of 3: G system:

RI2/3�t� = e−te−2t�1− e−3t�+ e−3te−2t�1− e−t�+ e−te−2te−3t

= e−3t + e−5t − e−6t�

(2) Reliability of the dependent system.

The reliability of the dependent 2 out of 3: G system is calculated in Sec. 4.1:

RC2/3�t� = P�X1 > t�X2 > t�X3 < t�+ P�X1 < t�X2 > t�X3 > t�

+ P�X1 > t�X2 > t�X3 > t�

Figure 9. The comparing curves for RC2/3�t� and RI

2/3�t�.

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A Study on the Reliability Based on Copula 2471

= 1− P�X2 < t�− P�X1 < t�X3 < t�+ P�X1 < t�X2 < t�X3 < t�

= e−2t − {�1− e−t�−1 + �1− e−3t�−1 − 1

}−1

+ {�1− e−t�−1 + �1− e−2t�−1 + �1− e−3t�−1 − 2

}−1�

Using Maple software, we get the comparing curves for RI2/3�t� and RC

2/3�t�; seeFig. 9

From the comparing curves we can get that, for the consecutive 2 out of3: G supply chain system, the reliability of dependent system is lower than theindependent system at the beginning and higher as t increases.

6. Conclusions

The methodology developed in this article resorts to copula functions for measuringthe reliability of consecutive k-out-of-n: G systems. In the system, the componentfailures are dependent or partially dependent. Copula contains the informationabout the dependence structure of a vector of random variables and can capturethe nonlinear dependence. So the component failures need not be independent,and that component failure probabilities need not be equal. It is an extension ofindependence and linear correlative parts and could be used in other models andfields.

Acknowledgment

This research is supported by the 211 project of Minzu University of China, theeconomic and social development in ethnic: minority areas analysis and predictionin China (021211030312).

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