On the Use of the Importance Measure for Multi-State Repairable k ...

Communications in Statistics—Theory and Methods, 43: 2766–2781, 2014Copyright © Taylor & Francis Group, LLCISSN: 0361-0926 print / 1532-415X onlineDOI: 10.1080/03610926.2012.685549

On the Use of the Importance Measure forMulti-State Repairable k-out-of-n: G Systems

HONGYAN DUI, SHUBIN SI, ZHIQIANG CAI,SHUDONG SUN, AND YINGFENG ZHANG

Ministry of Education Key Laboratory of Contemporary Design and IntegratedManufacturing Technology, School of Mechatronics, NorthwesternPolytechnical University, Xi’an, Shaanxi, P. R. China

Importance measures in reliability systems are used to identify weak components incontributing to proper functioning of the system. Traditional importance measuresmainly concern the change of the system reliability as the change of the reliability ofone component and seldom consider the expected number of repairs of the objectivecomponent in unit time. This paper proposes an improvement potential rate importance(IPR) to verify the effectiveness of the improvement in system reliability for multi-staterepairable k-out-of-n: G systems. Then the comparisons between IPR and Birnbaumimportance are discussed. Finally, a case study is given to demonstrate the proposedIPR.

Keywords Reliability; Importance measure; Multi-state k-out-of-n: G systems;Improvement; Repair.

Mathematics Subject Classification 90B25.

1. Introduction

Assumptions.

1) The systems are monotone coherent.2) The state space of each component and system is {0, 1, 2, . . . ,M}, where 0 corresponds

to complete failure of the system or its components and M is the perfect functioningstate of the system or components. The states are ordered from 0 to M.

3) The lifetimes of n components of the k-out-of-n system are independent and followingnon identical exponential distributions. The repair times of components are independentand non identical exponential random variables.

Received August 15, 2011; Accepted April 12, 2012.Address correspondence to Shubin Si, Ministry of Education Key Laboratory of Contemporary

Design and Integrated Manufacturing Technology, School of Mechatronics, Northwestern Polytech-nical University, P.O. Box 554, Xi’an, Shaanxi 710072, China. E-mail: [email protected]

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Use of the Importance Measure 2767

The k-out-of-n systems have attracted a great deal of attention in the field of reliability.The k-out-of-n system structure is a very popular type of redundancy in fault-tolerantsystems. It finds wide applications in both industrial and military systems such as the multi-engine system in an airplane and the multi-pump system in a hydraulic control system (Kuoand Zuo, 2003).

An n-component system that works if and only if at least k of the n components workis called a k-out-of-n: G system. For example, it may be possible to drive a car with a V8engine if only four cylinders are firing. However, when less than four cylinders fire, theautomobile cannot be driven. Thus, the functioning of the engine may be represented bya 4-out-of-8: G system. In a data processing system with five video displays, a minimumof three displays operable may be sufficient for full data display. In this case the displaysubsystem behaves as a 3-out-of-5: G system.

For binary systems, the properties of the reliability of k-out-of-n systems have beeninvestigated (Boland and Proschan, 1983; Barlow and Heidtmann, 1984; Risse, 1987; Sarjeand Prasad, 1989; Coit and Liu, 2000; Levitin, 2005).

Compared with binary systems, multi-state systems provide a more flexible tool forrepresenting engineering systems. For example, such systems appear in communicationnetworks, power generation, computer systems, logic circuits, transportation of oil and gas,etc. (Natvig, 2011). As Barlow and Wu (1978) generalized the theory of binary coherentsystems for multi-state components, a few researchers (Boedigheimer and Kapur, 1994;El-Neweihi, Proschan and Sethuraman, 1978) extended the definition of a binary k-out-of-n system to the multi-state case which has the same k values with respect to differentsystem state levels. Huang et al. (2000) proposed a generalized definition of the multi-state k-out-of-n system which allows different k values for different system state levelsand provided an algorithm to calculate the state distribution of generalized multi-statek-out-of-n: G systems. Zuo and Tian (2006) proposed recursive algorithms to evaluateperformance of generalized multi-state k-out-of-n systems. Zhao and Cui (2010) used thefinite Markov chain imbedding approach to present a unified formula with the productof matrices for evaluating the system state distribution for generalized multi-state k-out-of-n systems. In repairable k-out-of-n systems, Frostig and Levikson (2002) exploitedMarkov renewal processes to analyze a k-out-of-n system where components repair andlifetimes are assumed to be either exponentially or generally distributed. Li et al. (2006)proposed an approach for the analysis of system reliability with independent exponentialcomponents. Barron et al. (2006) studied a k-out-of-n system with several repair facilities.Khatab et al. (2009) studied the reliability of k-out-of-n: G systems with non identicalcomponents subject to repair priorities. Ding et al. (2010a,b) proposed a multi-state systemstructure and framework for the weighted multi-state k-out-of-n systems. Shrestha et al.(2010) presented three forms of decision diagrams for the modeling and analysis of multi-state systems. The reliabilities of generalized multi-state k-out-of-n system, generalizedconsecutive-k systems and consecutive-k-out-of-n system are analyzed by using the finiteMarkov chain imbedding approach (Zhao et al., 2007, 2011). Li et al. (2011) developed theinterval universal generating function to estimate the interval-valued reliability of multi-state systems. Eryilmaz (2010) evaluated mean residual, and mean past lifetime functionsof one unit multi-state systems, and multi-state k-out-of-n: G systems under the assumptionthat the degradations in systems and components follow an acyclic Markov process whichhas a discrete state space.

To achieve high reliability, it is necessary to identify the components that have thegreatest effect on the system reliability. Such items can be identified using importancemeasures. Importance measures reflect the effect of the individual component reliability on

2768 Dui et al.

the system reliability. They have been widely used for identifying system weaknesses andsupporting system improvement activities. With the importance values of all components,proper actions can be applied on the weakest component to improve system performanceat the minimum cost or effort for a k-out-of-n system.

The most fundamental importance measure of system components is the one introducedby Birnbaum (1969). The Birnbaum importance measure gives the contributions to thesystem reliability due to the reliability of the system component. Vasseur and Llory (1999)reviewed Reliability Achievement Worth (RAW), Reliability Reduction Worth (RRW),Fussell-Vesely (FV), and Birnbaum importance measures as the most valuable importancemeasures for binary systems. Currently, most reliability work has focused on multi-statesystems. Lisnianski and Levitin (2003) wrote the first book on the multi-state systemreliability analysis and optimization. According to different views of the influence of thecomponents on the system performance level in multi-state systems, Meng (1993) proposedtwo kinds of component importance measures for multi-state systems with the minimumsatisfactory system performance level. Zio and Podofillini (2003) generalized the Birnbaumimportance measure with the system performance level of multi-state systems from binarysystems by Monte Carlo simulation. Huang et al. (2000) generalized the multi-state k-out-of-n system model based on different system state levels.

In repairable k-out-of-n: G systems, once components fail, repair work is performedon the failed components. Traditional importance measures mainly concern the changeof the system reliability as the change of the reliability of one component and seldomconsider the expected number of repairs of the objective component in unit time. However,it is important to the improvement rate of the system reliability. In this article, we willpresent an improvement potential rate importance (IPR) to verify the effectiveness of theimprovement rate in system reliability, for multi-state repairable k-out-of-n: G systemsbased on the system performance level.

The rest of this article is organized as follows. The traditional importance measures areintroduced in Sec. 2. Section 3 describes the definition and physical meaning of the newimportance measure and discusses its characteristics. A case study is given to demonstratethe proposed importance in Sec. 4. At last, Sec. 5 draws some conclusions.

2. Traditional Importance Measures

Define 1 (Huang et al., 2000): �(X(t)) ≥ j (j =1, 2, . . . ,M) if there exists an integervalue l (j ≤ l ≤ M) such that at least kl components are in states at least as good as l attime t. An n-component system with such a property is a multi-state k-out-of-n: G systemin which kl do not have to be the same for different system states l.

Birnbaum first introduced the reliability importance of component i for binary systemsas follows (Birnbaum, 1969):

IBMi (t) = ∂ Pr{�(X(t)) = 1}

∂pi,1(t)= Pr {�(X(t)) = 1|Xi(t) = 1}

− Pr {�(X(t)) = 1|Xi(t) = 0} . (1)

Zio and Podofillini (2003) generalized the Birnbaum importance from the binarysystems to the multi-state ones. For one multi-state system of m level, the system workswhen �(X(t)) ≥ m and fails when �(X(t)) <m. So the Birnbaum importance of component


i with the system performance level m of multi-state systems is (Zio and Podofillini, 2003)

IBMi (t) = Pr {�(X(t)) ≥ m|Xi(t) ≥ m} − Pr {�(X(t)) ≥ m|Xi(t) < m} . (2)

So the multi-state k-out-of-n: G system based on the fixed system performance level m

at time t is equivalent to: �(X(t)) ≥ m if and only if at least km components have Xi(t) ≥ m.The reliability of multi-state k-out-of-n: G system at time t is

Pr(�(X(t)) ≥ m) =∑

X1+···+Xn≥km

n∏i=1

(pi,m(t))Xi

n∏i=1

(1 − pi,m(t))1−Xi . (3)

So, according to Eq. (2), the Birnbaum importance of component i of multi-state k-out-of-n:G system at time t is

IBMi (t) = Pr {�(X(t)) ≥ m|Xi(t) ≥ m} − Pr {�(X(t)) ≥ m|Xi(t) < m}

=∑

X1+···+Xi−1+Xi+1+···+Xn≥km−1

n∏j=1,j �=i

(pj,m(t))Xj

n∏j=1,j �=i

(1 − pj,m(t))1−Xj

−∑

X1+···+Xi−1+Xi+1+···+Xn≥km

n∏j=1,j �=i

(pj,m(t))Xj

n∏j=1,j �=i

(1 − pj,m(t))1−Xj

=∑

X1+···+Xi−1+Xi+1+···+Xn=km−1

n∏j=1,j �=i

(pj,m(t))Xj

n∏j=1,j �=i

(1 − pj,m(t))1−Xj . (4)

3. New Importance Measure for Multi-State Repairable k-out-of-n:G Systems

3.1. Definition and Physical Meaning of the New Importance

A repairable k-out-of-n: G system functions if and only if at least k components work,i.e., it fails only if the total number of failed components at any instant of time reachesn-k+1. Once components fail, repair work is performed on the failed components. Assumethat all of the system’s components are working at time 0 and at time t, components havedeteriorated or failed, and the number of components in the failed state does not reachn-k+1.

The transition rate matrix of states of component i is listed as follows:

� =

⎡⎢⎢⎢⎢⎣

aiM,M ai

M,M−1 · · · aiM,1 ai

M,0

aiM−1,M ai

M−1,M−1 · · · aiM−1,1 ai

M−1,0

......

. . ....

...

ai0,M ai

0,M−1 · · · ai0,1 ai

0,0

⎤⎥⎥⎥⎥⎦

,

where aiu,v represents the expected times that component i transits from state u to state v in

unit time. The repair rate μi represents the expected times that component i improves fromstate m below to state m and above in unit time, and μi = ∑m−1

x=0

∑My=m ai

x,y .

2770 Dui et al.

So the improvement potential rate importance (shortly, IPR) in system reliability byrepairing the failed component i from all possible failed current states to the working statesis as follows:

I IPRi (t) = μi (Pr{�(X(t)) ≥ m|Xi(t) ≥ m}− Pr{�(X(t)) ≥ m}) , (5)

where μi also represents the rate of the event of Pr{�(X(t)) ≥ m|Xi(t) ≥m}− Pr{�(X(t)) ≥ m}.Fact. The optimal expected improvement of system reliability in unit time is to improvecomponent i with the maximal I IPR

i (t) for all components in a multi-state repairable k-out-of-n: G system.

Proof. The latter of Eq. (5), Pr{�(X(t)) ≥ m|Xi(t) ≥ m}− Pr{�(X(t)) ≥ m} representsthe improvement of the system reliability by repairing a failed component i from the currentstate to the working state. The μi represents the expected times that component i improvesfrom failed states to working states in unit time. Then the I IPR

i (t) represents the expectedimprovement of system reliability in unit time. So the maximal I IPR

i (t) for all componentswill lead to the optimal expected improvement of system reliability in unit time. �

Theorem 3.1. I IPRi (t) = μi(1 − pi,m(t))IBM

i (t)

Proof. The latter of Eq. (5) can be written as:

Pr{�(X(t)) ≥ m|Xi(t) = M}− Pr{�(X(t)) ≥ m}

=∑

X1+···+Xi−1+Xi+1+···+Xn≥km−1

n∏j=1,j �=i

(pj,m(t))Xj

n∏j=1,j �=i

(1 − pj,m(t))1−Xj

−∑

X1+···+Xn≥km

n∏i=1

(pi,m(t))Xi

n∏i=1

(1 − pi,m(t))1−Xi

=∑

X1+···+Xi−1+Xi+1+···+Xn≥km−1

n∏j=1,j �=i

(pj,m(t))Xj

n∏j=1,j �=i

(1 − pj,m(t))1−Xj

−pi,m(t)∑

X1+···+Xi−1+Xi+1+···+Xn≥km−1

n∏j=1,j �=i

(pj,m(t))Xj

n∏j=1,j �=i

(1 − pj,m(t))1−Xj

− (1 − pi,m(t))∑

X1+···+Xi−1+Xi+1+···+Xn≥km

n∏j=1,j �=i

(pj,m(t))Xj

n∏j=1,j �=i

(1 − pj,m(t))1−Xj

= (1−pi,m(t))

⎡⎣ ∑

X1+···+Xi−1+Xi+1+···+Xn≥km−1

n∏j=1,j �=i

(pj,m(t))Xj

n∏j=1,j �=i

(1 − pj,m(t))1−Xj

−∑

X1+···+Xi−1+Xi+1+···+Xn≥km

n∏j=1,j �=i

(pj,m(t))Xj

n∏j=1,j �=i

(1 − pj,m(t))1−Xj

⎤⎦

= (1 − pi,m(t))∑

X1+···+Xi−1+Xi+1+···+Xn=km−1

n∏j=1,j �=i

(pj,m(t))Xj

n∏j=1,j �=i

(1 − pj,m(t))1−Xj .


According to Eq. (4), we get

Pr{�(X(t)) ≥ m|Xi(t) = M}− Pr{�(X(t)) ≥ m} = (1 − pi,m(t))IBMi (t).

So we have

I IPRi (t) = μi(1 − pi,m(t))IBM

i (t). (6)

This completes the proof of Theorem 3.1. �

From Theorem 3.1, we get that I IPRi (t) generalizes the Birnbaum importance. This is

reasonable since the improvement rate in system reliability due to the improvement of acomponent does depend on the component reliability and repair rate in practice.

3.2. Characteristics of I IPRi (t)

If repair rates of all components are the same, i.e., μ1 = μ2 = · · · = μn = μ, then

I IPRi (t) = μ(1 − pi,m(t))IBM

i (t). (7)

Theorem 3.2. If pj,m(t) > pi,m(t), then I IPRi (t) > I IPR

j (t) when repair times of all compo-nents are identical.

Proof. Equation (7) can be converted to

I IPRi (t)

= μ(1 − pi,m(t)) [Pr {�(X(t)) ≥ m|Xi(t) ≥ m} − Pr {�(X(t)) ≥ m|Xi(t) < m}]

= μ(1 − pi,m(t))∑

X1+···+Xi−1+Xi+1+···+Xn=km−1

n∏j=1,j �=i

(pj,m(t))Xj

n∏j=1,j �=i

(1 − pj,m(t))1−Xj

= μ(1 − pi,m(t))

⎡⎢⎢⎢⎣pj,m(t)

∑n∑

q=1,q �=i,j

Xq=km−2

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1 − pl,m(t))1−Xl

+ (1 − pj,m(t))∑

n∑q=1,q �=i,j

Xq=km−1

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1 − pl,m(t))1−Xl

⎤⎥⎥⎥⎦ .

Similarly, we can get

I IPRj (t) = μ(1 − pj,m(t))

⎡⎢⎢⎢⎣pi,m(t)

∑n∑

q=1,q �=i,j

Xq=km−2

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1−pl,m(t))1−Xl

2772 Dui et al.

+ (1 − pi,m(t))∑

n∑q=1,q �=i,j

Xq=km−1

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1 − pl,m(t))1−Xl

⎤⎥⎥⎥⎦ .

So we can get

I IPRi (t) − I IPR

j (t)

= μ(pj,m(t) − pi,m(t))∑

n∑q=1,q �=i,j

Xq=km−2

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1 − pl,m(t))1−Xl .

Therefore, if pj,m(t) > pi,m(t), then I IPRi (t) > I IPR

j (t).This completes the proof of Theorem 3.2. �

Corollary 3.1. I IPRi (t) = I IPR

j (t) when all components are independent identically dis-tributed and repair times of all components are identical.

Proof. If all components are independent identically distributed, then pj,m(t) = pi,m(t).According to Theorem 3.2, we can prove Corollary 3.1. �

From Corollary 3.1, we can get that all components are the same important when allcomponents are independent identically distributed and repair times of all components areidentical.

All failures and repair times are exponentially distributed according to Assumptions(3) and (4), so the performance stochastic process will have a Markov property and can berepresented by a Markov model (Trivedi, 2002).

The state-space diagram of component i is as in Fig. 1. The failure rate λi of componenti represents the expected times that component i deteriorates from state m and above tostate m below in unit time, and λi = ∑M

x=m

∑m−1y=0 ai

x,y .The following equation for the Markov process can be written (Trivedi, 2002) as

d Pr{Xi(t) ≥ m}dt

= Pr{Xi(t) ≥ m} · λi − Pr{Xi(t) < m} · μi. (8)

When components are at steady state, d Pr{Xi (t)≥m}dt

= 0, so the following algebraic linearequation can be used to evaluate the steady state distribution for component i as follows:

Pr{Xi(t) ≥ k} · λi − Pr{Xi(t) < k} · μi = 0. (9)

Pr{ ( ) }i ix t m( )ix t m( )<ix t m

Pr{ ( )< }i ix t m

Figure 1. State-space diagram of component i.


Theorem 3.3. If λi > λj , μi > μj , then I IPRi (t) > I IPR

j (t) when components are at steadystate.

Proof. Equation (6) can be transformed to

I IPRi (t)

= μi(1 − pi,m(t)) [Pr {�(X(t)) ≥ m|Xi(t) ≥ m} − Pr {�(X(t)) ≥ m|Xi(t) < m}]

= μi(1−pi,m(t))∑

X1+···+Xi−1+Xi+1+···+Xn=km−1

n∏j=1,j �=i

(pj,m(t))Xj

n∏j=1,j �=i

(1 − pj,m(t))1−Xj

= μi(1 − pi,m(t))

⎡⎢⎢⎢⎣pj,m(t)

∑n∑

q=1,q �=i,j

Xq=km−2

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1 − pl,m(t))1−Xl

+ (1 − pj,m(t))∑

n∑q=1,q �=i,j

Xq=km−1

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1 − pl,m(t))1−Xl

⎤⎥⎥⎥⎦ .


I IPRj (t) = μj (1−pj,m(t))

⎡⎢⎢⎢⎣pi,m(t)

∑n∑

q=1,q �=i,j

Xq=km−2

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1−pl,m(t))1−Xl

+ (1 − pi,m(t))∑

n∑q=1,q �=i,j

Xq=km−1

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1 − pl,m(t))1−Xl

⎤⎥⎥⎥⎦ .

So we can get


j (t)

= [μi(1 − pi,m(t))pj,m(t) − μj (1 − pj,m(t))pi,m(t)]

×∑

n∑q=1,q �=i,j

Xq=km−2

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1 − pl,m(t))1−Xl

+ (μi−μj )(1 − pi,m(t))(1 − pj,m(t))

×∑

n∑q=1,q �=i,j

Xq=km−1

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1 − pl,m(t))1−Xl .

2774 Dui et al.

According to Eq. (9), μi(1 − pi,m(t)) = λipi,m(t) and μj (1 − pj,m(t)) = λjpj,m(t).So we have


j (t)

= (λi − λj )pi,m(t)pj,m(t)∑

n∑q=1,q �=i,j

Xq=km−2

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1 − pl,m(t))1−Xl

+ (μi−μj )(1 − pi,m(t))(1 − pj,m(t))∑

n∑q=1,q �=i,j

Xq=km−1

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

× (1 − pl,m(t))1−Xl .

Therefore, if λi > λj , μi > μj , then I IPRi (t) > I IPR

j (t).This completes the proof of Theorem 3.3. �

Corollary 3.2. If pj,m(t) > pi,m(t) and μi > μj , then I IPRi (t) > I IPR

j (t) when componentsare at steady state.

Proof. We have pj,m(t)=e−λj t , pi,m(t) = e−λi t , so pj,m(t) > pi,m(t). According to Theo-rem 3.3, we can prove Corollary 3.2. �

3.3. Comparisons between I IPRi (t) and IBM

i (t)

Theorem 3.4. If pj,m(t) > pi,m(t) and 0 < pl,m(t) < km−1n−1 for l �= i, j , then IBM

i (t) >

IBMj (t).

Proof. Equation (4) can be transformed to

IBMi (t) =

∑X1+···+Xi−1+Xi+1+···+Xn=km−1

n∏j=1,j �=i

(pj,m(t))Xj

n∏j=1,j �=i

(1 − pj,m(t))1−Xj

= pj,m(t)∑

n∑q=1,q �=i,j

Xq=km−2

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1 − pl,m(t))1−Xl

+ (1 − pj,m(t))∑

n∑q=1,q �=i,j

Xq=km−1

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1 − pl,m(t))1−Xl .



IBMj (t) = pi,m(t)

∑n∑

q=1,q �=i,j

Xq=km−2

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1 − pl,m(t))1−Xl

+ (1 − pi,m(t))∑

n∑q=1,q �=i,j

Xq=km−1

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1 − pl,m(t))1−Xl .

So we can get

IBMi (t) − IBM

j (t) = (pj,m(t) − pi,m(t))

⎡⎢⎢⎢⎣

∑n∑

q=1,q �=i,j

Xq=km−2

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

× (1 − pl,m(t))1−Xl

−∑

n∑q=1,q �=i,j

Xq=km−1

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1 − pl,m(t))1−Xl

⎤⎥⎥⎥⎦ .

We consider the pl,m(t) a parameter, similar to the Lemma 2.3 of the reference (Boland andProschan, 1983), we can get that if 0 < pl,m(t) < km−1

n−1 , then

∑n∑

q=1,q �=i,j

Xq=km−2

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1 − pl,m(t))1−Xl

>∑

n∑q=1,q �=i,j

Xq=km−1

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1 − pl,m(t))1−Xl ;

if 1 > pl,m(t) > km−1n−1 , then

∑n∑

q=1,q �=i,j

Xq=km−2

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1 − pl,m(t))1−Xl

<∑

n∑q=1,q �=i,j

Xq=km−1

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1 − pl,m(t))1−Xl .

So if pj,m(t) > pi,m(t) and 0 < pl,m(t) < km−1n−1 for l �= i, j , then IBM

i (t) > IBMj (t).

This completes the proof of Theorem 3.4. �

2776 Dui et al.

Corollary 3.3. If IBMi (t) > IBM

j (t) and 0 < pl,m(t) < km−1n−1 for l �= i, j , then pj,m(t) >

pi,m(t).

Proof. Immediate from Theorem 3.4. �

Theorem 3.5. Suppose that repair times of all components are identical and the improve-ment rate of system reliability is �Ri(t) when we improve component i to the perfectfunctioning, and is �Rj (t) when we improve component j to the perfect functioning for amulti-state repairable k-out-of-n: G systems at time t.

(a) If I IPRi (t) > I IPR

j (t) > 0, then �Ri(t) > �Rj (t).

(b) If IBMi (t) > IBM

j (t) > 0 and 0 < pl,m(t) < km−1n−1 , for l �= i, j , then �Ri(t) >

�Rj (t).

Proof. Assume the original reliability of the system is R(t). Then we have:

�Ri(t)

= μ (Pr{�(X(t)) ≥ m|Xi(t) = M}−R(t))

= μ∑

X1+···+Xi−1+Xi+1+···+Xn≥km−1

n∏j=1,j �=i

(pj,m(t))Xj

n∏j=1,j �=i

(1 − pj,m(t))1−Xj

−μ∑

X1+···+Xn≥km

n∏i=1

(pi,m(t))Xi

n∏i=1

(1 − pi,m(t))1−Xi ,

�Rj (t)

= μ(Pr{�(Xj (t)) ≥ m|Xi(t) = M}−R(t)

)

= μ∑

X1+···+Xj−1+Xj+1+···+Xn≥km−1

n∏i=1,i �=j

(pj,m(t))Xi

n∏i=1,i �=j

(1 − pi,m(t))1−Xi

−μ∑

X1+···+Xn≥k

n∏i=1

(pi,m(t))Xi

n∏i=1

(1 − pi,m(t))1−Xi .

So, we can get

�Ri(t) − �Rj (t)

= μ∑

n∑q=1,q �=i

Xq≥km−1

n∏j=1,j �=i

(pj,m(t))Xj

n∏j=1,j �=i

(1 − pj,m(t))1−Xj

−μ∑

n∑q=1,q �=j

Xq≥km−1

n∏i=1,i �=j

(pi,m(t))Xi

n∏i=1,i �=j

(1 − pi,m(t))1−Xi


= μpj,m(t)∑

n∑q=1,q �=i,j

Xq≥km−2

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1 − pl,m(t))1−Xl

+μ(1 − pj,m(t))∑

n∑q=1,q �=i,j

Xq≥km−1

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1 − pl,m(t))1−Xl

−μpi,m(t)∑

n∑q=1,q �=i,j

Xq≥km−2

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1 − pl,m(t))1−Xl

−μ(1 − pi,m(t))∑

n∑q=1,q �=i,j

Xq≥km−1

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1 − pl,m(t))1−Xl

= μ(pj,m(t) − pi,m(t))∑

n∑q=1,q �=i,j

Xq≥km−2

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1 − pl,m(t))1−Xl

−μ(pj,m(t) − pi,m(t))∑

n∑q=1,q �=i,j

Xq≥km−1

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1 − pl,m(t))1−Xl

= μ(pj,m(t) − pi,m(t))∑

n∑q=1,q �=i,j

Xq=km−2

n∏l=1,l �=i,j

(pl,m(t))Xl

n∏l=1,l �=i,j

(1 − pl,m(t))1−Xl .

(a) From Fact, we can get �Ri(t) = I IPRi (t),�Rj (t) = I IPR

j (t), so if I IPRi (t) >

I IPRj (t) > 0, then �Ri(t) > �Rj (t).

(b) From Corollary 3.3, if IBMi (t) > IBM

j (t) and 0 < pl,m(t) < km−1n−1 for l �= i, j , then

pj,m(t) > pi,m(t). So we have �Ri(t) > �Rj (t).�

This completes the proof of Theorem 3.5.From Theorem 3.5, we can get that from the point of view of the improvement of

system reliability by improving component i to the perfect functioning, I IPRi (t) can exactly

measure the change of the system reliability, but the Birnbaum importance cannot.

4. A Case Study

In a communications system with three transmitters (k-out-of-3 system) based on the systemperformance level m, the average message load may be such that at least two transmittersmust be operational at all times or critical messages may be lost. Thus, the transmissionsubsystem functions as a 2-out-of-3: G system (km = 2). Table 1 shows the failure andrepair rate of components of the communications system.

2778 Dui et al.

Table 1Failure and repair rates of system components

Component Failure rate λi Repair rate μi

1 0.0038 0.11092 0.0427 0.53233 0.0159 0.1187

The reliability of multi-state 2-out-of-3: G system at time t is

Pr(�(X(t)) ≥ m) = p1,m(t)p2,m(t)p3,m(t) + p1,m(t)p2,m(t)(1 − p3,m(t))

+ (1 − p1,m(t))p2,m(t)p3,m(t) + p1,m(t)(1 − p2,m(t))p3,m(t).

The Birnbaum importance of component i at time t is

IBM1 (t) = p2,m(t)(1 − p3,m(t)) + (1 − p2,m(t))p3,m(t),

IBM2 (t) = p1,m(t)(1 − p3,m(t)) + (1 − p1,m(t))p3,m(t),

IBM3 (t) = p2,m(t)(1 − p1,m(t)) + (1 − p2,m(t))p1,m(t).

(10)

4.1. The Case of Time t = 30

The reliability of component i at time t ispi,m(t) = e−λi t , so at time t = 30, p1,m(30) =0.8923, p2,m(30) = 0.2778, p3,m(30) = 0.6206. According to Eqs. (6) and (10), we can getTable 2 as follows.

From Table 2, component 3 has the most importance according to the Birnbaum im-portance and IBM

3 (30) > IBM1 (30). This is in accordance with Theorem 3.4. Because the

km−1n−1 = 1

2 = 0.5, p2,m(30) = 0.2778 < 0.5 and p1,m(30) = 0.8923 > p3,m(30) = 0.6206,we can get IBM

3 (30) > IBM1 (30). From the point of view of the improvement potential rate

in system reliability, component 2 has the most importance according to IPR because com-ponent 2 has the largest repair rate. Further, IBM

3 (30) > IBM1 (30) and p2,m(30) = 0.2778 <

0.5, soI IPR3 (30) > I IPR

1 (30). This is in accordance with Theorem 3.5(b).From Table 2, when repair times of all components are identical, we have I IPR

1 (30) =0.0596 × μ, I IPR

2 (30) = 0.2928 × μ, I IPR3 (30) = 0.2558 × μ. ThenI IPR

2 (30) > I IPR3 (30) >

I IPR1 (30) and p2,m(30) < p3,m(30) < p1,m(30) which is in accordance with Theorem 3.2.

Table 2Component importance at time t = 30

Component Birnbaum importance Rank IPR Rank

1 0.5536 2 0.0596 × μ1 = 0.0066 32 0.4054 3 0.2928 × μ2 = 0.1558 13 0.6743 1 0.2558 × μ3 = 0.0304 2


Table 3Component importance at the steady state

Component Birnbaum importance Rank IPR Rank

1 0.1749 1 0.00064202 32 0.1434 2 0.00570000 13 0.1025 3 0.00140000 2

4.2. The Case of the Steady State (t = +∞)

For the steady state, according to Eq. (9), the reliability of component i is pi,m = μi/(λi +μi). So we can get p1,m(+∞) = 0.9669, p2,m(+∞) = 0.9257, p3,m(+∞) = 0.8819. Allthe parameters needed in the simulations can be shown in the following Table 3.

From Table 3, component 1 has the most importance according to the Birnbaumimportance. This is different from the rank of Table 2. The rank of the Birnbaum importanceof components varies with the time. While the ranks of IPR at time t = 30 and the steadystate are the same, and component 2 has the most importance according to IPR.

From Table 1, λ2 > λ3, μ2 > μ3 and λ3 > λ1, μ3 > μ1, so I IPR2 (+∞) > I IPR

3 (+∞) >

I IPR1 (+∞) in Table 3. This is in accordance with Theorem 3.3.

5. Conclusions

This article discussed the IPR for multi-state repairable k-out-of-n: G systems based on thesystem performance level m. The main contributions can be summarized as follows.

(1) In multi-state k-out-of-n: G systems, the expected number of repairs of the objectivecomponent is important to the improvement rate of the system reliability. From thepoint of view of this, IPR can describe the effectiveness of the improvement rate insystem reliability.

(2) In multi-state repairable k-out-of-n: G systems, the optimal expected improvement ofsystem reliability in unit time is to improve component i with the maximal IPR for allcomponents.

(3) IPR generalizes the Birnbaum importance. This is reasonable since the improvementrate in system reliability due to the improvement of a component depends on thecomponent reliability and repair rate in practice.

(4) From the point of view of the improvement of system reliability in unit time byimproving component i, I IPR

i (t) can exactly measure the change of the system reliability,but the Birnbaum importance cannot.

Funding

The authors gratefully acknowledge the financial support for this research from the 973Program of China (Grant No. 2010CB328000), the China Civil Aircraft Advanced ResearchProject, the China Aeronautical Science Foundations (Grant No. 2009ZE53052), the Sci-ence and Technology Project of Shaanxi Province (Grant No. 2010K8-11), the Grant ofNorthwestern Polytechnical University (No. 11GH0134).

2780 Dui et al.

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