Research ArticleReliability Analysis of Load-Sharing 119870-out-of-119873 SystemConsidering Component Degradation

Chunbo Yang1 Shengkui Zeng12 and Jianbin Guo12

1School of Reliability and Systems Engineering Beihang University Beijing 100191 China2Science and Technology on Reliability and Environmental Engineering Laboratory Beijing 100191 China

Correspondence should be addressed to Jianbin Guo guojianbinbuaaeducn

Received 29 October 2014 Accepted 2 April 2015

Academic Editor Enrico Zio

Copyright copy 2015 Chunbo Yang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The119870-out-of-119873 configuration is a typical form of redundancy techniques to improve system reliability where at least119870-out-of-119873components must work for successful operation of system When the components are degraded more components are needed tomeet the system requirement whichmeans that the value of119870 has to increaseThe current reliability analysis methods overestimatethe reliability because using constant 119870 ignores the degradation effect In a load-sharing system with degrading componentsthe workload shared on each surviving component will increase after a random component failure resulting in higher failurerate and increased performance degradation rate This paper proposes a method combining a tampered failure rate model witha performance degradation model to analyze the reliability of load-sharing 119870-out-of-119873 system with degrading components Theproposed method considers the value of 119870 as a variable which is derived by the performance degradation model Also the load-sharing effect is evaluated by the tampered failure rate model Monte-Carlo simulation procedure is used to estimate the discreteprobability distribution of 119870 The case of a solar panel is studied in this paper and the result shows that the reliability consideringcomponent degradation is less than that ignoring component degradation

1 Introduction

Redundancy technique is widely used to improve systemreliability A typical form of redundancy is a 119870-out-of-119873configuration in which at least 119870 out of119873 components mustwork for normal operation of systemWhen using traditionalmethods [1ndash4] to analyze reliability of 119870-out-of-119873 systemindependence is assumed within the system which meansthat a component failure does not affect the failure rate orperformance of surviving components However in the realworld many systems are load-sharing such as electric gen-erators sharing an electrical load in a power plant cables in asuspension bridge and valves or pumps in a hydraulic systemIn the load-sharing system the workload has to be sharedby the remaining components resulting in an increased loadshared on each surviving component [5] Many empiricalstudies of mechanical systems [6] and computer systems [7]have proved that theworkload strongly affects the componentfailure rate Scheuer [8] studied the reliability of 119870-out-of-119873 system when component failure induces higher failure

rate in survivors The method is limited in system composedof 119904-independent and identically distributed componentswith exponential lifetimes Liu [9] proposed a generalizedaccelerated failure-timemodel (AFTM) for reliability analysisof load-sharing 119870-out-of-119873 system with arbitrary distri-bution load-dependent component lifetime distributionsAmari et al [10] provided a closed-form analytical solutionfor the reliability of tampered failure rate load-sharing 119870-out-of-119873 system and Amari and Bergman [11] also used thecumulative exposure model to account for the effect of load-ing history The mentioned reliability analysis methods arebased on the assumption of binary components that is to saythe component is either failed or working in perfect state Infact the performance of component is degrading in lifetimeIn order to meet the system performance requirement thevalue of 119870 has to monotonously increase in service time Inthat case the reliability of 119870-out-of-119873 system consideringcomponent degradationmay be less than the reliability basedon assumption of binary component

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 726853 10 pageshttpdxdoiorg1011552015726853

2 Mathematical Problems in Engineering

When components have several degraded performancestates the multistate system (MSS) model is introducedto analyze the reliability of 119870-out-of-119873 system [12ndash15]Amari et al [16] presented a fast and robust reliabilityevaluation algorithm for very large multistate 119870-out-of-119873systems Levitin [17] introduced a new model named multi-state vector-119870-out-of-119873 system which is a generalization ofexistingmultistate119870-out-of-119873 systemmodels Using currentMSS algorithms to calculate reliability we should know theprobability that component performance is located in eachstate Many experimental and theoretical researches [18ndash20]indicate that the performance degradation rate is strangelyaffected by the load shared on components In the load-sharing system the shared load is increasing and then theperformance degradation rate will increase correspondinglyConsequently it is difficult to get the probability of eachperformance state

In the degradation process the value of 119870 is increasingbecause of component degradation The phased-missionsystem (PMS) model [21ndash23] is introduced to calculate thereliability of 119870-out-of-119873 system with variable 119870 in differentphases Xing et al [24] proposed an efficient method for reli-ability evaluation of 119870-out-of-119873 systems subject to phased-mission requirements Unfortunately when using the PMSmodel to evaluate reliability we need to know the specified119870in each phase However in degradation process the value of119870 is determined by the system performance requirement andthe degraded performance of each component As a resultthe value of 119870 in each phase is variable Considering thecomplexity of degradation law in load-sharing system it ishard to know the specified 119870 in each phase

In conclusion there are some special characteristics ofload-sharing 119870-out-of-119873 system with degrading compo-nents (1) a component random failure increases the loadshared on each remaining component (2) the failure rate ofsurviving components will increase after component failures(3) the rise of load will raise the degradation rate of com-ponent (4) the duration when components have a specifieddegradation rate is stochastic because a component failureoccurs at random time (5) the value of 119870 is stochasticbecause the component performance is a random variableat any given time Some mentioned load-sharing methodsdeal with characteristics (1) and (2) effectively What ismore the MSS model is suggested to compute the reliabilitywhen component has degraded states and the PMS modelis used to deal with phase-mission requirement Howevercharacteristics (3) (4) and (5) make the degradation law ofcomponent performance very complicated consequently it isvery hard to get some conditions of existed methods

This paper proposes a method combining a tamperedfailure rate model with a performance degradation modelto analyze the reliability of load-sharing 119870-out-of-119873 sys-tem with degrading components The tampered failure ratemodel is introduced to evaluate the reliability where therise of load makes the failure rate of surviving componentsincrease The performance degradation model is derived tocalculate degraded component performance in service timeFurthermore the load-sharing effect on degradation rate and

the randomness of duration are included in the performancedegradation model In this way the discrete probability dis-tribution of 119870 is obtained through performance degradationmodel coupled with load-sharing effect Using the discreteprobability distribution of 119870 and the reliability computedby the tampered failure rate model when the system has aspecified119870 the reliability of load-sharing119870-out-of-119873 systemwith degrading components is calculated accurately

The remainder of this paper is divided into five sectionsIn Section 2 the tampered failure rate model is introduced toanalyze reliability of load-sharing 119870-out-of-119873 system whenthe value of 119870 is constant In Section 3 the performancedegradation model coupled with load-sharing effect is for-mulated to evaluate the discrete probability distribution of119870 In Section 4Monte-Carlo simulation procedure is used toestimate the discrete probability distribution of119870The case ofa solar panel is studied in Section 5 Conclusions are drawn inSection 6

2 Reliability of Load-Sharing119870-out-of-119873 System

In a load-sharing system theworkload has to be redistributedamong the remaining components after a component failureMostly the load is equally shared by each surviving compo-nent Let the total workload be 119871 and let the total numberof components be 119899 Let 119911

119894be the load on each surviving

component when 119894 components have failed Hence

1199110 =119871

119899

119911119894=

119871

(119899 minus 119894)

(1)

In order to analyze the reliability of load-sharing systemthe load-sharing effect that the rise of shared load on asurviving component raises the failure rate has to be eval-uated The tampered failure rate (TFR) model proposed byBhattacharyya and Soejoeti [25] can be applied for the load-sharing systemThe acceleration of failure when load is raisedfrom lower level to a higher level is reflected in the failure ratefunction

Let the component be subject to an ordered sequence ofloads where load 119911

119894(119894 = 0 1 119899 minus 119896) is applied during the

time interval [120591119894 120591119894+1] (1205910 = 0) According to the TFR model

the failure rate of the component at 119905 is

120582 (119905) = 120582119894 (119905) = 120575119894 sdot 1205820 (119905) = 120575 (119911119894) sdot 1205820 (119905)

for 120591119894minus1 le 119905 le 120591119894

(2)

where 1205820(119905) is the baseline failure rate which has nothing todo with load 120575

119894is the tampered factor at load 119911

119894 and 119911

119894is the

load shared on component at 119905With the assumptions that the load is equally distributed

among all surviving identical components and the failurerate of a component varies as described in the TFR modelthe reliability of load-sharing 119870-out-of-119873 system withoutcomponent performance degradation could be calculated byanalytical method

Mathematical Problems in Engineering 3

21 Exponential Case Firstly consider a load-sharing 119870-out-of-119873 system with components following the exponentiallifetime distribution [8] in other words the baseline failurerate of the TFR model 1205820(119905) is constant

When the system is put into operation the failure rateof every component is denoted by 1205820 Because there are119899 working components in the system the first componentfailure occurs at failure rate 1205721 = 119899 sdot 1205820 After the first failurethe remaining (119899 minus 1) working components must carry thesame workload of system As a result the failure rate of eachsurviving component becomes1205821 which is commonly higherthan 1205820 The second component failure occurs at rate 1205722 =(119899 minus 1) sdot 1205821 When 119894 components have failed the failure rateof each (119899 minus 119894) remaining component is denoted by 120582

119894(0 le

119894 le 119899 minus 119896) The 119894th component failure occurs at failure rate120572119894= (119899 minus 119894 + 1) sdot 120582

119894minus1 The system is failed when more than(119899 minus 119896) components are failed

The time when the 119894th component failure occurs in thesystem is denoted by 119879

119894(1198790 equiv 0) and the time interval

between (119894 minus 1)th and 119894th component failure is representedby 119883119894= 119879119894minus 119879119894minus1 (1 le 119894 le 119899 minus 119896 + 1) Since all identical

components are following the exponential distributions the119883119894follows the exponential distribution with parameter 120572

119894

Hence the lifetime of system is the (119899 minus 119896 + 1)th failure time

119879 = 119879119899minus119896+1 =

119899minus119896+1sum

119894=1119883119894 (3)

Then the reliability of119870-out-of-119873 system at 1199050 is

119877 (1199050) = 119875 119879gt 1199050 (4)

In order to calculate the distribution of 119879 and thereliability function of load-sharing 119870-out-of-119873 system twotypical formulas which can be used are as follows

Case a All 120572119894are equal (say 120572) [8]

119877 (119905) =

119899minus119896

sum

119894=0

(120572119905)119894 exp (minus120572119905)119894

= 119892119886119898119891119888 (120572119905 119899 minus 119896 + 1) (5)

This case arises when the failure rate of each survivingcomponent is directly proportional to the load it carrieswhich means that 120575(119911) prop 119911 in the TFR model

Case b All 120572119894are distinct [8]

119877 (119905) =

119899minus119896+1sum

119894=1119860119894sdot exp (minus120572

119894119905)

119860119894equiv

119899minus119896+1prod

119895=1119895 =119894

120572119895

120572119895minus 120572119894

119894 = 1 2 119899 minus 119896 + 1(6)

22 General Case In the case of a load-sharing 119870-out-of-119873 system with components following arbitrary lifetimedistributions the baseline failure rate of the TFR model is nolonger a constant A closed-form analytical solution for TFR

model with an arbitrary baseline distribution is introducedThe basic idea is to use a time-transformation to convert TFRmodel with an arbitrary baseline distribution into an equiva-lent problem with an exponential baseline distribution [10]

Lemma 1 (1) For any failure distribution 119865(119905) the reliabilityfunction is 119877(119905) = 1 minus 119865(119905)

(2) The cumulative failure rate function is

Λ (119905) = int

119905

0120582 (119905) 119889119905 = int

119905

0

119889119865 (119905)

119877 (119905)= int

119905

0

minus119889119877 (119905)

119877 (119905)

= minus ln119877 (119905) (7)

(3) The random variable 119910 = 119877(119905) follows a uniformdistribution in interval [0 1] The random variable 119897 = Λ(119905)follows an exponential distribution with mean 1

(4) For a TFR model with a standard exponential (120582 = 1)baseline failure time distribution

120582 (119905) = 120575119894

Λ (119905) = Λ (120591119894minus1) + 120575119894 (119905 minus 120591119894minus1) 120591119894minus1 le 119905 lt 120591119894

(8)

(5) For a TFR model with a baseline failure rate of 1205820(119905)and a baseline cumulative failure rate of Λ 0(119905)

(a) under the regular scale 119905

120582 (119905) = 120575119894 sdot 1205820 (119905)

Λ (119905) = Λ (120591119894minus1) + 120575119894 sdot [Λ 0 (119905) minusΛ 0 (120591119894minus1)]

120591119894minus1 le 119905 lt 120591119894

(9)

(b) under the transformed scale 119897 = Λ 0(119905)Let

V119894= Λ 0 (120591119894)

120582119897 (119897) = 120575119894

Λ119897 (119897) = Λ 119897 (V119894minus1) + 120575119894 sdot [119897 minus V119894minus1] V

119894minus1 le 119897 lt V119894

(10)

where Λ119897(119897) is the cumulative failure rate in the transformed

scale

According to the above lemmas it becomes obvious thatif the load-sharing effect on the failure rate of an individualcomponent follows a TFR model with 120582(119905) = 120575

119894sdot 1205820(119905) the

reliability of a load-sharing system at 119905 is equivalent to thereliability of corresponding exponential load-sharing modelat time 119897 = Λ 0(119905) where the failure rate of a component is120582119894= 120575119894when 119894 components have failed for 119894 = 0 (119899 minus 119896)

3 Degradation Effect in Load-Sharing System

In a load-sharing 119870-out-of-119873 system with degrading com-ponents a random component failure raises the load sharedon remaining components leading to the rise of the failurerate and performance degradation rate Besides the durationwhen component has a specified degradation rate is variablebecause a component failure occurs at random time There-fore the component performance is a random variable at anygiven time thus the value of119870 is stochastic in service time

4 Mathematical Problems in Engineering

31 Independent Degradation Effect At the first step considerthe component degradation rate to be independent of load-sharing effect which means that the load-sharing effect justmakes the failure rate higher but the degradation rate hasnothing to do with shared load When the system is justput into operation the value of 119870 is determined by 119870 =

lceil1198621198880rceil where 119862 is the system performance requirement1198880 is the initial performance of each component and 119870 isthe minimum integer that is greater than or equal to thequotient When the system has operated for a period of time119905 the degraded performance of component is denoted by119888(119905) = 1198880119863(119905) where 119863(119905) is degradation law of componentperformance 119863(119905) is a monotone decreasing function and119863(0) = 1 119863(+infin) = 0 For example the commondegradation laws are exponential law denoted by 119863(119905) = 1 minus119886119905119898 and power law denoted by119863(119905) = exp(minus119886119905) At time 119905 the

value of 119870 is determined by 119870(119905) = lceil119862119888(119905)rceil Since 119863(119905) is amonotone decreasing function it is quite clear that 119888(119905) lt 1198880Thus119870(119905) ge 119870 while the condition for equality is lceil119862119888(119905)rceil =lceil1198621198880rceil Hence the reliability of 119870-out-of-119873 system withdegrading component is calculated in a general expression as

1198771015840(119905) =

119899minus119870(119905)

sum

119894=0119862119894

119899[1minus1198770 (119905)]

119894[1198770 (119905)]

119899minus119894 (11)

Correspondingly using the TFR model to analyze theload-sharing 119870-out-of-119873 system with degrading compo-nents the value of119870 should be replaced by 119870(119905)

Obviously 1198771015840(119905) le 119877(119905) because of 119870(119905) ge 119870 mean-ing that the reliability of 119870-out-of-119873 system consideringcomponent degradation is less than the reliability ignoringcomponent degradation

32 Degradation Coupled with Load-Sharing Effect Whenthe component performance degradation is related to load-sharing effect a random failure of component will raise thedegradation rate of surviving component Since a componentfailure occurs at random time the duration when compo-nents have a specified degradation rate is variable Thereforethe variable119870 denoted by (119905) is stochastic at any given time

In a load-sharing system the degradation law of compo-nent performance119863(119905) is not only a function of time but alsoa function of the shared load The generalization of degrada-tion law is expressed as a function of time and load denotedby 119863(119905 119911) The degradation rate of component performancewith load-sharing effect is 119889(119905 119911) = 120597119863(119905 119911)120597119905 After119894th component failure the performance degradation rate ofsurviving components is denoted by 119889

119894(119905 119911119894) = 119889[119905 119871(119899minus 119894)]

In order to analyze the reliability of load-sharing 119870-out-of-119873 system with degrading components at 1199050 (1199050) shouldbe evaluated previously which is determined by the degradedperformance of component at 1199050 With the understandingof degradation rate 119889

119894(119905 119911119894) after 119894th component failure the

degraded performance at 1199050 is estimated by using the dura-tion when the component has a specified degradation ratedenoted by119883

119894Theduration119883

119894is the interval between (119894minus1)th

component failure and 119894th component failure In other words119883119894is the minimum order statistic of the components failure

time of the surviving (119899 minus 119894 + 1) components under thecondition that the (119899 minus 119894 + 1) components do not fail at 119879

119894minus1

Lemma 2(1) Conditional Distribution Function

(a) The reliability of component at (119879119894+ Δ119905) is

119877 (119879119894+Δ119905) = exp [minusint

119879119894+Δ119905

0120582 (119905) 119889119905]

= exp [minusint1198791

01205820 (119905) 119889119905 minusint

1198792

1198791

1205821 (119905) 119889119905 minus sdot sdot sdot

minus int

119879119894

119879119894minus1

120582119894minus1 (119905) 119889119905 minusint

119879119894+Δ119905

119879119894

120582119894 (119905) 119889119905]

= exp[

[

minus

119894minus1sum

119895=0int

119879119895+1

119879119895

120582119895 (119905) 119889119905 minusint

119879119894+Δ119905

119879119894

120582119894 (119905) 119889119905

]

]

(12)

(b) The reliability at 119879119894is

119877 (119879119894) = exp[

[

minus

119894minus1sum

119895=0int

119879119895+1

119879119895

120582119895 (119905) 119889119905

]

]

(13)

(c) The cumulative distribution function of componentfailure time under the condition that the component didnot fail at 119879

119894is

119865 (119879119894+Δ119905 | 119879

119894) = 1minus119877 (119879

119894+Δ119905 | 119879

119894)

= 1minus119877 (119879119894+ Δ119905)

119877 (119879119894)

= 1minus exp [minusint119879119894+Δ119905

119879119894

120582119894 (119905) 119889119905]

= 1minus exp [minusintΔ119905

0120582119894(119905 + 119879119894) 119889119905]

(14)

(2) Minimum Order Statistic

(a) Let the cumulative distribution function of a popula-tion be 119865(119905) and probability distribution function is119891(119905) For samples with size 119899 probability distributionfunction of 119895th order statistic 119884

(119895)is

119891119884(119895) (119905) = 119899119862

119895minus1119899minus1 [119865 (119905)]

119895minus1[1minus119865 (119905)]119899minus119895 119891 (119905) (15)

(b) The probability distribution function of the minimumorder statistic is

119891119884(1) (119905) = 119899119891 (119905) [1minus119865 (119905)]

119899minus1 (16)

Mathematical Problems in Engineering 5

(c) The cumulative distribution function of the minimumorder statistic is

119865119884(1) (119905) = int

119905

0119891119884(1) (119905) 119889119905 = int

119905

0119899119891 (119905) [1minus119865 (119905)]119899minus1 119889119905

= 1minus [1minus119865 (119905)]119899 (17)

Hence the cumulative distribution function of 119883119894is

expressed as

119865119883119894(Δ119905)

= 1minus [1minus119865 (119879119894minus1 +Δ119905 | 119879119894minus1)]

119899minus119894+1

= 1minus [119877 (119879119894minus1 +Δ119905 | 119879119894minus1)]

119899minus119894+1

= 1minus exp [minus (119899 minus 119894 + 1) intΔ119905

0120582119894minus1 (119905 + 119879119894minus1) 119889119905]

(18)

In a special situation that the baseline distribution ofTFR is exponential distribution the distribution of 119883

119894is an

exponential distribution with parameter (119899minus119894+1)120582119894minus1 which

agrees with the conclusion in previous paper Section 21As 119894 components have failed in the system the degrada-

tion rate of each surviving component is 119889119894(119905 119911119894) Moreover

the time interval between 119894th component failure and (119894 +1)th failure is 119883

119894+1 Therefore the component performancedegradation in time interval [119879

119894 119879119894+1] is

Δ119888119894= 1198880 int

119879119894+1

119879119894

119889119894(119905 119911119894) 119889119905 = 1198880 int

119883119894+1

0119889119894(119879119894+ 119905 119911119894) 119889119905 (19)

The degraded component performance at 1199050 is calculatedas

119888 (1199050) = 1198880 +119898minus1sum

119895=0Δ119888119895+ 1198880 int

1199050

119879119898

119889119898(119905 119911119898) 119889119905

= 1198880 + 1198880

119898minus1sum

119895=0int

119879119895+1

119879119895

119889119895(119905 119911119895) 119889119905

+ 1198880 int1199050

119879119898

119889119898(119905 119911119898) 119889119905

(20)

According to 119883119895+1 = 119879119895+1 minus 119879119895 119879119895 = sum

119895

119894=0119883119894+1 (1198790 equiv 0)and 119911119895= 119871(119899 minus 119895) for 119895 = 0 1 2 119898 119898 is determined by

119879119898le 1199050 lt 119879119898+1

int

119879119895+1

119879119895

119889119895(119905 119911119895) 119889119905 = int

119883119895+1

0119889119895(119905 +119879

119895119871

119899 minus 119895) 119889119905 (21)

Hence

119888 (1199050) = 1198880 + 1198880

119898minus1sum

119895=0int

119883119895+1

0119889119895(119905 +119879

119895119871

119899 minus 119895) 119889119905

+ 1198880 int1199050minus119879119898

0119889119898(119905 +119879

119898119871

119899 minus 119898)119889119905

(22)

where the distribution of 119883119895is 119865119883119895(Δ119905) and the degradation

rate at each phase is 119889119895(119905 119911119895)

According to (22) the distribution of the degradedcomponent performance 119888(1199050) is calculated on basis of thedistribution of 119883

119895denoted by 119865

119883119895(Δ119905) and degradation rate

119889119895(119905 119911119895) Then the discrete probability distribution of (1199050)

could be obtained through

(1199050) = lceil119862

119888 (1199050)rceil = [

[[[

119862

1198880 + 1198880sum119898minus1119895=0 int119883119895+1

0 119889119895(119905 + 119879

119895 119871119899 minus 119895) 119889119905 + 1198880 int

1199050minus119879119898

0 119889119898(119905 + 119879

119898 119871119899 minus 119898) 119889119905

]]]]

(23)

Using the probability distribution of (1199050) estimatedby the performance degradation model and the reliabilitycalculated by the TFRmodel when the system has a specified119870 the reliability of load-sharing 119870-out-of-119873 system withdegrading components is computed in formula as

(1199050) = sum119901(119895) 119877119895 (1199050) (24)

where 119901(119895) is the probability when the (1199050) is equal to

119895and 119877

119895(1199050) is the reliability of 119870-out-of-119873 system when

the (1199050) is equal to 119895 Using the proposed formula to

evaluate the reliability of load-sharing 119870-out-of-119873 systemwith degrading components the load-sharing effect on com-ponent failure rate and the degradation effect coupled withload-sharing effect are all included in the model Thereforethe reliability is calculated more accurately

4 Monte-Carlo Simulation

In the load-sharing system the failure rate and degradationrate are variable during the lifetime and the duration in eachphase is stochastic because a component failure occurs at ran-dom time The analytic expression of degraded performancecould be obtained only if the failure rate or degradation rateis subject to some specified formats such as componentsfollowing the exponential lifetime distribution or degrada-tion rate being constant This paper employs Monte-Carlosimulation to estimate the discrete probability distribution of119870 The simulation procedure is shown in Figure 1

In order to evaluate the discrete probability distributionof 119870 the system configuration must be set firstly as thetotal component number denoted by 119899 the total workloaddenoted by 119871 the system performance requirement denotedby 119862 the initial performance of each component denoted by

6 Mathematical Problems in Engineering

Start

Initialization

Develop FX119894(Δt)

Sample the failure time interval Xi

Calculate Ti

Calculate the probability of each K(t0)

Compute the component performance degradation in [Timinus1 t0]

Compute the component performance

Set i = i + 1

No

No

Yes

Calculate the K(t0)Count the number of different K(t0)

End

Yes

Set j = j + 1

Ti gt t0

j = S

degradation in [Timinus1 Ti]

Set initial value n L C c0 t0 S j

Formulate d(t z) 1205820(t) 120575(z)

Reset T0 = 0 i = 1

Calculate ziminus1 and refresh 120575(ziminus1) d(t ziminus1)

Figure 1 The Monte-Carlo simulation procedure

1198880 and the system required operation time denoted by 1199050Furthermore the baseline failure rate function denoted by1205820(119905) the degradation rate function denoted by 119889(119905 119911) andthe tampered factor function denoted by 120575(119911) all need to beformulated in advance Set the required number of simulationcycle 119878 and initialize cyclic variable 119895 = 1 and then carry outthe Monte-Carlo simulation

At the beginning of a simulation cycle reset the startmoment 1198790 = 0 and sampling variable 119894 = 1 At the 119894th sam-pling in a cycle of the simulation there are (119899minus119894+1) remainingcomponents The workload shared on each component isdenoted by 119911

119894minus1 Then the tampered factor can be calculatedby 120575(119911) and the failure rate under current load is obtainedthrough the baseline failure rate function multiplied by thetampered factor Besides the degradation rate under currentload is computed by 119889(119905 119911) According to the current failurerate and last component failure time 119879

119894minus1 the distributionof 119883119894between (119894 minus 1)th and 119894th component failure denoted

by 119865119883119894(Δ119905) is obtained Using the distribution 119865

119883119894(Δ119905) the

sampling formula of119883119894denoted by 119865minus1

119883119894(120578) is derived where 120578

follows uniform distribution in the interval [0 1] Generatingthe 119883

119894from 119865minus1

119883119894(120578) the 119894th component failure time is 119879

119894=

119879119894minus1 + 119883119894 If 119879119894 is less than 1199050 compute the component

performance degradation in time interval [119879119894minus1 119879119894] and then

start the next sampling When 119879119894reaches system required

operation time 1199050 compute the component performancedegradation in time interval [119879

119894minus1 1199050] and then the degradedcomponent performance at 1199050 is obtained The (1199050) isdetermined by the system performance requirement 119862 andthe degraded component performance 119888(1199050) and then carryout the next cycle of the simulation When the number ofsimulation cycles meets the requirement denoted by 119878 countthe number when (1199050) is equal to different values Then theprobability of each (1199050) is estimated by dividing the numberof different (1199050) by the number of simulation cycles 119878

Mathematical Problems in Engineering 7

5 Case Study

Satellites or space station which needs to work for long peri-ods in aerospace must be equipped with suitable and reliablesolar panels to provide power for the normal operation ofequipment Once a solar panel cannot supply required powerthe spacecraft will lose its function and it may become ldquospacejunkrdquo As the critical module there are strict requirementsabout the performance and reliability of solar panel A solarpanelmodule consists ofmany solar cells Factors of the spaceenvironment like space radiation temperature cycling andcharge-discharge cycles will destroy some solar cells Besidesthe performance of solar cells will degrade in lifetime Thefailure solar cell cannot be repaired in outer space timely sothe workload of solar panel is shared by the remaining solarcells and surviving solar cells should supply the requiredpower of system

A kind of solar panel of geostationary orbit satellite has 20solar cellsThe satellite needs 10 kWoutput power (the systemperformance requirement) from solar panel to ensure normaloperation of other subsystems The initial output power ofeach solar cell is 1 kW (initial performance of component)The solar panel is subject to 10A steady current (the totalworkload) and it is shared by 20 solar cells evenly Thebaseline failure time distribution is Weibull one with shapeparameter 120573 = 2 and location parameter 120578 = 200000 h Thetampered factor is 120575(119911) = 11991115 under the operating current 119911The degradation law of solar cell output power is 119863(119905 119911) =exp(1198861199051199114) where 119886 = minus16000000 According to therequirement the solar panel should supply sufficient powerfor 15 years (consider one year as 365 days and then 15 yearsare equal to 131400 hours) The output power of a solar cellis degrading with service time The random failure of a solarcell will increase the current shared on remaining solar cellsresulting in higher failure rate of solar cells and the degra-dation rate of solar cells output power will increase corre-spondinglyThe value of119870 is determined by the output powerof surviving solar cells and the output power requirement ofsolar panel It is obvious that the solar panels are a typicalload-sharing 119870-out-of-119873 system with component perfor-mance degradationWith the proposedmethod in this paperthe reliability of solar panels could be estimated accurately

Firstly we should calculate the discrete probability of 119870by performance degradation model

According to the Monte-Carlo simulation method thetime interval 119883

119894between 119894th and (119894 minus 1)th solar cell failure

is subject to

119865119883119894(Δ119905)

= 1minus exp [minus (119899 minus 119894 + 1) intΔ119905

0120582119894minus1 (119905 + 119879119894minus1) 119889119905]

(25)

Using the given value 119899 = 20 119871 = 10 1205820(119905) and 120575(119911119894minus1)the distribution of119883

119894is expressed as

Table 1 The count and possibility of119870(131400)

119870(131400) 12 13 14 15Count 29 1710 6749 1512Possibility 029 171 6749 1512

119865119883119894(Δ119905) = 1minus expminus1015 sdot (21minus 119894)minus05

sdot [(119879119894minus1 + Δ119905

200000)

2minus(

119879119894minus1

200000)

2]

(26)

Hence the sampling formula of119883119894is derived

Δ119905 = 200000

sdot [(119879119894minus1

200000)

2minus(21 minus 119894)05

1015ln (1minus 120578)]

12

minus119879119894minus1

(27)

where 120578 is subject to uniform distribution in [0 1]The degradation rate of a solar cell output power is

119889 (119905 119911) =120597119863 (119905 119911)

120597119905=119886

1199114exp(119886119905

1199114) (28)

where 119886 = minus16000000 The required output power of thesolar panel is 119862 = 10 and the initial output power of a solarcell is 1198880 = 1

The output power degradation in time interval [119879119894 119879119894+1]

is

Δ119888119894= 1198880 int

119879119894

119879119894minus1

119889119894minus1 (119905 119911119894minus1) 119889119905

= 1198880 int119879119894

119879119894minus1

119889 (119905119871

119899 + 1 minus 119894) 119889119905

(29)

With the given value

Δ119888119894

= minusint

119879119894

119879119894minus1

16000000 (1021 minus 119894)4

exp[minus 119905

6000000 (1021 minus 119894)4]119889119905

(30)

After a cycle of the simulation the output power of a solarcell 119888(131400) at 119905 = 131400 h can be calculated Hence thevalue of119870 is expressed as

(131400) = lceil 10119888 (131400)

rceil (31)

Set the number of cycle as 119878 = 10000 the count andpossibility of (131400) are shown in Table 1

If the output power of a solar cell does not degrade thevalue of119870 is

119870 = lceil101198880rceil = lceil

101rceil = 10 (32)

8 Mathematical Problems in Engineering

Table 2 The coefficients of 120572119894

1205721 1205722 1205723 1205724 1205725 1205726 1205727 1205728 1205729 12057210

70711 72548 74536 76696 79057 81650 84515 87706 91287 9534612057211 12057212 12057213 12057214 12057215 12057216 12057217 12057218 12057219 12057220

10 105409 111803 119523 129099 141421 158114 182574 223607 316228

Table 3 The coefficients of 119860119894when 119870(131400) = 12

1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609

977231198907 minus578581198908 147681198909 minus211931198909 186671198909 minus103111198909 347871198908 minus653321198907 520861198906

From the above results because the output power of asolar cell is degrading the value of119870 at 119905 = 131400 h is greaterthan the situation ignoring solar cell degradation

Secondly after getting the discrete probability distri-bution of (131400) the reliability when (131400) is aspecified value should be calculated by the TFR model

TFR model of solar panel is shown as follows

Model 119899 = 20 119871 = 10 119905 = 131400 h 120573 = 2 and120578 = 200000Baseline failure rate

1205820 (119905) =120573

120578sdot (119905

120578)

120573minus1=

1100000

sdot119905

200000 (33)

The cumulative failure rate

Λ 0 (119905) = (119905

120578)

120573

= (119905

200000)

2 (34)

The TFR model of failure rate caused by operatingcurrent

120582 (119905) = 120575 (119911) sdot 1205820 (119905) (35)

where the tampered factor is 120575(119911) = 11991115

Solution is as follows

The tampered factor

120575119894= 120575 (119911

119894) = 119911

15119894= (

119871

119899 minus 119894)

15= (

1020 minus 119894

)

15 (36)

The corresponding TFR model with exponentialbaseline failure distribution is as followsTransformed scale

119897 = Λ 0 (119905) = (131400200000

)

2 (37)

Because 120582119894= 120575119894= (10(20 minus 119894))15 120572

119894= (119899 minus 119894 + 1)120582

119894minus1 =

1015(21 minus 119894)minus05 the coefficients of 120572119894are shown in Table 2

When (131400) = 12

119860119894equiv

20minus12+1prod

119895=1119895 =119894

120572119895

120572119895minus 120572119894

=

9prod

119895=1119895 =119894

120572119895

120572119895minus 120572119894

119894 = 1 2 9 (38)

The coefficients of119860119894when (131400) = 12 are shown in

Table 3

=12 (131400) =

20minus12+1sum

119894=1119860119894sdot exp [minus120572

119894Λ 0 (131400)]

=

9sum

119894=1119860119894sdot exp [minus(1314

2000)

2120572119894]

= 09912

(39)

In the same way the reliability when the system has other119870 is calculated as follows

=13 (131400) = 09778

=14 (131400) = 09477

=15 (131400) = 08866

(40)

Therefore the reliability of solar panel that it could supplysufficient power for 15 years is computed as

(131400) = sum119901 (119895) 119877119895 (131400) = 09437 (41)

If the output power of solar cells does not degrade mean-ing that119870 = 10 the coefficients of 119860

119894are shown in Table 4

Hence the reliability is

119877 (131400) =20minus10+1sum

119894=1119860119894sdot exp [minus120572

119894Λ 0 (131400)]

=

11sum

119894=1119860119894sdot exp [minus(1314

2000)

2120572119894] = 09988

(42)

Comparing 119877(131400) and (131400) we can see thatrandom failures of some solar cells make the operatingcurrent of the remaining solar cells increase also leading toan increase of failure rate and degradation rate Due to thedegradation of solar cells output power the reliability of solarpanel (131400) is less than 119877(131400) which ignores thedegradation of solar cell output power

6 Conclusion

In a load-sharing119870-out-of-119873 systemwith degrading compo-nents a component random failure raises the load shared on

Mathematical Problems in Engineering 9

Table 4 The coefficients of 119860119894when 119870(131400) = 10

1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 11986011

129131198909 minus881411198909 2657111989010 minus4649511989010 5217111989010 minus3911511989010 1977711989010 minus663151198909 140421198909 minus168481198908 862831198906

each remaining component resulting in a higher failure rateand increased degradation rate of surviving componentsThispaper proposes amethod combining a TFRmodel with a per-formance degradationmodel to analyze the reliability of load-sharing119870-out-of-119873 systemwith degrading componentsTheTFR model deals with load-sharing effect on failure rateand the reliability when the system has a specified 119870 iscalculated by the TFR model The performance degradationmodel is derived to evaluate degradation effect coupledwith load-sharing effect and then the degraded componentperformance is estimated considering the load-sharing effecton degradation rateThe case of a solar panel is a typical load-sharing119870-out-of-119873 systemwith degrading componentsTheresults calculated by the proposed method show that thereliability considering component degradation is less thanthat ignoring component degradation With utilization ofthe proposed method the degradation effect is quantitativelyevaluated and then the reliability of load-sharing 119870-out-of-119873 system can be calculated more accurately

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was funded by National Natural Science Foun-dation of China (NSFC 61304218) In addition the authorswould like to thank the anonymous reviewers who havehelped to improve the paper

References

[1] S V Amari H Pham and R B Misra ldquoReliability characteris-tics of k-out-of-n warm standby systemsrdquo IEEE Transactions onReliability vol 61 no 4 pp 1007ndash1018 2012

[2] R Moghaddass M J Zuo and W Wang ldquoAvailability of ageneral k-out-of-nG system with non-identical componentsconsidering shut-off rules using quasi-birthdeath processrdquo Reli-ability Engineering and SystemSafety vol 96 no 4 pp 489ndash4962011

[3] P Boddu and L Xing ldquoReliability evaluation and optimizationof series-parallel systems with k-out-of-n G subsystems andmixed redundancy typesrdquo Proceedings of the Institution ofMechanical Engineers Part O Journal of Risk and Reliability vol227 no 2 pp 187ndash198 2013

[4] RMoghaddassM J Zuo and J Qu ldquoReliability and availabilityanalysis of a repairable k-out-of-n G system with R repairmensubject to shut-off rulesrdquo IEEE Transactions on Reliability vol60 no 3 pp 658ndash666 2011

[5] S V Amari K B Misra and H Pham ldquoTampered failure rateload-sharing systems status and perspectivesrdquo in Handbook on

Performability Engineering chapter 20 Springer New York NYUSA 2007

[6] K C Kapur and L R Lambberson Reliability in EngineeringDesign John Wiley amp Sons 1977

[7] R K Iyer and D J Rossetti ldquoA measurement-based model forworkload dependency of CPU errorsrdquo IEEE Transactions onComputers vol C-35 no 6 pp 511ndash519 1986

[8] E M Scheuer ldquoReliability of an m-out-of-n system whencomponent failure induces higher failure rates in survivorsrdquoIEEE Transactions on Reliability vol 37 no 1 pp 73ndash74 1988

[9] H Liu ldquoReliability of a load-sharing fc-out-of-nG system non-iid components with arbitrary distributionsrdquo IEEE Transactionson Reliability vol 47 no 3 pp 279ndash284 1998

[10] S V Amari K B Misra and H Pham ldquoReliability analysisof tampered failure rate load-sharing k-out-of-n G systemsrdquoin Proceedings of the 12th ISSAT International Conference onReliability and Quality in Design pp 30ndash35 August 2006

[11] S V Amari and R Bergman ldquoReliability analysis of k-out-of-n load-sharing systemsrdquo in Proceedings of the 54th AnnualReliability and Maintainability Symposium (RAMS rsquo08) pp440ndash445 IEEE Las Vegas Nev USA January 2008

[12] G Levitin and S V Amari ldquoMulti-state systemswithmulti-faultcoveragerdquo Reliability Engineering and System Safety vol 93 no11 pp 1730ndash1739 2008

[13] Y Ding M J Zuo A Lisnianski and W Li ldquoA frameworkfor reliability approximation of multi-state weighted k-out-of-n systemsrdquo IEEE Transactions on Reliability vol 59 no 2 pp297ndash308 2010

[14] S K Chaturvedi S H Bahsa S V Amari et al ldquoReliability anal-ysis of generalized multi-state k-out-of-n systemsrdquo Proceedingsof the Institution of Mechanical Engineers Part O Journal of Riskand Reliability vol 226 pp 327ndash336 2011

[15] S Eryilmaz ldquoOn reliability analysis of a k-out-of-n system withcomponents having random weightsrdquo Reliability Engineeringand System Safety vol 109 pp 41ndash44 2013

[16] S V Amari M J Zuo and G Dill ldquoA fast and robust reliabilityevaluation algorithm for generalized multi-state k-out-of-nsystemsrdquo IEEE Transactions on Reliability vol 58 no 1 pp 88ndash97 2009

[17] G Levitin ldquoMulti-state vector-k-out-of-n systemsrdquo IEEE Trans-actions on Reliability vol 62 no 3 pp 648ndash657 2013

[18] J Fan K C Yung and M Pecht ldquoPhysics-of-failure-basedprognostics and health management for high-power whitelight-emitting diode lightingrdquo IEEE Transactions on Device andMaterials Reliability vol 11 no 3 pp 407ndash416 2011

[19] R Baillot Y Deshayes L Bechou et al ldquoEffects of siliconecoating degradation on GaN MQW LEDs performances usingphysical and chemical analysesrdquoMicroelectronics Reliability vol50 no 9ndash11 pp 1568ndash1573 2010

[20] Y-F Li E Zio and Y-H Lin ldquoA multistate physics modelof component degradation based on stochastic petri nets andsimulationrdquo IEEE Transactions on Reliability vol 61 no 4 pp921ndash931 2012

[21] L Xing and J B Dugan ldquoAnalysis of generalized phased-mission system reliability performance and sensitivityrdquo IEEETransactions on Reliability vol 51 no 2 pp 199ndash211 2002

10 Mathematical Problems in Engineering

[22] A Shrestha L Xing and Y Dai ldquoReliability analysis ofmultistate phased-mission systemswith unordered and orderedstatesrdquo IEEETransactions on SystemsMan andCybernetics PartASystems and Humans vol 41 no 4 pp 625ndash636 2011

[23] G Levitin L Xing S V Amari and Y Dai ldquoReliability ofnonrepairable phased-mission systems with common causefailuresrdquo IEEE Transactions on Systems Man and CyberneticsPart ASystems and Humans vol 43 no 4 pp 967ndash978 2013

[24] L Xing S V Amari and C Wang ldquoReliability of k-out-of-nsystems with phased-mission requirements and imperfect faultcoveragerdquo Reliability Engineering and System Safety vol 103 pp45ndash50 2012

[25] G K Bhattacharyya and Z Soejoeti ldquoA tampered failure ratemodel for step-stress accelerated life testrdquo Communications inStatisticsTheory andMethods vol 18 no 5 pp 1627ndash1643 1989

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

When components have several degraded performancestates the multistate system (MSS) model is introducedto analyze the reliability of 119870-out-of-119873 system [12ndash15]Amari et al [16] presented a fast and robust reliabilityevaluation algorithm for very large multistate 119870-out-of-119873systems Levitin [17] introduced a new model named multi-state vector-119870-out-of-119873 system which is a generalization ofexistingmultistate119870-out-of-119873 systemmodels Using currentMSS algorithms to calculate reliability we should know theprobability that component performance is located in eachstate Many experimental and theoretical researches [18ndash20]indicate that the performance degradation rate is strangelyaffected by the load shared on components In the load-sharing system the shared load is increasing and then theperformance degradation rate will increase correspondinglyConsequently it is difficult to get the probability of eachperformance state

In the degradation process the value of 119870 is increasingbecause of component degradation The phased-missionsystem (PMS) model [21ndash23] is introduced to calculate thereliability of 119870-out-of-119873 system with variable 119870 in differentphases Xing et al [24] proposed an efficient method for reli-ability evaluation of 119870-out-of-119873 systems subject to phased-mission requirements Unfortunately when using the PMSmodel to evaluate reliability we need to know the specified119870in each phase However in degradation process the value of119870 is determined by the system performance requirement andthe degraded performance of each component As a resultthe value of 119870 in each phase is variable Considering thecomplexity of degradation law in load-sharing system it ishard to know the specified 119870 in each phase

In conclusion there are some special characteristics ofload-sharing 119870-out-of-119873 system with degrading compo-nents (1) a component random failure increases the loadshared on each remaining component (2) the failure rate ofsurviving components will increase after component failures(3) the rise of load will raise the degradation rate of com-ponent (4) the duration when components have a specifieddegradation rate is stochastic because a component failureoccurs at random time (5) the value of 119870 is stochasticbecause the component performance is a random variableat any given time Some mentioned load-sharing methodsdeal with characteristics (1) and (2) effectively What ismore the MSS model is suggested to compute the reliabilitywhen component has degraded states and the PMS modelis used to deal with phase-mission requirement Howevercharacteristics (3) (4) and (5) make the degradation law ofcomponent performance very complicated consequently it isvery hard to get some conditions of existed methods

This paper proposes a method combining a tamperedfailure rate model with a performance degradation modelto analyze the reliability of load-sharing 119870-out-of-119873 sys-tem with degrading components The tampered failure ratemodel is introduced to evaluate the reliability where therise of load makes the failure rate of surviving componentsincrease The performance degradation model is derived tocalculate degraded component performance in service timeFurthermore the load-sharing effect on degradation rate and

the randomness of duration are included in the performancedegradation model In this way the discrete probability dis-tribution of 119870 is obtained through performance degradationmodel coupled with load-sharing effect Using the discreteprobability distribution of 119870 and the reliability computedby the tampered failure rate model when the system has aspecified119870 the reliability of load-sharing119870-out-of-119873 systemwith degrading components is calculated accurately

The remainder of this paper is divided into five sectionsIn Section 2 the tampered failure rate model is introduced toanalyze reliability of load-sharing 119870-out-of-119873 system whenthe value of 119870 is constant In Section 3 the performancedegradation model coupled with load-sharing effect is for-mulated to evaluate the discrete probability distribution of119870 In Section 4Monte-Carlo simulation procedure is used toestimate the discrete probability distribution of119870The case ofa solar panel is studied in Section 5 Conclusions are drawn inSection 6

2 Reliability of Load-Sharing119870-out-of-119873 System

In a load-sharing system theworkload has to be redistributedamong the remaining components after a component failureMostly the load is equally shared by each surviving compo-nent Let the total workload be 119871 and let the total numberof components be 119899 Let 119911

119894be the load on each surviving

component when 119894 components have failed Hence

1199110 =119871

119899

119911119894=

119871

(119899 minus 119894)

(1)

In order to analyze the reliability of load-sharing systemthe load-sharing effect that the rise of shared load on asurviving component raises the failure rate has to be eval-uated The tampered failure rate (TFR) model proposed byBhattacharyya and Soejoeti [25] can be applied for the load-sharing systemThe acceleration of failure when load is raisedfrom lower level to a higher level is reflected in the failure ratefunction

Let the component be subject to an ordered sequence ofloads where load 119911

119894(119894 = 0 1 119899 minus 119896) is applied during the

time interval [120591119894 120591119894+1] (1205910 = 0) According to the TFR model

the failure rate of the component at 119905 is

120582 (119905) = 120582119894 (119905) = 120575119894 sdot 1205820 (119905) = 120575 (119911119894) sdot 1205820 (119905)

for 120591119894minus1 le 119905 le 120591119894

(2)

where 1205820(119905) is the baseline failure rate which has nothing todo with load 120575

119894is the tampered factor at load 119911

119894 and 119911

119894is the

load shared on component at 119905With the assumptions that the load is equally distributed

among all surviving identical components and the failurerate of a component varies as described in the TFR modelthe reliability of load-sharing 119870-out-of-119873 system withoutcomponent performance degradation could be calculated byanalytical method

Mathematical Problems in Engineering 3

21 Exponential Case Firstly consider a load-sharing 119870-out-of-119873 system with components following the exponentiallifetime distribution [8] in other words the baseline failurerate of the TFR model 1205820(119905) is constant

When the system is put into operation the failure rateof every component is denoted by 1205820 Because there are119899 working components in the system the first componentfailure occurs at failure rate 1205721 = 119899 sdot 1205820 After the first failurethe remaining (119899 minus 1) working components must carry thesame workload of system As a result the failure rate of eachsurviving component becomes1205821 which is commonly higherthan 1205820 The second component failure occurs at rate 1205722 =(119899 minus 1) sdot 1205821 When 119894 components have failed the failure rateof each (119899 minus 119894) remaining component is denoted by 120582

119894(0 le

119894 le 119899 minus 119896) The 119894th component failure occurs at failure rate120572119894= (119899 minus 119894 + 1) sdot 120582

119894minus1 The system is failed when more than(119899 minus 119896) components are failed

The time when the 119894th component failure occurs in thesystem is denoted by 119879

119894(1198790 equiv 0) and the time interval

between (119894 minus 1)th and 119894th component failure is representedby 119883119894= 119879119894minus 119879119894minus1 (1 le 119894 le 119899 minus 119896 + 1) Since all identical

components are following the exponential distributions the119883119894follows the exponential distribution with parameter 120572

119894

Hence the lifetime of system is the (119899 minus 119896 + 1)th failure time

119879 = 119879119899minus119896+1 =

119899minus119896+1sum

119894=1119883119894 (3)

Then the reliability of119870-out-of-119873 system at 1199050 is

119877 (1199050) = 119875 119879gt 1199050 (4)

In order to calculate the distribution of 119879 and thereliability function of load-sharing 119870-out-of-119873 system twotypical formulas which can be used are as follows

Case a All 120572119894are equal (say 120572) [8]

119877 (119905) =

119899minus119896

sum

119894=0

(120572119905)119894 exp (minus120572119905)119894

= 119892119886119898119891119888 (120572119905 119899 minus 119896 + 1) (5)

This case arises when the failure rate of each survivingcomponent is directly proportional to the load it carrieswhich means that 120575(119911) prop 119911 in the TFR model

Case b All 120572119894are distinct [8]

119877 (119905) =

119899minus119896+1sum

119894=1119860119894sdot exp (minus120572

119894119905)

119860119894equiv

119899minus119896+1prod

119895=1119895 =119894

120572119895

120572119895minus 120572119894

119894 = 1 2 119899 minus 119896 + 1(6)

22 General Case In the case of a load-sharing 119870-out-of-119873 system with components following arbitrary lifetimedistributions the baseline failure rate of the TFR model is nolonger a constant A closed-form analytical solution for TFR

model with an arbitrary baseline distribution is introducedThe basic idea is to use a time-transformation to convert TFRmodel with an arbitrary baseline distribution into an equiva-lent problem with an exponential baseline distribution [10]

Lemma 1 (1) For any failure distribution 119865(119905) the reliabilityfunction is 119877(119905) = 1 minus 119865(119905)

(2) The cumulative failure rate function is

Λ (119905) = int

119905

0120582 (119905) 119889119905 = int

119905

0

119889119865 (119905)

119877 (119905)= int

119905

0

minus119889119877 (119905)

119877 (119905)

= minus ln119877 (119905) (7)

(3) The random variable 119910 = 119877(119905) follows a uniformdistribution in interval [0 1] The random variable 119897 = Λ(119905)follows an exponential distribution with mean 1

(4) For a TFR model with a standard exponential (120582 = 1)baseline failure time distribution

120582 (119905) = 120575119894

Λ (119905) = Λ (120591119894minus1) + 120575119894 (119905 minus 120591119894minus1) 120591119894minus1 le 119905 lt 120591119894

(8)

(5) For a TFR model with a baseline failure rate of 1205820(119905)and a baseline cumulative failure rate of Λ 0(119905)

(a) under the regular scale 119905

120582 (119905) = 120575119894 sdot 1205820 (119905)

Λ (119905) = Λ (120591119894minus1) + 120575119894 sdot [Λ 0 (119905) minusΛ 0 (120591119894minus1)]

120591119894minus1 le 119905 lt 120591119894

(9)

(b) under the transformed scale 119897 = Λ 0(119905)Let

V119894= Λ 0 (120591119894)

120582119897 (119897) = 120575119894

Λ119897 (119897) = Λ 119897 (V119894minus1) + 120575119894 sdot [119897 minus V119894minus1] V

119894minus1 le 119897 lt V119894

(10)

where Λ119897(119897) is the cumulative failure rate in the transformed

scale

According to the above lemmas it becomes obvious thatif the load-sharing effect on the failure rate of an individualcomponent follows a TFR model with 120582(119905) = 120575

119894sdot 1205820(119905) the

reliability of a load-sharing system at 119905 is equivalent to thereliability of corresponding exponential load-sharing modelat time 119897 = Λ 0(119905) where the failure rate of a component is120582119894= 120575119894when 119894 components have failed for 119894 = 0 (119899 minus 119896)

3 Degradation Effect in Load-Sharing System

In a load-sharing 119870-out-of-119873 system with degrading com-ponents a random component failure raises the load sharedon remaining components leading to the rise of the failurerate and performance degradation rate Besides the durationwhen component has a specified degradation rate is variablebecause a component failure occurs at random time There-fore the component performance is a random variable at anygiven time thus the value of119870 is stochastic in service time

4 Mathematical Problems in Engineering

31 Independent Degradation Effect At the first step considerthe component degradation rate to be independent of load-sharing effect which means that the load-sharing effect justmakes the failure rate higher but the degradation rate hasnothing to do with shared load When the system is justput into operation the value of 119870 is determined by 119870 =

lceil1198621198880rceil where 119862 is the system performance requirement1198880 is the initial performance of each component and 119870 isthe minimum integer that is greater than or equal to thequotient When the system has operated for a period of time119905 the degraded performance of component is denoted by119888(119905) = 1198880119863(119905) where 119863(119905) is degradation law of componentperformance 119863(119905) is a monotone decreasing function and119863(0) = 1 119863(+infin) = 0 For example the commondegradation laws are exponential law denoted by 119863(119905) = 1 minus119886119905119898 and power law denoted by119863(119905) = exp(minus119886119905) At time 119905 the

value of 119870 is determined by 119870(119905) = lceil119862119888(119905)rceil Since 119863(119905) is amonotone decreasing function it is quite clear that 119888(119905) lt 1198880Thus119870(119905) ge 119870 while the condition for equality is lceil119862119888(119905)rceil =lceil1198621198880rceil Hence the reliability of 119870-out-of-119873 system withdegrading component is calculated in a general expression as

1198771015840(119905) =

119899minus119870(119905)

sum

119894=0119862119894

119899[1minus1198770 (119905)]

119894[1198770 (119905)]

119899minus119894 (11)

Correspondingly using the TFR model to analyze theload-sharing 119870-out-of-119873 system with degrading compo-nents the value of119870 should be replaced by 119870(119905)

Obviously 1198771015840(119905) le 119877(119905) because of 119870(119905) ge 119870 mean-ing that the reliability of 119870-out-of-119873 system consideringcomponent degradation is less than the reliability ignoringcomponent degradation

32 Degradation Coupled with Load-Sharing Effect Whenthe component performance degradation is related to load-sharing effect a random failure of component will raise thedegradation rate of surviving component Since a componentfailure occurs at random time the duration when compo-nents have a specified degradation rate is variable Thereforethe variable119870 denoted by (119905) is stochastic at any given time

In a load-sharing system the degradation law of compo-nent performance119863(119905) is not only a function of time but alsoa function of the shared load The generalization of degrada-tion law is expressed as a function of time and load denotedby 119863(119905 119911) The degradation rate of component performancewith load-sharing effect is 119889(119905 119911) = 120597119863(119905 119911)120597119905 After119894th component failure the performance degradation rate ofsurviving components is denoted by 119889

119894(119905 119911119894) = 119889[119905 119871(119899minus 119894)]

In order to analyze the reliability of load-sharing 119870-out-of-119873 system with degrading components at 1199050 (1199050) shouldbe evaluated previously which is determined by the degradedperformance of component at 1199050 With the understandingof degradation rate 119889

119894(119905 119911119894) after 119894th component failure the

degraded performance at 1199050 is estimated by using the dura-tion when the component has a specified degradation ratedenoted by119883

119894Theduration119883

119894is the interval between (119894minus1)th

component failure and 119894th component failure In other words119883119894is the minimum order statistic of the components failure

time of the surviving (119899 minus 119894 + 1) components under thecondition that the (119899 minus 119894 + 1) components do not fail at 119879

119894minus1

Lemma 2(1) Conditional Distribution Function

(a) The reliability of component at (119879119894+ Δ119905) is

119877 (119879119894+Δ119905) = exp [minusint

119879119894+Δ119905

0120582 (119905) 119889119905]

= exp [minusint1198791

01205820 (119905) 119889119905 minusint

1198792

1198791

1205821 (119905) 119889119905 minus sdot sdot sdot

minus int

119879119894

119879119894minus1

120582119894minus1 (119905) 119889119905 minusint

119879119894+Δ119905

119879119894

120582119894 (119905) 119889119905]

= exp[

[

minus

119894minus1sum

119895=0int

119879119895+1

119879119895

120582119895 (119905) 119889119905 minusint

119879119894+Δ119905

119879119894

120582119894 (119905) 119889119905

]

]

(12)

(b) The reliability at 119879119894is

119877 (119879119894) = exp[

[

minus

119894minus1sum

119895=0int

119879119895+1

119879119895

120582119895 (119905) 119889119905

]

]

(13)

(c) The cumulative distribution function of componentfailure time under the condition that the component didnot fail at 119879

119894is

119865 (119879119894+Δ119905 | 119879

119894) = 1minus119877 (119879

119894+Δ119905 | 119879

119894)

= 1minus119877 (119879119894+ Δ119905)

119877 (119879119894)

= 1minus exp [minusint119879119894+Δ119905

119879119894

120582119894 (119905) 119889119905]

= 1minus exp [minusintΔ119905

0120582119894(119905 + 119879119894) 119889119905]

(14)

(2) Minimum Order Statistic

(a) Let the cumulative distribution function of a popula-tion be 119865(119905) and probability distribution function is119891(119905) For samples with size 119899 probability distributionfunction of 119895th order statistic 119884

(119895)is

119891119884(119895) (119905) = 119899119862

119895minus1119899minus1 [119865 (119905)]

119895minus1[1minus119865 (119905)]119899minus119895 119891 (119905) (15)

(b) The probability distribution function of the minimumorder statistic is

119891119884(1) (119905) = 119899119891 (119905) [1minus119865 (119905)]

119899minus1 (16)

Mathematical Problems in Engineering 5

(c) The cumulative distribution function of the minimumorder statistic is

119865119884(1) (119905) = int

119905

0119891119884(1) (119905) 119889119905 = int

119905

0119899119891 (119905) [1minus119865 (119905)]119899minus1 119889119905

= 1minus [1minus119865 (119905)]119899 (17)

Hence the cumulative distribution function of 119883119894is

expressed as

119865119883119894(Δ119905)

= 1minus [1minus119865 (119879119894minus1 +Δ119905 | 119879119894minus1)]

119899minus119894+1

= 1minus [119877 (119879119894minus1 +Δ119905 | 119879119894minus1)]

119899minus119894+1

= 1minus exp [minus (119899 minus 119894 + 1) intΔ119905

0120582119894minus1 (119905 + 119879119894minus1) 119889119905]

(18)

In a special situation that the baseline distribution ofTFR is exponential distribution the distribution of 119883

119894is an

exponential distribution with parameter (119899minus119894+1)120582119894minus1 which

agrees with the conclusion in previous paper Section 21As 119894 components have failed in the system the degrada-

tion rate of each surviving component is 119889119894(119905 119911119894) Moreover

the time interval between 119894th component failure and (119894 +1)th failure is 119883

119894+1 Therefore the component performancedegradation in time interval [119879

119894 119879119894+1] is

Δ119888119894= 1198880 int

119879119894+1

119879119894

119889119894(119905 119911119894) 119889119905 = 1198880 int

119883119894+1

0119889119894(119879119894+ 119905 119911119894) 119889119905 (19)

The degraded component performance at 1199050 is calculatedas

119888 (1199050) = 1198880 +119898minus1sum

119895=0Δ119888119895+ 1198880 int

1199050

119879119898

119889119898(119905 119911119898) 119889119905

= 1198880 + 1198880

119898minus1sum

119895=0int

119879119895+1

119879119895

119889119895(119905 119911119895) 119889119905

+ 1198880 int1199050

119879119898

119889119898(119905 119911119898) 119889119905

(20)

According to 119883119895+1 = 119879119895+1 minus 119879119895 119879119895 = sum

119895

119894=0119883119894+1 (1198790 equiv 0)and 119911119895= 119871(119899 minus 119895) for 119895 = 0 1 2 119898 119898 is determined by

119879119898le 1199050 lt 119879119898+1

int

119879119895+1

119879119895

119889119895(119905 119911119895) 119889119905 = int

119883119895+1

0119889119895(119905 +119879

119895119871

119899 minus 119895) 119889119905 (21)

Hence

119888 (1199050) = 1198880 + 1198880

119898minus1sum

119895=0int

119883119895+1

0119889119895(119905 +119879

119895119871

119899 minus 119895) 119889119905

+ 1198880 int1199050minus119879119898

0119889119898(119905 +119879

119898119871

119899 minus 119898)119889119905

(22)

where the distribution of 119883119895is 119865119883119895(Δ119905) and the degradation

rate at each phase is 119889119895(119905 119911119895)

According to (22) the distribution of the degradedcomponent performance 119888(1199050) is calculated on basis of thedistribution of 119883

119895denoted by 119865

119883119895(Δ119905) and degradation rate

119889119895(119905 119911119895) Then the discrete probability distribution of (1199050)

could be obtained through

(1199050) = lceil119862

119888 (1199050)rceil = [

[[[

119862

1198880 + 1198880sum119898minus1119895=0 int119883119895+1

0 119889119895(119905 + 119879

119895 119871119899 minus 119895) 119889119905 + 1198880 int

1199050minus119879119898

0 119889119898(119905 + 119879

119898 119871119899 minus 119898) 119889119905

]]]]

(23)

Using the probability distribution of (1199050) estimatedby the performance degradation model and the reliabilitycalculated by the TFRmodel when the system has a specified119870 the reliability of load-sharing 119870-out-of-119873 system withdegrading components is computed in formula as

(1199050) = sum119901(119895) 119877119895 (1199050) (24)

where 119901(119895) is the probability when the (1199050) is equal to

119895and 119877

119895(1199050) is the reliability of 119870-out-of-119873 system when

the (1199050) is equal to 119895 Using the proposed formula to

evaluate the reliability of load-sharing 119870-out-of-119873 systemwith degrading components the load-sharing effect on com-ponent failure rate and the degradation effect coupled withload-sharing effect are all included in the model Thereforethe reliability is calculated more accurately

4 Monte-Carlo Simulation

In the load-sharing system the failure rate and degradationrate are variable during the lifetime and the duration in eachphase is stochastic because a component failure occurs at ran-dom time The analytic expression of degraded performancecould be obtained only if the failure rate or degradation rateis subject to some specified formats such as componentsfollowing the exponential lifetime distribution or degrada-tion rate being constant This paper employs Monte-Carlosimulation to estimate the discrete probability distribution of119870 The simulation procedure is shown in Figure 1

In order to evaluate the discrete probability distributionof 119870 the system configuration must be set firstly as thetotal component number denoted by 119899 the total workloaddenoted by 119871 the system performance requirement denotedby 119862 the initial performance of each component denoted by

6 Mathematical Problems in Engineering

Start

Initialization

Develop FX119894(Δt)

Sample the failure time interval Xi

Calculate Ti

Calculate the probability of each K(t0)

Compute the component performance degradation in [Timinus1 t0]

Compute the component performance

Set i = i + 1

No

No

Yes

Calculate the K(t0)Count the number of different K(t0)

End

Yes

Set j = j + 1

Ti gt t0

j = S

degradation in [Timinus1 Ti]

Set initial value n L C c0 t0 S j

Formulate d(t z) 1205820(t) 120575(z)

Reset T0 = 0 i = 1

Calculate ziminus1 and refresh 120575(ziminus1) d(t ziminus1)

Figure 1 The Monte-Carlo simulation procedure

1198880 and the system required operation time denoted by 1199050Furthermore the baseline failure rate function denoted by1205820(119905) the degradation rate function denoted by 119889(119905 119911) andthe tampered factor function denoted by 120575(119911) all need to beformulated in advance Set the required number of simulationcycle 119878 and initialize cyclic variable 119895 = 1 and then carry outthe Monte-Carlo simulation

At the beginning of a simulation cycle reset the startmoment 1198790 = 0 and sampling variable 119894 = 1 At the 119894th sam-pling in a cycle of the simulation there are (119899minus119894+1) remainingcomponents The workload shared on each component isdenoted by 119911

119894minus1 Then the tampered factor can be calculatedby 120575(119911) and the failure rate under current load is obtainedthrough the baseline failure rate function multiplied by thetampered factor Besides the degradation rate under currentload is computed by 119889(119905 119911) According to the current failurerate and last component failure time 119879

119894minus1 the distributionof 119883119894between (119894 minus 1)th and 119894th component failure denoted

by 119865119883119894(Δ119905) is obtained Using the distribution 119865

119883119894(Δ119905) the

sampling formula of119883119894denoted by 119865minus1

119883119894(120578) is derived where 120578

follows uniform distribution in the interval [0 1] Generatingthe 119883

119894from 119865minus1

119883119894(120578) the 119894th component failure time is 119879

119894=

119879119894minus1 + 119883119894 If 119879119894 is less than 1199050 compute the component

performance degradation in time interval [119879119894minus1 119879119894] and then

start the next sampling When 119879119894reaches system required

operation time 1199050 compute the component performancedegradation in time interval [119879

119894minus1 1199050] and then the degradedcomponent performance at 1199050 is obtained The (1199050) isdetermined by the system performance requirement 119862 andthe degraded component performance 119888(1199050) and then carryout the next cycle of the simulation When the number ofsimulation cycles meets the requirement denoted by 119878 countthe number when (1199050) is equal to different values Then theprobability of each (1199050) is estimated by dividing the numberof different (1199050) by the number of simulation cycles 119878

Mathematical Problems in Engineering 7

5 Case Study

Satellites or space station which needs to work for long peri-ods in aerospace must be equipped with suitable and reliablesolar panels to provide power for the normal operation ofequipment Once a solar panel cannot supply required powerthe spacecraft will lose its function and it may become ldquospacejunkrdquo As the critical module there are strict requirementsabout the performance and reliability of solar panel A solarpanelmodule consists ofmany solar cells Factors of the spaceenvironment like space radiation temperature cycling andcharge-discharge cycles will destroy some solar cells Besidesthe performance of solar cells will degrade in lifetime Thefailure solar cell cannot be repaired in outer space timely sothe workload of solar panel is shared by the remaining solarcells and surviving solar cells should supply the requiredpower of system

A kind of solar panel of geostationary orbit satellite has 20solar cellsThe satellite needs 10 kWoutput power (the systemperformance requirement) from solar panel to ensure normaloperation of other subsystems The initial output power ofeach solar cell is 1 kW (initial performance of component)The solar panel is subject to 10A steady current (the totalworkload) and it is shared by 20 solar cells evenly Thebaseline failure time distribution is Weibull one with shapeparameter 120573 = 2 and location parameter 120578 = 200000 h Thetampered factor is 120575(119911) = 11991115 under the operating current 119911The degradation law of solar cell output power is 119863(119905 119911) =exp(1198861199051199114) where 119886 = minus16000000 According to therequirement the solar panel should supply sufficient powerfor 15 years (consider one year as 365 days and then 15 yearsare equal to 131400 hours) The output power of a solar cellis degrading with service time The random failure of a solarcell will increase the current shared on remaining solar cellsresulting in higher failure rate of solar cells and the degra-dation rate of solar cells output power will increase corre-spondinglyThe value of119870 is determined by the output powerof surviving solar cells and the output power requirement ofsolar panel It is obvious that the solar panels are a typicalload-sharing 119870-out-of-119873 system with component perfor-mance degradationWith the proposedmethod in this paperthe reliability of solar panels could be estimated accurately

Firstly we should calculate the discrete probability of 119870by performance degradation model

According to the Monte-Carlo simulation method thetime interval 119883

119894between 119894th and (119894 minus 1)th solar cell failure

is subject to

119865119883119894(Δ119905)

= 1minus exp [minus (119899 minus 119894 + 1) intΔ119905

0120582119894minus1 (119905 + 119879119894minus1) 119889119905]

(25)

Using the given value 119899 = 20 119871 = 10 1205820(119905) and 120575(119911119894minus1)the distribution of119883

119894is expressed as

Table 1 The count and possibility of119870(131400)

119870(131400) 12 13 14 15Count 29 1710 6749 1512Possibility 029 171 6749 1512

119865119883119894(Δ119905) = 1minus expminus1015 sdot (21minus 119894)minus05

sdot [(119879119894minus1 + Δ119905

200000)

2minus(

119879119894minus1

200000)

2]

(26)

Hence the sampling formula of119883119894is derived

Δ119905 = 200000

sdot [(119879119894minus1

200000)

2minus(21 minus 119894)05

1015ln (1minus 120578)]

12

minus119879119894minus1

(27)

where 120578 is subject to uniform distribution in [0 1]The degradation rate of a solar cell output power is

119889 (119905 119911) =120597119863 (119905 119911)

120597119905=119886

1199114exp(119886119905

1199114) (28)

where 119886 = minus16000000 The required output power of thesolar panel is 119862 = 10 and the initial output power of a solarcell is 1198880 = 1

The output power degradation in time interval [119879119894 119879119894+1]

is

Δ119888119894= 1198880 int

119879119894

119879119894minus1

119889119894minus1 (119905 119911119894minus1) 119889119905

= 1198880 int119879119894

119879119894minus1

119889 (119905119871

119899 + 1 minus 119894) 119889119905

(29)

With the given value

Δ119888119894

= minusint

119879119894

119879119894minus1

16000000 (1021 minus 119894)4

exp[minus 119905

6000000 (1021 minus 119894)4]119889119905

(30)

After a cycle of the simulation the output power of a solarcell 119888(131400) at 119905 = 131400 h can be calculated Hence thevalue of119870 is expressed as

(131400) = lceil 10119888 (131400)

rceil (31)

Set the number of cycle as 119878 = 10000 the count andpossibility of (131400) are shown in Table 1

If the output power of a solar cell does not degrade thevalue of119870 is

119870 = lceil101198880rceil = lceil

101rceil = 10 (32)

8 Mathematical Problems in Engineering

Table 2 The coefficients of 120572119894

1205721 1205722 1205723 1205724 1205725 1205726 1205727 1205728 1205729 12057210

70711 72548 74536 76696 79057 81650 84515 87706 91287 9534612057211 12057212 12057213 12057214 12057215 12057216 12057217 12057218 12057219 12057220

10 105409 111803 119523 129099 141421 158114 182574 223607 316228

Table 3 The coefficients of 119860119894when 119870(131400) = 12

1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609

977231198907 minus578581198908 147681198909 minus211931198909 186671198909 minus103111198909 347871198908 minus653321198907 520861198906

From the above results because the output power of asolar cell is degrading the value of119870 at 119905 = 131400 h is greaterthan the situation ignoring solar cell degradation

Secondly after getting the discrete probability distri-bution of (131400) the reliability when (131400) is aspecified value should be calculated by the TFR model

TFR model of solar panel is shown as follows

Model 119899 = 20 119871 = 10 119905 = 131400 h 120573 = 2 and120578 = 200000Baseline failure rate

1205820 (119905) =120573

120578sdot (119905

120578)

120573minus1=

1100000

sdot119905

200000 (33)

The cumulative failure rate

Λ 0 (119905) = (119905

120578)

120573

= (119905

200000)

2 (34)

The TFR model of failure rate caused by operatingcurrent

120582 (119905) = 120575 (119911) sdot 1205820 (119905) (35)

where the tampered factor is 120575(119911) = 11991115

Solution is as follows

The tampered factor

120575119894= 120575 (119911

119894) = 119911

15119894= (

119871

119899 minus 119894)

15= (

1020 minus 119894

)

15 (36)

The corresponding TFR model with exponentialbaseline failure distribution is as followsTransformed scale

119897 = Λ 0 (119905) = (131400200000

)

2 (37)

Because 120582119894= 120575119894= (10(20 minus 119894))15 120572

119894= (119899 minus 119894 + 1)120582

119894minus1 =

1015(21 minus 119894)minus05 the coefficients of 120572119894are shown in Table 2

When (131400) = 12

119860119894equiv

20minus12+1prod

119895=1119895 =119894

120572119895

120572119895minus 120572119894

=

9prod

119895=1119895 =119894

120572119895

120572119895minus 120572119894

119894 = 1 2 9 (38)

The coefficients of119860119894when (131400) = 12 are shown in

Table 3

=12 (131400) =

20minus12+1sum

119894=1119860119894sdot exp [minus120572

119894Λ 0 (131400)]

=

9sum

119894=1119860119894sdot exp [minus(1314

2000)

2120572119894]

= 09912

(39)

In the same way the reliability when the system has other119870 is calculated as follows

=13 (131400) = 09778

=14 (131400) = 09477

=15 (131400) = 08866

(40)

Therefore the reliability of solar panel that it could supplysufficient power for 15 years is computed as

(131400) = sum119901 (119895) 119877119895 (131400) = 09437 (41)

If the output power of solar cells does not degrade mean-ing that119870 = 10 the coefficients of 119860

119894are shown in Table 4

Hence the reliability is

119877 (131400) =20minus10+1sum

119894=1119860119894sdot exp [minus120572

119894Λ 0 (131400)]

=

11sum

119894=1119860119894sdot exp [minus(1314

2000)

2120572119894] = 09988

(42)

Comparing 119877(131400) and (131400) we can see thatrandom failures of some solar cells make the operatingcurrent of the remaining solar cells increase also leading toan increase of failure rate and degradation rate Due to thedegradation of solar cells output power the reliability of solarpanel (131400) is less than 119877(131400) which ignores thedegradation of solar cell output power

6 Conclusion

In a load-sharing119870-out-of-119873 systemwith degrading compo-nents a component random failure raises the load shared on

Mathematical Problems in Engineering 9

Table 4 The coefficients of 119860119894when 119870(131400) = 10

1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 11986011

129131198909 minus881411198909 2657111989010 minus4649511989010 5217111989010 minus3911511989010 1977711989010 minus663151198909 140421198909 minus168481198908 862831198906

each remaining component resulting in a higher failure rateand increased degradation rate of surviving componentsThispaper proposes amethod combining a TFRmodel with a per-formance degradationmodel to analyze the reliability of load-sharing119870-out-of-119873 systemwith degrading componentsTheTFR model deals with load-sharing effect on failure rateand the reliability when the system has a specified 119870 iscalculated by the TFR model The performance degradationmodel is derived to evaluate degradation effect coupledwith load-sharing effect and then the degraded componentperformance is estimated considering the load-sharing effecton degradation rateThe case of a solar panel is a typical load-sharing119870-out-of-119873 systemwith degrading componentsTheresults calculated by the proposed method show that thereliability considering component degradation is less thanthat ignoring component degradation With utilization ofthe proposed method the degradation effect is quantitativelyevaluated and then the reliability of load-sharing 119870-out-of-119873 system can be calculated more accurately

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was funded by National Natural Science Foun-dation of China (NSFC 61304218) In addition the authorswould like to thank the anonymous reviewers who havehelped to improve the paper

References

[1] S V Amari H Pham and R B Misra ldquoReliability characteris-tics of k-out-of-n warm standby systemsrdquo IEEE Transactions onReliability vol 61 no 4 pp 1007ndash1018 2012

[2] R Moghaddass M J Zuo and W Wang ldquoAvailability of ageneral k-out-of-nG system with non-identical componentsconsidering shut-off rules using quasi-birthdeath processrdquo Reli-ability Engineering and SystemSafety vol 96 no 4 pp 489ndash4962011

[3] P Boddu and L Xing ldquoReliability evaluation and optimizationof series-parallel systems with k-out-of-n G subsystems andmixed redundancy typesrdquo Proceedings of the Institution ofMechanical Engineers Part O Journal of Risk and Reliability vol227 no 2 pp 187ndash198 2013

[4] RMoghaddassM J Zuo and J Qu ldquoReliability and availabilityanalysis of a repairable k-out-of-n G system with R repairmensubject to shut-off rulesrdquo IEEE Transactions on Reliability vol60 no 3 pp 658ndash666 2011

[5] S V Amari K B Misra and H Pham ldquoTampered failure rateload-sharing systems status and perspectivesrdquo in Handbook on

Performability Engineering chapter 20 Springer New York NYUSA 2007

[6] K C Kapur and L R Lambberson Reliability in EngineeringDesign John Wiley amp Sons 1977

[7] R K Iyer and D J Rossetti ldquoA measurement-based model forworkload dependency of CPU errorsrdquo IEEE Transactions onComputers vol C-35 no 6 pp 511ndash519 1986

[8] E M Scheuer ldquoReliability of an m-out-of-n system whencomponent failure induces higher failure rates in survivorsrdquoIEEE Transactions on Reliability vol 37 no 1 pp 73ndash74 1988

[9] H Liu ldquoReliability of a load-sharing fc-out-of-nG system non-iid components with arbitrary distributionsrdquo IEEE Transactionson Reliability vol 47 no 3 pp 279ndash284 1998

[10] S V Amari K B Misra and H Pham ldquoReliability analysisof tampered failure rate load-sharing k-out-of-n G systemsrdquoin Proceedings of the 12th ISSAT International Conference onReliability and Quality in Design pp 30ndash35 August 2006

[11] S V Amari and R Bergman ldquoReliability analysis of k-out-of-n load-sharing systemsrdquo in Proceedings of the 54th AnnualReliability and Maintainability Symposium (RAMS rsquo08) pp440ndash445 IEEE Las Vegas Nev USA January 2008

[12] G Levitin and S V Amari ldquoMulti-state systemswithmulti-faultcoveragerdquo Reliability Engineering and System Safety vol 93 no11 pp 1730ndash1739 2008

[13] Y Ding M J Zuo A Lisnianski and W Li ldquoA frameworkfor reliability approximation of multi-state weighted k-out-of-n systemsrdquo IEEE Transactions on Reliability vol 59 no 2 pp297ndash308 2010

[14] S K Chaturvedi S H Bahsa S V Amari et al ldquoReliability anal-ysis of generalized multi-state k-out-of-n systemsrdquo Proceedingsof the Institution of Mechanical Engineers Part O Journal of Riskand Reliability vol 226 pp 327ndash336 2011

[15] S Eryilmaz ldquoOn reliability analysis of a k-out-of-n system withcomponents having random weightsrdquo Reliability Engineeringand System Safety vol 109 pp 41ndash44 2013

[16] S V Amari M J Zuo and G Dill ldquoA fast and robust reliabilityevaluation algorithm for generalized multi-state k-out-of-nsystemsrdquo IEEE Transactions on Reliability vol 58 no 1 pp 88ndash97 2009

[17] G Levitin ldquoMulti-state vector-k-out-of-n systemsrdquo IEEE Trans-actions on Reliability vol 62 no 3 pp 648ndash657 2013

[18] J Fan K C Yung and M Pecht ldquoPhysics-of-failure-basedprognostics and health management for high-power whitelight-emitting diode lightingrdquo IEEE Transactions on Device andMaterials Reliability vol 11 no 3 pp 407ndash416 2011

[19] R Baillot Y Deshayes L Bechou et al ldquoEffects of siliconecoating degradation on GaN MQW LEDs performances usingphysical and chemical analysesrdquoMicroelectronics Reliability vol50 no 9ndash11 pp 1568ndash1573 2010

[20] Y-F Li E Zio and Y-H Lin ldquoA multistate physics modelof component degradation based on stochastic petri nets andsimulationrdquo IEEE Transactions on Reliability vol 61 no 4 pp921ndash931 2012

[21] L Xing and J B Dugan ldquoAnalysis of generalized phased-mission system reliability performance and sensitivityrdquo IEEETransactions on Reliability vol 51 no 2 pp 199ndash211 2002

10 Mathematical Problems in Engineering

[22] A Shrestha L Xing and Y Dai ldquoReliability analysis ofmultistate phased-mission systemswith unordered and orderedstatesrdquo IEEETransactions on SystemsMan andCybernetics PartASystems and Humans vol 41 no 4 pp 625ndash636 2011

[23] G Levitin L Xing S V Amari and Y Dai ldquoReliability ofnonrepairable phased-mission systems with common causefailuresrdquo IEEE Transactions on Systems Man and CyberneticsPart ASystems and Humans vol 43 no 4 pp 967ndash978 2013

[24] L Xing S V Amari and C Wang ldquoReliability of k-out-of-nsystems with phased-mission requirements and imperfect faultcoveragerdquo Reliability Engineering and System Safety vol 103 pp45ndash50 2012

[25] G K Bhattacharyya and Z Soejoeti ldquoA tampered failure ratemodel for step-stress accelerated life testrdquo Communications inStatisticsTheory andMethods vol 18 no 5 pp 1627ndash1643 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

21 Exponential Case Firstly consider a load-sharing 119870-out-of-119873 system with components following the exponentiallifetime distribution [8] in other words the baseline failurerate of the TFR model 1205820(119905) is constant

When the system is put into operation the failure rateof every component is denoted by 1205820 Because there are119899 working components in the system the first componentfailure occurs at failure rate 1205721 = 119899 sdot 1205820 After the first failurethe remaining (119899 minus 1) working components must carry thesame workload of system As a result the failure rate of eachsurviving component becomes1205821 which is commonly higherthan 1205820 The second component failure occurs at rate 1205722 =(119899 minus 1) sdot 1205821 When 119894 components have failed the failure rateof each (119899 minus 119894) remaining component is denoted by 120582

119894(0 le

119894 le 119899 minus 119896) The 119894th component failure occurs at failure rate120572119894= (119899 minus 119894 + 1) sdot 120582

119894minus1 The system is failed when more than(119899 minus 119896) components are failed

The time when the 119894th component failure occurs in thesystem is denoted by 119879

119894(1198790 equiv 0) and the time interval

between (119894 minus 1)th and 119894th component failure is representedby 119883119894= 119879119894minus 119879119894minus1 (1 le 119894 le 119899 minus 119896 + 1) Since all identical

components are following the exponential distributions the119883119894follows the exponential distribution with parameter 120572

119894

Hence the lifetime of system is the (119899 minus 119896 + 1)th failure time

119879 = 119879119899minus119896+1 =

119899minus119896+1sum

119894=1119883119894 (3)

Then the reliability of119870-out-of-119873 system at 1199050 is

119877 (1199050) = 119875 119879gt 1199050 (4)

In order to calculate the distribution of 119879 and thereliability function of load-sharing 119870-out-of-119873 system twotypical formulas which can be used are as follows

Case a All 120572119894are equal (say 120572) [8]

119877 (119905) =

119899minus119896

sum

119894=0

(120572119905)119894 exp (minus120572119905)119894

= 119892119886119898119891119888 (120572119905 119899 minus 119896 + 1) (5)

This case arises when the failure rate of each survivingcomponent is directly proportional to the load it carrieswhich means that 120575(119911) prop 119911 in the TFR model

Case b All 120572119894are distinct [8]

119877 (119905) =

119899minus119896+1sum

119894=1119860119894sdot exp (minus120572

119894119905)

119860119894equiv

119899minus119896+1prod

119895=1119895 =119894

120572119895

120572119895minus 120572119894

119894 = 1 2 119899 minus 119896 + 1(6)

22 General Case In the case of a load-sharing 119870-out-of-119873 system with components following arbitrary lifetimedistributions the baseline failure rate of the TFR model is nolonger a constant A closed-form analytical solution for TFR

model with an arbitrary baseline distribution is introducedThe basic idea is to use a time-transformation to convert TFRmodel with an arbitrary baseline distribution into an equiva-lent problem with an exponential baseline distribution [10]

Lemma 1 (1) For any failure distribution 119865(119905) the reliabilityfunction is 119877(119905) = 1 minus 119865(119905)

(2) The cumulative failure rate function is

Λ (119905) = int

119905

0120582 (119905) 119889119905 = int

119905

0

119889119865 (119905)

119877 (119905)= int

119905

0

minus119889119877 (119905)

119877 (119905)

= minus ln119877 (119905) (7)

(3) The random variable 119910 = 119877(119905) follows a uniformdistribution in interval [0 1] The random variable 119897 = Λ(119905)follows an exponential distribution with mean 1

(4) For a TFR model with a standard exponential (120582 = 1)baseline failure time distribution

120582 (119905) = 120575119894

Λ (119905) = Λ (120591119894minus1) + 120575119894 (119905 minus 120591119894minus1) 120591119894minus1 le 119905 lt 120591119894

(8)

(5) For a TFR model with a baseline failure rate of 1205820(119905)and a baseline cumulative failure rate of Λ 0(119905)

(a) under the regular scale 119905

120582 (119905) = 120575119894 sdot 1205820 (119905)

Λ (119905) = Λ (120591119894minus1) + 120575119894 sdot [Λ 0 (119905) minusΛ 0 (120591119894minus1)]

120591119894minus1 le 119905 lt 120591119894

(9)

(b) under the transformed scale 119897 = Λ 0(119905)Let

V119894= Λ 0 (120591119894)

120582119897 (119897) = 120575119894

Λ119897 (119897) = Λ 119897 (V119894minus1) + 120575119894 sdot [119897 minus V119894minus1] V

119894minus1 le 119897 lt V119894

(10)

where Λ119897(119897) is the cumulative failure rate in the transformed

scale

According to the above lemmas it becomes obvious thatif the load-sharing effect on the failure rate of an individualcomponent follows a TFR model with 120582(119905) = 120575

119894sdot 1205820(119905) the

reliability of a load-sharing system at 119905 is equivalent to thereliability of corresponding exponential load-sharing modelat time 119897 = Λ 0(119905) where the failure rate of a component is120582119894= 120575119894when 119894 components have failed for 119894 = 0 (119899 minus 119896)

3 Degradation Effect in Load-Sharing System

In a load-sharing 119870-out-of-119873 system with degrading com-ponents a random component failure raises the load sharedon remaining components leading to the rise of the failurerate and performance degradation rate Besides the durationwhen component has a specified degradation rate is variablebecause a component failure occurs at random time There-fore the component performance is a random variable at anygiven time thus the value of119870 is stochastic in service time

4 Mathematical Problems in Engineering

31 Independent Degradation Effect At the first step considerthe component degradation rate to be independent of load-sharing effect which means that the load-sharing effect justmakes the failure rate higher but the degradation rate hasnothing to do with shared load When the system is justput into operation the value of 119870 is determined by 119870 =

lceil1198621198880rceil where 119862 is the system performance requirement1198880 is the initial performance of each component and 119870 isthe minimum integer that is greater than or equal to thequotient When the system has operated for a period of time119905 the degraded performance of component is denoted by119888(119905) = 1198880119863(119905) where 119863(119905) is degradation law of componentperformance 119863(119905) is a monotone decreasing function and119863(0) = 1 119863(+infin) = 0 For example the commondegradation laws are exponential law denoted by 119863(119905) = 1 minus119886119905119898 and power law denoted by119863(119905) = exp(minus119886119905) At time 119905 the

value of 119870 is determined by 119870(119905) = lceil119862119888(119905)rceil Since 119863(119905) is amonotone decreasing function it is quite clear that 119888(119905) lt 1198880Thus119870(119905) ge 119870 while the condition for equality is lceil119862119888(119905)rceil =lceil1198621198880rceil Hence the reliability of 119870-out-of-119873 system withdegrading component is calculated in a general expression as

1198771015840(119905) =

119899minus119870(119905)

sum

119894=0119862119894

119899[1minus1198770 (119905)]

119894[1198770 (119905)]

119899minus119894 (11)

Correspondingly using the TFR model to analyze theload-sharing 119870-out-of-119873 system with degrading compo-nents the value of119870 should be replaced by 119870(119905)

Obviously 1198771015840(119905) le 119877(119905) because of 119870(119905) ge 119870 mean-ing that the reliability of 119870-out-of-119873 system consideringcomponent degradation is less than the reliability ignoringcomponent degradation

32 Degradation Coupled with Load-Sharing Effect Whenthe component performance degradation is related to load-sharing effect a random failure of component will raise thedegradation rate of surviving component Since a componentfailure occurs at random time the duration when compo-nents have a specified degradation rate is variable Thereforethe variable119870 denoted by (119905) is stochastic at any given time

In a load-sharing system the degradation law of compo-nent performance119863(119905) is not only a function of time but alsoa function of the shared load The generalization of degrada-tion law is expressed as a function of time and load denotedby 119863(119905 119911) The degradation rate of component performancewith load-sharing effect is 119889(119905 119911) = 120597119863(119905 119911)120597119905 After119894th component failure the performance degradation rate ofsurviving components is denoted by 119889

119894(119905 119911119894) = 119889[119905 119871(119899minus 119894)]

In order to analyze the reliability of load-sharing 119870-out-of-119873 system with degrading components at 1199050 (1199050) shouldbe evaluated previously which is determined by the degradedperformance of component at 1199050 With the understandingof degradation rate 119889

119894(119905 119911119894) after 119894th component failure the

degraded performance at 1199050 is estimated by using the dura-tion when the component has a specified degradation ratedenoted by119883

119894Theduration119883

119894is the interval between (119894minus1)th

component failure and 119894th component failure In other words119883119894is the minimum order statistic of the components failure

time of the surviving (119899 minus 119894 + 1) components under thecondition that the (119899 minus 119894 + 1) components do not fail at 119879

119894minus1

Lemma 2(1) Conditional Distribution Function

(a) The reliability of component at (119879119894+ Δ119905) is

119877 (119879119894+Δ119905) = exp [minusint

119879119894+Δ119905

0120582 (119905) 119889119905]

= exp [minusint1198791

01205820 (119905) 119889119905 minusint

1198792

1198791

1205821 (119905) 119889119905 minus sdot sdot sdot

minus int

119879119894

119879119894minus1

120582119894minus1 (119905) 119889119905 minusint

119879119894+Δ119905

119879119894

120582119894 (119905) 119889119905]

= exp[

[

minus

119894minus1sum

119895=0int

119879119895+1

119879119895

120582119895 (119905) 119889119905 minusint

119879119894+Δ119905

119879119894

120582119894 (119905) 119889119905

]

]

(12)

(b) The reliability at 119879119894is

119877 (119879119894) = exp[

[

minus

119894minus1sum

119895=0int

119879119895+1

119879119895

120582119895 (119905) 119889119905

]

]

(13)

(c) The cumulative distribution function of componentfailure time under the condition that the component didnot fail at 119879

119894is

119865 (119879119894+Δ119905 | 119879

119894) = 1minus119877 (119879

119894+Δ119905 | 119879

119894)

= 1minus119877 (119879119894+ Δ119905)

119877 (119879119894)

= 1minus exp [minusint119879119894+Δ119905

119879119894

120582119894 (119905) 119889119905]

= 1minus exp [minusintΔ119905

0120582119894(119905 + 119879119894) 119889119905]

(14)

(2) Minimum Order Statistic

(a) Let the cumulative distribution function of a popula-tion be 119865(119905) and probability distribution function is119891(119905) For samples with size 119899 probability distributionfunction of 119895th order statistic 119884

(119895)is

119891119884(119895) (119905) = 119899119862

119895minus1119899minus1 [119865 (119905)]

119895minus1[1minus119865 (119905)]119899minus119895 119891 (119905) (15)

(b) The probability distribution function of the minimumorder statistic is

119891119884(1) (119905) = 119899119891 (119905) [1minus119865 (119905)]

119899minus1 (16)

Mathematical Problems in Engineering 5

(c) The cumulative distribution function of the minimumorder statistic is

119865119884(1) (119905) = int

119905

0119891119884(1) (119905) 119889119905 = int

119905

0119899119891 (119905) [1minus119865 (119905)]119899minus1 119889119905

= 1minus [1minus119865 (119905)]119899 (17)

Hence the cumulative distribution function of 119883119894is

expressed as

119865119883119894(Δ119905)

= 1minus [1minus119865 (119879119894minus1 +Δ119905 | 119879119894minus1)]

119899minus119894+1

= 1minus [119877 (119879119894minus1 +Δ119905 | 119879119894minus1)]

119899minus119894+1

= 1minus exp [minus (119899 minus 119894 + 1) intΔ119905

0120582119894minus1 (119905 + 119879119894minus1) 119889119905]

(18)

In a special situation that the baseline distribution ofTFR is exponential distribution the distribution of 119883

119894is an

exponential distribution with parameter (119899minus119894+1)120582119894minus1 which

agrees with the conclusion in previous paper Section 21As 119894 components have failed in the system the degrada-

tion rate of each surviving component is 119889119894(119905 119911119894) Moreover

the time interval between 119894th component failure and (119894 +1)th failure is 119883

119894+1 Therefore the component performancedegradation in time interval [119879

119894 119879119894+1] is

Δ119888119894= 1198880 int

119879119894+1

119879119894

119889119894(119905 119911119894) 119889119905 = 1198880 int

119883119894+1

0119889119894(119879119894+ 119905 119911119894) 119889119905 (19)

The degraded component performance at 1199050 is calculatedas

119888 (1199050) = 1198880 +119898minus1sum

119895=0Δ119888119895+ 1198880 int

1199050

119879119898

119889119898(119905 119911119898) 119889119905

= 1198880 + 1198880

119898minus1sum

119895=0int

119879119895+1

119879119895

119889119895(119905 119911119895) 119889119905

+ 1198880 int1199050

119879119898

119889119898(119905 119911119898) 119889119905

(20)

According to 119883119895+1 = 119879119895+1 minus 119879119895 119879119895 = sum

119895

119894=0119883119894+1 (1198790 equiv 0)and 119911119895= 119871(119899 minus 119895) for 119895 = 0 1 2 119898 119898 is determined by

119879119898le 1199050 lt 119879119898+1

int

119879119895+1

119879119895

119889119895(119905 119911119895) 119889119905 = int

119883119895+1

0119889119895(119905 +119879

119895119871

119899 minus 119895) 119889119905 (21)

Hence

119888 (1199050) = 1198880 + 1198880

119898minus1sum

119895=0int

119883119895+1

0119889119895(119905 +119879

119895119871

119899 minus 119895) 119889119905

+ 1198880 int1199050minus119879119898

0119889119898(119905 +119879

119898119871

119899 minus 119898)119889119905

(22)

where the distribution of 119883119895is 119865119883119895(Δ119905) and the degradation

rate at each phase is 119889119895(119905 119911119895)

According to (22) the distribution of the degradedcomponent performance 119888(1199050) is calculated on basis of thedistribution of 119883

119895denoted by 119865

119883119895(Δ119905) and degradation rate

119889119895(119905 119911119895) Then the discrete probability distribution of (1199050)

could be obtained through

(1199050) = lceil119862

119888 (1199050)rceil = [

[[[

119862

1198880 + 1198880sum119898minus1119895=0 int119883119895+1

0 119889119895(119905 + 119879

119895 119871119899 minus 119895) 119889119905 + 1198880 int

1199050minus119879119898

0 119889119898(119905 + 119879

119898 119871119899 minus 119898) 119889119905

]]]]

(23)

Using the probability distribution of (1199050) estimatedby the performance degradation model and the reliabilitycalculated by the TFRmodel when the system has a specified119870 the reliability of load-sharing 119870-out-of-119873 system withdegrading components is computed in formula as

(1199050) = sum119901(119895) 119877119895 (1199050) (24)

where 119901(119895) is the probability when the (1199050) is equal to

119895and 119877

119895(1199050) is the reliability of 119870-out-of-119873 system when

the (1199050) is equal to 119895 Using the proposed formula to

evaluate the reliability of load-sharing 119870-out-of-119873 systemwith degrading components the load-sharing effect on com-ponent failure rate and the degradation effect coupled withload-sharing effect are all included in the model Thereforethe reliability is calculated more accurately

4 Monte-Carlo Simulation

In the load-sharing system the failure rate and degradationrate are variable during the lifetime and the duration in eachphase is stochastic because a component failure occurs at ran-dom time The analytic expression of degraded performancecould be obtained only if the failure rate or degradation rateis subject to some specified formats such as componentsfollowing the exponential lifetime distribution or degrada-tion rate being constant This paper employs Monte-Carlosimulation to estimate the discrete probability distribution of119870 The simulation procedure is shown in Figure 1

In order to evaluate the discrete probability distributionof 119870 the system configuration must be set firstly as thetotal component number denoted by 119899 the total workloaddenoted by 119871 the system performance requirement denotedby 119862 the initial performance of each component denoted by

6 Mathematical Problems in Engineering

Start

Initialization

Develop FX119894(Δt)

Sample the failure time interval Xi

Calculate Ti

Calculate the probability of each K(t0)

Compute the component performance degradation in [Timinus1 t0]

Compute the component performance

Set i = i + 1

No

No

Yes

Calculate the K(t0)Count the number of different K(t0)

End

Yes

Set j = j + 1

Ti gt t0

j = S

degradation in [Timinus1 Ti]

Set initial value n L C c0 t0 S j

Formulate d(t z) 1205820(t) 120575(z)

Reset T0 = 0 i = 1

Calculate ziminus1 and refresh 120575(ziminus1) d(t ziminus1)

Figure 1 The Monte-Carlo simulation procedure

1198880 and the system required operation time denoted by 1199050Furthermore the baseline failure rate function denoted by1205820(119905) the degradation rate function denoted by 119889(119905 119911) andthe tampered factor function denoted by 120575(119911) all need to beformulated in advance Set the required number of simulationcycle 119878 and initialize cyclic variable 119895 = 1 and then carry outthe Monte-Carlo simulation

At the beginning of a simulation cycle reset the startmoment 1198790 = 0 and sampling variable 119894 = 1 At the 119894th sam-pling in a cycle of the simulation there are (119899minus119894+1) remainingcomponents The workload shared on each component isdenoted by 119911

119894minus1 Then the tampered factor can be calculatedby 120575(119911) and the failure rate under current load is obtainedthrough the baseline failure rate function multiplied by thetampered factor Besides the degradation rate under currentload is computed by 119889(119905 119911) According to the current failurerate and last component failure time 119879

119894minus1 the distributionof 119883119894between (119894 minus 1)th and 119894th component failure denoted

by 119865119883119894(Δ119905) is obtained Using the distribution 119865

119883119894(Δ119905) the

sampling formula of119883119894denoted by 119865minus1

119883119894(120578) is derived where 120578

follows uniform distribution in the interval [0 1] Generatingthe 119883

119894from 119865minus1

119883119894(120578) the 119894th component failure time is 119879

119894=

119879119894minus1 + 119883119894 If 119879119894 is less than 1199050 compute the component

performance degradation in time interval [119879119894minus1 119879119894] and then

start the next sampling When 119879119894reaches system required

operation time 1199050 compute the component performancedegradation in time interval [119879

119894minus1 1199050] and then the degradedcomponent performance at 1199050 is obtained The (1199050) isdetermined by the system performance requirement 119862 andthe degraded component performance 119888(1199050) and then carryout the next cycle of the simulation When the number ofsimulation cycles meets the requirement denoted by 119878 countthe number when (1199050) is equal to different values Then theprobability of each (1199050) is estimated by dividing the numberof different (1199050) by the number of simulation cycles 119878

Mathematical Problems in Engineering 7

5 Case Study

Satellites or space station which needs to work for long peri-ods in aerospace must be equipped with suitable and reliablesolar panels to provide power for the normal operation ofequipment Once a solar panel cannot supply required powerthe spacecraft will lose its function and it may become ldquospacejunkrdquo As the critical module there are strict requirementsabout the performance and reliability of solar panel A solarpanelmodule consists ofmany solar cells Factors of the spaceenvironment like space radiation temperature cycling andcharge-discharge cycles will destroy some solar cells Besidesthe performance of solar cells will degrade in lifetime Thefailure solar cell cannot be repaired in outer space timely sothe workload of solar panel is shared by the remaining solarcells and surviving solar cells should supply the requiredpower of system

A kind of solar panel of geostationary orbit satellite has 20solar cellsThe satellite needs 10 kWoutput power (the systemperformance requirement) from solar panel to ensure normaloperation of other subsystems The initial output power ofeach solar cell is 1 kW (initial performance of component)The solar panel is subject to 10A steady current (the totalworkload) and it is shared by 20 solar cells evenly Thebaseline failure time distribution is Weibull one with shapeparameter 120573 = 2 and location parameter 120578 = 200000 h Thetampered factor is 120575(119911) = 11991115 under the operating current 119911The degradation law of solar cell output power is 119863(119905 119911) =exp(1198861199051199114) where 119886 = minus16000000 According to therequirement the solar panel should supply sufficient powerfor 15 years (consider one year as 365 days and then 15 yearsare equal to 131400 hours) The output power of a solar cellis degrading with service time The random failure of a solarcell will increase the current shared on remaining solar cellsresulting in higher failure rate of solar cells and the degra-dation rate of solar cells output power will increase corre-spondinglyThe value of119870 is determined by the output powerof surviving solar cells and the output power requirement ofsolar panel It is obvious that the solar panels are a typicalload-sharing 119870-out-of-119873 system with component perfor-mance degradationWith the proposedmethod in this paperthe reliability of solar panels could be estimated accurately

Firstly we should calculate the discrete probability of 119870by performance degradation model

According to the Monte-Carlo simulation method thetime interval 119883

119894between 119894th and (119894 minus 1)th solar cell failure

is subject to

119865119883119894(Δ119905)

= 1minus exp [minus (119899 minus 119894 + 1) intΔ119905

0120582119894minus1 (119905 + 119879119894minus1) 119889119905]

(25)

Using the given value 119899 = 20 119871 = 10 1205820(119905) and 120575(119911119894minus1)the distribution of119883

119894is expressed as

Table 1 The count and possibility of119870(131400)

119870(131400) 12 13 14 15Count 29 1710 6749 1512Possibility 029 171 6749 1512

119865119883119894(Δ119905) = 1minus expminus1015 sdot (21minus 119894)minus05

sdot [(119879119894minus1 + Δ119905

200000)

2minus(

119879119894minus1

200000)

2]

(26)

Hence the sampling formula of119883119894is derived

Δ119905 = 200000

sdot [(119879119894minus1

200000)

2minus(21 minus 119894)05

1015ln (1minus 120578)]

12

minus119879119894minus1

(27)

where 120578 is subject to uniform distribution in [0 1]The degradation rate of a solar cell output power is

119889 (119905 119911) =120597119863 (119905 119911)

120597119905=119886

1199114exp(119886119905

1199114) (28)

where 119886 = minus16000000 The required output power of thesolar panel is 119862 = 10 and the initial output power of a solarcell is 1198880 = 1

The output power degradation in time interval [119879119894 119879119894+1]

is

Δ119888119894= 1198880 int

119879119894

119879119894minus1

119889119894minus1 (119905 119911119894minus1) 119889119905

= 1198880 int119879119894

119879119894minus1

119889 (119905119871

119899 + 1 minus 119894) 119889119905

(29)

With the given value

Δ119888119894

= minusint

119879119894

119879119894minus1

16000000 (1021 minus 119894)4

exp[minus 119905

6000000 (1021 minus 119894)4]119889119905

(30)

After a cycle of the simulation the output power of a solarcell 119888(131400) at 119905 = 131400 h can be calculated Hence thevalue of119870 is expressed as

(131400) = lceil 10119888 (131400)

rceil (31)

Set the number of cycle as 119878 = 10000 the count andpossibility of (131400) are shown in Table 1

If the output power of a solar cell does not degrade thevalue of119870 is

119870 = lceil101198880rceil = lceil

101rceil = 10 (32)

8 Mathematical Problems in Engineering

Table 2 The coefficients of 120572119894

1205721 1205722 1205723 1205724 1205725 1205726 1205727 1205728 1205729 12057210

70711 72548 74536 76696 79057 81650 84515 87706 91287 9534612057211 12057212 12057213 12057214 12057215 12057216 12057217 12057218 12057219 12057220

10 105409 111803 119523 129099 141421 158114 182574 223607 316228

Table 3 The coefficients of 119860119894when 119870(131400) = 12

1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609

977231198907 minus578581198908 147681198909 minus211931198909 186671198909 minus103111198909 347871198908 minus653321198907 520861198906

From the above results because the output power of asolar cell is degrading the value of119870 at 119905 = 131400 h is greaterthan the situation ignoring solar cell degradation

Secondly after getting the discrete probability distri-bution of (131400) the reliability when (131400) is aspecified value should be calculated by the TFR model

TFR model of solar panel is shown as follows

Model 119899 = 20 119871 = 10 119905 = 131400 h 120573 = 2 and120578 = 200000Baseline failure rate

1205820 (119905) =120573

120578sdot (119905

120578)

120573minus1=

1100000

sdot119905

200000 (33)

The cumulative failure rate

Λ 0 (119905) = (119905

120578)

120573

= (119905

200000)

2 (34)

The TFR model of failure rate caused by operatingcurrent

120582 (119905) = 120575 (119911) sdot 1205820 (119905) (35)

where the tampered factor is 120575(119911) = 11991115

Solution is as follows

The tampered factor

120575119894= 120575 (119911

119894) = 119911

15119894= (

119871

119899 minus 119894)

15= (

1020 minus 119894

)

15 (36)

The corresponding TFR model with exponentialbaseline failure distribution is as followsTransformed scale

119897 = Λ 0 (119905) = (131400200000

)

2 (37)

Because 120582119894= 120575119894= (10(20 minus 119894))15 120572

119894= (119899 minus 119894 + 1)120582

119894minus1 =

1015(21 minus 119894)minus05 the coefficients of 120572119894are shown in Table 2

When (131400) = 12

119860119894equiv

20minus12+1prod

119895=1119895 =119894

120572119895

120572119895minus 120572119894

=

9prod

119895=1119895 =119894

120572119895

120572119895minus 120572119894

119894 = 1 2 9 (38)

The coefficients of119860119894when (131400) = 12 are shown in

Table 3

=12 (131400) =

20minus12+1sum

119894=1119860119894sdot exp [minus120572

119894Λ 0 (131400)]

=

9sum

119894=1119860119894sdot exp [minus(1314

2000)

2120572119894]

= 09912

(39)

In the same way the reliability when the system has other119870 is calculated as follows

=13 (131400) = 09778

=14 (131400) = 09477

=15 (131400) = 08866

(40)

Therefore the reliability of solar panel that it could supplysufficient power for 15 years is computed as

(131400) = sum119901 (119895) 119877119895 (131400) = 09437 (41)

If the output power of solar cells does not degrade mean-ing that119870 = 10 the coefficients of 119860

119894are shown in Table 4

Hence the reliability is

119877 (131400) =20minus10+1sum

119894=1119860119894sdot exp [minus120572

119894Λ 0 (131400)]

=

11sum

119894=1119860119894sdot exp [minus(1314

2000)

2120572119894] = 09988

(42)

Comparing 119877(131400) and (131400) we can see thatrandom failures of some solar cells make the operatingcurrent of the remaining solar cells increase also leading toan increase of failure rate and degradation rate Due to thedegradation of solar cells output power the reliability of solarpanel (131400) is less than 119877(131400) which ignores thedegradation of solar cell output power

6 Conclusion

In a load-sharing119870-out-of-119873 systemwith degrading compo-nents a component random failure raises the load shared on

Mathematical Problems in Engineering 9

Table 4 The coefficients of 119860119894when 119870(131400) = 10

1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 11986011

129131198909 minus881411198909 2657111989010 minus4649511989010 5217111989010 minus3911511989010 1977711989010 minus663151198909 140421198909 minus168481198908 862831198906

each remaining component resulting in a higher failure rateand increased degradation rate of surviving componentsThispaper proposes amethod combining a TFRmodel with a per-formance degradationmodel to analyze the reliability of load-sharing119870-out-of-119873 systemwith degrading componentsTheTFR model deals with load-sharing effect on failure rateand the reliability when the system has a specified 119870 iscalculated by the TFR model The performance degradationmodel is derived to evaluate degradation effect coupledwith load-sharing effect and then the degraded componentperformance is estimated considering the load-sharing effecton degradation rateThe case of a solar panel is a typical load-sharing119870-out-of-119873 systemwith degrading componentsTheresults calculated by the proposed method show that thereliability considering component degradation is less thanthat ignoring component degradation With utilization ofthe proposed method the degradation effect is quantitativelyevaluated and then the reliability of load-sharing 119870-out-of-119873 system can be calculated more accurately

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was funded by National Natural Science Foun-dation of China (NSFC 61304218) In addition the authorswould like to thank the anonymous reviewers who havehelped to improve the paper

References

[1] S V Amari H Pham and R B Misra ldquoReliability characteris-tics of k-out-of-n warm standby systemsrdquo IEEE Transactions onReliability vol 61 no 4 pp 1007ndash1018 2012

[2] R Moghaddass M J Zuo and W Wang ldquoAvailability of ageneral k-out-of-nG system with non-identical componentsconsidering shut-off rules using quasi-birthdeath processrdquo Reli-ability Engineering and SystemSafety vol 96 no 4 pp 489ndash4962011

[3] P Boddu and L Xing ldquoReliability evaluation and optimizationof series-parallel systems with k-out-of-n G subsystems andmixed redundancy typesrdquo Proceedings of the Institution ofMechanical Engineers Part O Journal of Risk and Reliability vol227 no 2 pp 187ndash198 2013

[4] RMoghaddassM J Zuo and J Qu ldquoReliability and availabilityanalysis of a repairable k-out-of-n G system with R repairmensubject to shut-off rulesrdquo IEEE Transactions on Reliability vol60 no 3 pp 658ndash666 2011

[5] S V Amari K B Misra and H Pham ldquoTampered failure rateload-sharing systems status and perspectivesrdquo in Handbook on

Performability Engineering chapter 20 Springer New York NYUSA 2007

[6] K C Kapur and L R Lambberson Reliability in EngineeringDesign John Wiley amp Sons 1977

[7] R K Iyer and D J Rossetti ldquoA measurement-based model forworkload dependency of CPU errorsrdquo IEEE Transactions onComputers vol C-35 no 6 pp 511ndash519 1986

[8] E M Scheuer ldquoReliability of an m-out-of-n system whencomponent failure induces higher failure rates in survivorsrdquoIEEE Transactions on Reliability vol 37 no 1 pp 73ndash74 1988

[9] H Liu ldquoReliability of a load-sharing fc-out-of-nG system non-iid components with arbitrary distributionsrdquo IEEE Transactionson Reliability vol 47 no 3 pp 279ndash284 1998

[10] S V Amari K B Misra and H Pham ldquoReliability analysisof tampered failure rate load-sharing k-out-of-n G systemsrdquoin Proceedings of the 12th ISSAT International Conference onReliability and Quality in Design pp 30ndash35 August 2006

[11] S V Amari and R Bergman ldquoReliability analysis of k-out-of-n load-sharing systemsrdquo in Proceedings of the 54th AnnualReliability and Maintainability Symposium (RAMS rsquo08) pp440ndash445 IEEE Las Vegas Nev USA January 2008

[12] G Levitin and S V Amari ldquoMulti-state systemswithmulti-faultcoveragerdquo Reliability Engineering and System Safety vol 93 no11 pp 1730ndash1739 2008

[13] Y Ding M J Zuo A Lisnianski and W Li ldquoA frameworkfor reliability approximation of multi-state weighted k-out-of-n systemsrdquo IEEE Transactions on Reliability vol 59 no 2 pp297ndash308 2010

[14] S K Chaturvedi S H Bahsa S V Amari et al ldquoReliability anal-ysis of generalized multi-state k-out-of-n systemsrdquo Proceedingsof the Institution of Mechanical Engineers Part O Journal of Riskand Reliability vol 226 pp 327ndash336 2011

[15] S Eryilmaz ldquoOn reliability analysis of a k-out-of-n system withcomponents having random weightsrdquo Reliability Engineeringand System Safety vol 109 pp 41ndash44 2013

[16] S V Amari M J Zuo and G Dill ldquoA fast and robust reliabilityevaluation algorithm for generalized multi-state k-out-of-nsystemsrdquo IEEE Transactions on Reliability vol 58 no 1 pp 88ndash97 2009

[17] G Levitin ldquoMulti-state vector-k-out-of-n systemsrdquo IEEE Trans-actions on Reliability vol 62 no 3 pp 648ndash657 2013

[18] J Fan K C Yung and M Pecht ldquoPhysics-of-failure-basedprognostics and health management for high-power whitelight-emitting diode lightingrdquo IEEE Transactions on Device andMaterials Reliability vol 11 no 3 pp 407ndash416 2011

[19] R Baillot Y Deshayes L Bechou et al ldquoEffects of siliconecoating degradation on GaN MQW LEDs performances usingphysical and chemical analysesrdquoMicroelectronics Reliability vol50 no 9ndash11 pp 1568ndash1573 2010

[20] Y-F Li E Zio and Y-H Lin ldquoA multistate physics modelof component degradation based on stochastic petri nets andsimulationrdquo IEEE Transactions on Reliability vol 61 no 4 pp921ndash931 2012

[21] L Xing and J B Dugan ldquoAnalysis of generalized phased-mission system reliability performance and sensitivityrdquo IEEETransactions on Reliability vol 51 no 2 pp 199ndash211 2002

10 Mathematical Problems in Engineering

[22] A Shrestha L Xing and Y Dai ldquoReliability analysis ofmultistate phased-mission systemswith unordered and orderedstatesrdquo IEEETransactions on SystemsMan andCybernetics PartASystems and Humans vol 41 no 4 pp 625ndash636 2011

[23] G Levitin L Xing S V Amari and Y Dai ldquoReliability ofnonrepairable phased-mission systems with common causefailuresrdquo IEEE Transactions on Systems Man and CyberneticsPart ASystems and Humans vol 43 no 4 pp 967ndash978 2013

[24] L Xing S V Amari and C Wang ldquoReliability of k-out-of-nsystems with phased-mission requirements and imperfect faultcoveragerdquo Reliability Engineering and System Safety vol 103 pp45ndash50 2012

[25] G K Bhattacharyya and Z Soejoeti ldquoA tampered failure ratemodel for step-stress accelerated life testrdquo Communications inStatisticsTheory andMethods vol 18 no 5 pp 1627ndash1643 1989

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

31 Independent Degradation Effect At the first step considerthe component degradation rate to be independent of load-sharing effect which means that the load-sharing effect justmakes the failure rate higher but the degradation rate hasnothing to do with shared load When the system is justput into operation the value of 119870 is determined by 119870 =

lceil1198621198880rceil where 119862 is the system performance requirement1198880 is the initial performance of each component and 119870 isthe minimum integer that is greater than or equal to thequotient When the system has operated for a period of time119905 the degraded performance of component is denoted by119888(119905) = 1198880119863(119905) where 119863(119905) is degradation law of componentperformance 119863(119905) is a monotone decreasing function and119863(0) = 1 119863(+infin) = 0 For example the commondegradation laws are exponential law denoted by 119863(119905) = 1 minus119886119905119898 and power law denoted by119863(119905) = exp(minus119886119905) At time 119905 the

value of 119870 is determined by 119870(119905) = lceil119862119888(119905)rceil Since 119863(119905) is amonotone decreasing function it is quite clear that 119888(119905) lt 1198880Thus119870(119905) ge 119870 while the condition for equality is lceil119862119888(119905)rceil =lceil1198621198880rceil Hence the reliability of 119870-out-of-119873 system withdegrading component is calculated in a general expression as

1198771015840(119905) =

119899minus119870(119905)

sum

119894=0119862119894

119899[1minus1198770 (119905)]

119894[1198770 (119905)]

119899minus119894 (11)

Correspondingly using the TFR model to analyze theload-sharing 119870-out-of-119873 system with degrading compo-nents the value of119870 should be replaced by 119870(119905)

Obviously 1198771015840(119905) le 119877(119905) because of 119870(119905) ge 119870 mean-ing that the reliability of 119870-out-of-119873 system consideringcomponent degradation is less than the reliability ignoringcomponent degradation

32 Degradation Coupled with Load-Sharing Effect Whenthe component performance degradation is related to load-sharing effect a random failure of component will raise thedegradation rate of surviving component Since a componentfailure occurs at random time the duration when compo-nents have a specified degradation rate is variable Thereforethe variable119870 denoted by (119905) is stochastic at any given time

In a load-sharing system the degradation law of compo-nent performance119863(119905) is not only a function of time but alsoa function of the shared load The generalization of degrada-tion law is expressed as a function of time and load denotedby 119863(119905 119911) The degradation rate of component performancewith load-sharing effect is 119889(119905 119911) = 120597119863(119905 119911)120597119905 After119894th component failure the performance degradation rate ofsurviving components is denoted by 119889

119894(119905 119911119894) = 119889[119905 119871(119899minus 119894)]

In order to analyze the reliability of load-sharing 119870-out-of-119873 system with degrading components at 1199050 (1199050) shouldbe evaluated previously which is determined by the degradedperformance of component at 1199050 With the understandingof degradation rate 119889

119894(119905 119911119894) after 119894th component failure the

degraded performance at 1199050 is estimated by using the dura-tion when the component has a specified degradation ratedenoted by119883

119894Theduration119883

119894is the interval between (119894minus1)th

component failure and 119894th component failure In other words119883119894is the minimum order statistic of the components failure

time of the surviving (119899 minus 119894 + 1) components under thecondition that the (119899 minus 119894 + 1) components do not fail at 119879

119894minus1

Lemma 2(1) Conditional Distribution Function

(a) The reliability of component at (119879119894+ Δ119905) is

119877 (119879119894+Δ119905) = exp [minusint

119879119894+Δ119905

0120582 (119905) 119889119905]

= exp [minusint1198791

01205820 (119905) 119889119905 minusint

1198792

1198791

1205821 (119905) 119889119905 minus sdot sdot sdot

minus int

119879119894

119879119894minus1

120582119894minus1 (119905) 119889119905 minusint

119879119894+Δ119905

119879119894

120582119894 (119905) 119889119905]

= exp[

[

minus

119894minus1sum

119895=0int

119879119895+1

119879119895

120582119895 (119905) 119889119905 minusint

119879119894+Δ119905

119879119894

120582119894 (119905) 119889119905

]

]

(12)

(b) The reliability at 119879119894is

119877 (119879119894) = exp[

[

minus

119894minus1sum

119895=0int

119879119895+1

119879119895

120582119895 (119905) 119889119905

]

]

(13)

(c) The cumulative distribution function of componentfailure time under the condition that the component didnot fail at 119879

119894is

119865 (119879119894+Δ119905 | 119879

119894) = 1minus119877 (119879

119894+Δ119905 | 119879

119894)

= 1minus119877 (119879119894+ Δ119905)

119877 (119879119894)

= 1minus exp [minusint119879119894+Δ119905

119879119894

120582119894 (119905) 119889119905]

= 1minus exp [minusintΔ119905

0120582119894(119905 + 119879119894) 119889119905]

(14)

(2) Minimum Order Statistic

(a) Let the cumulative distribution function of a popula-tion be 119865(119905) and probability distribution function is119891(119905) For samples with size 119899 probability distributionfunction of 119895th order statistic 119884

(119895)is

119891119884(119895) (119905) = 119899119862

119895minus1119899minus1 [119865 (119905)]

119895minus1[1minus119865 (119905)]119899minus119895 119891 (119905) (15)

(b) The probability distribution function of the minimumorder statistic is

119891119884(1) (119905) = 119899119891 (119905) [1minus119865 (119905)]

119899minus1 (16)

Mathematical Problems in Engineering 5

(c) The cumulative distribution function of the minimumorder statistic is

119865119884(1) (119905) = int

119905

0119891119884(1) (119905) 119889119905 = int

119905

0119899119891 (119905) [1minus119865 (119905)]119899minus1 119889119905

= 1minus [1minus119865 (119905)]119899 (17)

Hence the cumulative distribution function of 119883119894is

expressed as

119865119883119894(Δ119905)

= 1minus [1minus119865 (119879119894minus1 +Δ119905 | 119879119894minus1)]

119899minus119894+1

= 1minus [119877 (119879119894minus1 +Δ119905 | 119879119894minus1)]

119899minus119894+1

= 1minus exp [minus (119899 minus 119894 + 1) intΔ119905

0120582119894minus1 (119905 + 119879119894minus1) 119889119905]

(18)

In a special situation that the baseline distribution ofTFR is exponential distribution the distribution of 119883

119894is an

exponential distribution with parameter (119899minus119894+1)120582119894minus1 which

agrees with the conclusion in previous paper Section 21As 119894 components have failed in the system the degrada-

tion rate of each surviving component is 119889119894(119905 119911119894) Moreover

the time interval between 119894th component failure and (119894 +1)th failure is 119883

119894+1 Therefore the component performancedegradation in time interval [119879

119894 119879119894+1] is

Δ119888119894= 1198880 int

119879119894+1

119879119894

119889119894(119905 119911119894) 119889119905 = 1198880 int

119883119894+1

0119889119894(119879119894+ 119905 119911119894) 119889119905 (19)

The degraded component performance at 1199050 is calculatedas

119888 (1199050) = 1198880 +119898minus1sum

119895=0Δ119888119895+ 1198880 int

1199050

119879119898

119889119898(119905 119911119898) 119889119905

= 1198880 + 1198880

119898minus1sum

119895=0int

119879119895+1

119879119895

119889119895(119905 119911119895) 119889119905

+ 1198880 int1199050

119879119898

119889119898(119905 119911119898) 119889119905

(20)

According to 119883119895+1 = 119879119895+1 minus 119879119895 119879119895 = sum

119895

119894=0119883119894+1 (1198790 equiv 0)and 119911119895= 119871(119899 minus 119895) for 119895 = 0 1 2 119898 119898 is determined by

119879119898le 1199050 lt 119879119898+1

int

119879119895+1

119879119895

119889119895(119905 119911119895) 119889119905 = int

119883119895+1

0119889119895(119905 +119879

119895119871

119899 minus 119895) 119889119905 (21)

Hence

119888 (1199050) = 1198880 + 1198880

119898minus1sum

119895=0int

119883119895+1

0119889119895(119905 +119879

119895119871

119899 minus 119895) 119889119905

+ 1198880 int1199050minus119879119898

0119889119898(119905 +119879

119898119871

119899 minus 119898)119889119905

(22)

where the distribution of 119883119895is 119865119883119895(Δ119905) and the degradation

rate at each phase is 119889119895(119905 119911119895)

According to (22) the distribution of the degradedcomponent performance 119888(1199050) is calculated on basis of thedistribution of 119883

119895denoted by 119865

119883119895(Δ119905) and degradation rate

119889119895(119905 119911119895) Then the discrete probability distribution of (1199050)

could be obtained through

(1199050) = lceil119862

119888 (1199050)rceil = [

[[[

119862

1198880 + 1198880sum119898minus1119895=0 int119883119895+1

0 119889119895(119905 + 119879

119895 119871119899 minus 119895) 119889119905 + 1198880 int

1199050minus119879119898

0 119889119898(119905 + 119879

119898 119871119899 minus 119898) 119889119905

]]]]

(23)

Using the probability distribution of (1199050) estimatedby the performance degradation model and the reliabilitycalculated by the TFRmodel when the system has a specified119870 the reliability of load-sharing 119870-out-of-119873 system withdegrading components is computed in formula as

(1199050) = sum119901(119895) 119877119895 (1199050) (24)

where 119901(119895) is the probability when the (1199050) is equal to

119895and 119877

119895(1199050) is the reliability of 119870-out-of-119873 system when

the (1199050) is equal to 119895 Using the proposed formula to

evaluate the reliability of load-sharing 119870-out-of-119873 systemwith degrading components the load-sharing effect on com-ponent failure rate and the degradation effect coupled withload-sharing effect are all included in the model Thereforethe reliability is calculated more accurately

4 Monte-Carlo Simulation

In the load-sharing system the failure rate and degradationrate are variable during the lifetime and the duration in eachphase is stochastic because a component failure occurs at ran-dom time The analytic expression of degraded performancecould be obtained only if the failure rate or degradation rateis subject to some specified formats such as componentsfollowing the exponential lifetime distribution or degrada-tion rate being constant This paper employs Monte-Carlosimulation to estimate the discrete probability distribution of119870 The simulation procedure is shown in Figure 1

In order to evaluate the discrete probability distributionof 119870 the system configuration must be set firstly as thetotal component number denoted by 119899 the total workloaddenoted by 119871 the system performance requirement denotedby 119862 the initial performance of each component denoted by

6 Mathematical Problems in Engineering

Start

Initialization

Develop FX119894(Δt)

Sample the failure time interval Xi

Calculate Ti

Calculate the probability of each K(t0)

Compute the component performance degradation in [Timinus1 t0]

Compute the component performance

Set i = i + 1

No

No

Yes

Calculate the K(t0)Count the number of different K(t0)

End

Yes

Set j = j + 1

Ti gt t0

j = S

degradation in [Timinus1 Ti]

Set initial value n L C c0 t0 S j

Formulate d(t z) 1205820(t) 120575(z)

Reset T0 = 0 i = 1

Calculate ziminus1 and refresh 120575(ziminus1) d(t ziminus1)

Figure 1 The Monte-Carlo simulation procedure

1198880 and the system required operation time denoted by 1199050Furthermore the baseline failure rate function denoted by1205820(119905) the degradation rate function denoted by 119889(119905 119911) andthe tampered factor function denoted by 120575(119911) all need to beformulated in advance Set the required number of simulationcycle 119878 and initialize cyclic variable 119895 = 1 and then carry outthe Monte-Carlo simulation

At the beginning of a simulation cycle reset the startmoment 1198790 = 0 and sampling variable 119894 = 1 At the 119894th sam-pling in a cycle of the simulation there are (119899minus119894+1) remainingcomponents The workload shared on each component isdenoted by 119911

119894minus1 Then the tampered factor can be calculatedby 120575(119911) and the failure rate under current load is obtainedthrough the baseline failure rate function multiplied by thetampered factor Besides the degradation rate under currentload is computed by 119889(119905 119911) According to the current failurerate and last component failure time 119879

119894minus1 the distributionof 119883119894between (119894 minus 1)th and 119894th component failure denoted

by 119865119883119894(Δ119905) is obtained Using the distribution 119865

119883119894(Δ119905) the

sampling formula of119883119894denoted by 119865minus1

119883119894(120578) is derived where 120578

follows uniform distribution in the interval [0 1] Generatingthe 119883

119894from 119865minus1

119883119894(120578) the 119894th component failure time is 119879

119894=

119879119894minus1 + 119883119894 If 119879119894 is less than 1199050 compute the component

performance degradation in time interval [119879119894minus1 119879119894] and then

start the next sampling When 119879119894reaches system required

operation time 1199050 compute the component performancedegradation in time interval [119879

119894minus1 1199050] and then the degradedcomponent performance at 1199050 is obtained The (1199050) isdetermined by the system performance requirement 119862 andthe degraded component performance 119888(1199050) and then carryout the next cycle of the simulation When the number ofsimulation cycles meets the requirement denoted by 119878 countthe number when (1199050) is equal to different values Then theprobability of each (1199050) is estimated by dividing the numberof different (1199050) by the number of simulation cycles 119878

Mathematical Problems in Engineering 7

5 Case Study

Satellites or space station which needs to work for long peri-ods in aerospace must be equipped with suitable and reliablesolar panels to provide power for the normal operation ofequipment Once a solar panel cannot supply required powerthe spacecraft will lose its function and it may become ldquospacejunkrdquo As the critical module there are strict requirementsabout the performance and reliability of solar panel A solarpanelmodule consists ofmany solar cells Factors of the spaceenvironment like space radiation temperature cycling andcharge-discharge cycles will destroy some solar cells Besidesthe performance of solar cells will degrade in lifetime Thefailure solar cell cannot be repaired in outer space timely sothe workload of solar panel is shared by the remaining solarcells and surviving solar cells should supply the requiredpower of system

A kind of solar panel of geostationary orbit satellite has 20solar cellsThe satellite needs 10 kWoutput power (the systemperformance requirement) from solar panel to ensure normaloperation of other subsystems The initial output power ofeach solar cell is 1 kW (initial performance of component)The solar panel is subject to 10A steady current (the totalworkload) and it is shared by 20 solar cells evenly Thebaseline failure time distribution is Weibull one with shapeparameter 120573 = 2 and location parameter 120578 = 200000 h Thetampered factor is 120575(119911) = 11991115 under the operating current 119911The degradation law of solar cell output power is 119863(119905 119911) =exp(1198861199051199114) where 119886 = minus16000000 According to therequirement the solar panel should supply sufficient powerfor 15 years (consider one year as 365 days and then 15 yearsare equal to 131400 hours) The output power of a solar cellis degrading with service time The random failure of a solarcell will increase the current shared on remaining solar cellsresulting in higher failure rate of solar cells and the degra-dation rate of solar cells output power will increase corre-spondinglyThe value of119870 is determined by the output powerof surviving solar cells and the output power requirement ofsolar panel It is obvious that the solar panels are a typicalload-sharing 119870-out-of-119873 system with component perfor-mance degradationWith the proposedmethod in this paperthe reliability of solar panels could be estimated accurately

Firstly we should calculate the discrete probability of 119870by performance degradation model

According to the Monte-Carlo simulation method thetime interval 119883

119894between 119894th and (119894 minus 1)th solar cell failure

is subject to

119865119883119894(Δ119905)

= 1minus exp [minus (119899 minus 119894 + 1) intΔ119905

0120582119894minus1 (119905 + 119879119894minus1) 119889119905]

(25)

Using the given value 119899 = 20 119871 = 10 1205820(119905) and 120575(119911119894minus1)the distribution of119883

119894is expressed as

Table 1 The count and possibility of119870(131400)

119870(131400) 12 13 14 15Count 29 1710 6749 1512Possibility 029 171 6749 1512

119865119883119894(Δ119905) = 1minus expminus1015 sdot (21minus 119894)minus05

sdot [(119879119894minus1 + Δ119905

200000)

2minus(

119879119894minus1

200000)

2]

(26)

Hence the sampling formula of119883119894is derived

Δ119905 = 200000

sdot [(119879119894minus1

200000)

2minus(21 minus 119894)05

1015ln (1minus 120578)]

12

minus119879119894minus1

(27)

where 120578 is subject to uniform distribution in [0 1]The degradation rate of a solar cell output power is

119889 (119905 119911) =120597119863 (119905 119911)

120597119905=119886

1199114exp(119886119905

1199114) (28)

where 119886 = minus16000000 The required output power of thesolar panel is 119862 = 10 and the initial output power of a solarcell is 1198880 = 1

The output power degradation in time interval [119879119894 119879119894+1]

is

Δ119888119894= 1198880 int

119879119894

119879119894minus1

119889119894minus1 (119905 119911119894minus1) 119889119905

= 1198880 int119879119894

119879119894minus1

119889 (119905119871

119899 + 1 minus 119894) 119889119905

(29)

With the given value

Δ119888119894

= minusint

119879119894

119879119894minus1

16000000 (1021 minus 119894)4

exp[minus 119905

6000000 (1021 minus 119894)4]119889119905

(30)

After a cycle of the simulation the output power of a solarcell 119888(131400) at 119905 = 131400 h can be calculated Hence thevalue of119870 is expressed as

(131400) = lceil 10119888 (131400)

rceil (31)

Set the number of cycle as 119878 = 10000 the count andpossibility of (131400) are shown in Table 1

If the output power of a solar cell does not degrade thevalue of119870 is

119870 = lceil101198880rceil = lceil

101rceil = 10 (32)

8 Mathematical Problems in Engineering

Table 2 The coefficients of 120572119894

1205721 1205722 1205723 1205724 1205725 1205726 1205727 1205728 1205729 12057210

70711 72548 74536 76696 79057 81650 84515 87706 91287 9534612057211 12057212 12057213 12057214 12057215 12057216 12057217 12057218 12057219 12057220

10 105409 111803 119523 129099 141421 158114 182574 223607 316228

Table 3 The coefficients of 119860119894when 119870(131400) = 12

1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609

977231198907 minus578581198908 147681198909 minus211931198909 186671198909 minus103111198909 347871198908 minus653321198907 520861198906

From the above results because the output power of asolar cell is degrading the value of119870 at 119905 = 131400 h is greaterthan the situation ignoring solar cell degradation

Secondly after getting the discrete probability distri-bution of (131400) the reliability when (131400) is aspecified value should be calculated by the TFR model

TFR model of solar panel is shown as follows

Model 119899 = 20 119871 = 10 119905 = 131400 h 120573 = 2 and120578 = 200000Baseline failure rate

1205820 (119905) =120573

120578sdot (119905

120578)

120573minus1=

1100000

sdot119905

200000 (33)

The cumulative failure rate

Λ 0 (119905) = (119905

120578)

120573

= (119905

200000)

2 (34)

The TFR model of failure rate caused by operatingcurrent

120582 (119905) = 120575 (119911) sdot 1205820 (119905) (35)

where the tampered factor is 120575(119911) = 11991115

Solution is as follows

The tampered factor

120575119894= 120575 (119911

119894) = 119911

15119894= (

119871

119899 minus 119894)

15= (

1020 minus 119894

)

15 (36)

The corresponding TFR model with exponentialbaseline failure distribution is as followsTransformed scale

119897 = Λ 0 (119905) = (131400200000

)

2 (37)

Because 120582119894= 120575119894= (10(20 minus 119894))15 120572

119894= (119899 minus 119894 + 1)120582

119894minus1 =

1015(21 minus 119894)minus05 the coefficients of 120572119894are shown in Table 2

When (131400) = 12

119860119894equiv

20minus12+1prod

119895=1119895 =119894

120572119895

120572119895minus 120572119894

=

9prod

119895=1119895 =119894

120572119895

120572119895minus 120572119894

119894 = 1 2 9 (38)

The coefficients of119860119894when (131400) = 12 are shown in

Table 3

=12 (131400) =

20minus12+1sum

119894=1119860119894sdot exp [minus120572

119894Λ 0 (131400)]

=

9sum

119894=1119860119894sdot exp [minus(1314

2000)

2120572119894]

= 09912

(39)

In the same way the reliability when the system has other119870 is calculated as follows

=13 (131400) = 09778

=14 (131400) = 09477

=15 (131400) = 08866

(40)

Therefore the reliability of solar panel that it could supplysufficient power for 15 years is computed as

(131400) = sum119901 (119895) 119877119895 (131400) = 09437 (41)

If the output power of solar cells does not degrade mean-ing that119870 = 10 the coefficients of 119860

119894are shown in Table 4

Hence the reliability is

119877 (131400) =20minus10+1sum

119894=1119860119894sdot exp [minus120572

119894Λ 0 (131400)]

=

11sum

119894=1119860119894sdot exp [minus(1314

2000)

2120572119894] = 09988

(42)

Comparing 119877(131400) and (131400) we can see thatrandom failures of some solar cells make the operatingcurrent of the remaining solar cells increase also leading toan increase of failure rate and degradation rate Due to thedegradation of solar cells output power the reliability of solarpanel (131400) is less than 119877(131400) which ignores thedegradation of solar cell output power

6 Conclusion

In a load-sharing119870-out-of-119873 systemwith degrading compo-nents a component random failure raises the load shared on

Mathematical Problems in Engineering 9

Table 4 The coefficients of 119860119894when 119870(131400) = 10

1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 11986011

129131198909 minus881411198909 2657111989010 minus4649511989010 5217111989010 minus3911511989010 1977711989010 minus663151198909 140421198909 minus168481198908 862831198906

each remaining component resulting in a higher failure rateand increased degradation rate of surviving componentsThispaper proposes amethod combining a TFRmodel with a per-formance degradationmodel to analyze the reliability of load-sharing119870-out-of-119873 systemwith degrading componentsTheTFR model deals with load-sharing effect on failure rateand the reliability when the system has a specified 119870 iscalculated by the TFR model The performance degradationmodel is derived to evaluate degradation effect coupledwith load-sharing effect and then the degraded componentperformance is estimated considering the load-sharing effecton degradation rateThe case of a solar panel is a typical load-sharing119870-out-of-119873 systemwith degrading componentsTheresults calculated by the proposed method show that thereliability considering component degradation is less thanthat ignoring component degradation With utilization ofthe proposed method the degradation effect is quantitativelyevaluated and then the reliability of load-sharing 119870-out-of-119873 system can be calculated more accurately

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was funded by National Natural Science Foun-dation of China (NSFC 61304218) In addition the authorswould like to thank the anonymous reviewers who havehelped to improve the paper

References

[1] S V Amari H Pham and R B Misra ldquoReliability characteris-tics of k-out-of-n warm standby systemsrdquo IEEE Transactions onReliability vol 61 no 4 pp 1007ndash1018 2012

[2] R Moghaddass M J Zuo and W Wang ldquoAvailability of ageneral k-out-of-nG system with non-identical componentsconsidering shut-off rules using quasi-birthdeath processrdquo Reli-ability Engineering and SystemSafety vol 96 no 4 pp 489ndash4962011

[3] P Boddu and L Xing ldquoReliability evaluation and optimizationof series-parallel systems with k-out-of-n G subsystems andmixed redundancy typesrdquo Proceedings of the Institution ofMechanical Engineers Part O Journal of Risk and Reliability vol227 no 2 pp 187ndash198 2013

[4] RMoghaddassM J Zuo and J Qu ldquoReliability and availabilityanalysis of a repairable k-out-of-n G system with R repairmensubject to shut-off rulesrdquo IEEE Transactions on Reliability vol60 no 3 pp 658ndash666 2011

[5] S V Amari K B Misra and H Pham ldquoTampered failure rateload-sharing systems status and perspectivesrdquo in Handbook on

Performability Engineering chapter 20 Springer New York NYUSA 2007

[6] K C Kapur and L R Lambberson Reliability in EngineeringDesign John Wiley amp Sons 1977

[7] R K Iyer and D J Rossetti ldquoA measurement-based model forworkload dependency of CPU errorsrdquo IEEE Transactions onComputers vol C-35 no 6 pp 511ndash519 1986

[8] E M Scheuer ldquoReliability of an m-out-of-n system whencomponent failure induces higher failure rates in survivorsrdquoIEEE Transactions on Reliability vol 37 no 1 pp 73ndash74 1988

[9] H Liu ldquoReliability of a load-sharing fc-out-of-nG system non-iid components with arbitrary distributionsrdquo IEEE Transactionson Reliability vol 47 no 3 pp 279ndash284 1998

[10] S V Amari K B Misra and H Pham ldquoReliability analysisof tampered failure rate load-sharing k-out-of-n G systemsrdquoin Proceedings of the 12th ISSAT International Conference onReliability and Quality in Design pp 30ndash35 August 2006

[11] S V Amari and R Bergman ldquoReliability analysis of k-out-of-n load-sharing systemsrdquo in Proceedings of the 54th AnnualReliability and Maintainability Symposium (RAMS rsquo08) pp440ndash445 IEEE Las Vegas Nev USA January 2008

[12] G Levitin and S V Amari ldquoMulti-state systemswithmulti-faultcoveragerdquo Reliability Engineering and System Safety vol 93 no11 pp 1730ndash1739 2008

[13] Y Ding M J Zuo A Lisnianski and W Li ldquoA frameworkfor reliability approximation of multi-state weighted k-out-of-n systemsrdquo IEEE Transactions on Reliability vol 59 no 2 pp297ndash308 2010

[14] S K Chaturvedi S H Bahsa S V Amari et al ldquoReliability anal-ysis of generalized multi-state k-out-of-n systemsrdquo Proceedingsof the Institution of Mechanical Engineers Part O Journal of Riskand Reliability vol 226 pp 327ndash336 2011

[15] S Eryilmaz ldquoOn reliability analysis of a k-out-of-n system withcomponents having random weightsrdquo Reliability Engineeringand System Safety vol 109 pp 41ndash44 2013

[16] S V Amari M J Zuo and G Dill ldquoA fast and robust reliabilityevaluation algorithm for generalized multi-state k-out-of-nsystemsrdquo IEEE Transactions on Reliability vol 58 no 1 pp 88ndash97 2009

[17] G Levitin ldquoMulti-state vector-k-out-of-n systemsrdquo IEEE Trans-actions on Reliability vol 62 no 3 pp 648ndash657 2013

[18] J Fan K C Yung and M Pecht ldquoPhysics-of-failure-basedprognostics and health management for high-power whitelight-emitting diode lightingrdquo IEEE Transactions on Device andMaterials Reliability vol 11 no 3 pp 407ndash416 2011

[19] R Baillot Y Deshayes L Bechou et al ldquoEffects of siliconecoating degradation on GaN MQW LEDs performances usingphysical and chemical analysesrdquoMicroelectronics Reliability vol50 no 9ndash11 pp 1568ndash1573 2010

[20] Y-F Li E Zio and Y-H Lin ldquoA multistate physics modelof component degradation based on stochastic petri nets andsimulationrdquo IEEE Transactions on Reliability vol 61 no 4 pp921ndash931 2012

[21] L Xing and J B Dugan ldquoAnalysis of generalized phased-mission system reliability performance and sensitivityrdquo IEEETransactions on Reliability vol 51 no 2 pp 199ndash211 2002

10 Mathematical Problems in Engineering

[22] A Shrestha L Xing and Y Dai ldquoReliability analysis ofmultistate phased-mission systemswith unordered and orderedstatesrdquo IEEETransactions on SystemsMan andCybernetics PartASystems and Humans vol 41 no 4 pp 625ndash636 2011

[23] G Levitin L Xing S V Amari and Y Dai ldquoReliability ofnonrepairable phased-mission systems with common causefailuresrdquo IEEE Transactions on Systems Man and CyberneticsPart ASystems and Humans vol 43 no 4 pp 967ndash978 2013

[24] L Xing S V Amari and C Wang ldquoReliability of k-out-of-nsystems with phased-mission requirements and imperfect faultcoveragerdquo Reliability Engineering and System Safety vol 103 pp45ndash50 2012

[25] G K Bhattacharyya and Z Soejoeti ldquoA tampered failure ratemodel for step-stress accelerated life testrdquo Communications inStatisticsTheory andMethods vol 18 no 5 pp 1627ndash1643 1989

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

(c) The cumulative distribution function of the minimumorder statistic is

119865119884(1) (119905) = int

119905

0119891119884(1) (119905) 119889119905 = int

119905

0119899119891 (119905) [1minus119865 (119905)]119899minus1 119889119905

= 1minus [1minus119865 (119905)]119899 (17)

Hence the cumulative distribution function of 119883119894is

expressed as

119865119883119894(Δ119905)

= 1minus [1minus119865 (119879119894minus1 +Δ119905 | 119879119894minus1)]

119899minus119894+1

= 1minus [119877 (119879119894minus1 +Δ119905 | 119879119894minus1)]

119899minus119894+1

= 1minus exp [minus (119899 minus 119894 + 1) intΔ119905

0120582119894minus1 (119905 + 119879119894minus1) 119889119905]

(18)

In a special situation that the baseline distribution ofTFR is exponential distribution the distribution of 119883

119894is an

exponential distribution with parameter (119899minus119894+1)120582119894minus1 which

agrees with the conclusion in previous paper Section 21As 119894 components have failed in the system the degrada-

tion rate of each surviving component is 119889119894(119905 119911119894) Moreover

the time interval between 119894th component failure and (119894 +1)th failure is 119883

119894+1 Therefore the component performancedegradation in time interval [119879

119894 119879119894+1] is

Δ119888119894= 1198880 int

119879119894+1

119879119894

119889119894(119905 119911119894) 119889119905 = 1198880 int

119883119894+1

0119889119894(119879119894+ 119905 119911119894) 119889119905 (19)

The degraded component performance at 1199050 is calculatedas

119888 (1199050) = 1198880 +119898minus1sum

119895=0Δ119888119895+ 1198880 int

1199050

119879119898

119889119898(119905 119911119898) 119889119905

= 1198880 + 1198880

119898minus1sum

119895=0int

119879119895+1

119879119895

119889119895(119905 119911119895) 119889119905

+ 1198880 int1199050

119879119898

119889119898(119905 119911119898) 119889119905

(20)

According to 119883119895+1 = 119879119895+1 minus 119879119895 119879119895 = sum

119895

119894=0119883119894+1 (1198790 equiv 0)and 119911119895= 119871(119899 minus 119895) for 119895 = 0 1 2 119898 119898 is determined by

119879119898le 1199050 lt 119879119898+1

int

119879119895+1

119879119895

119889119895(119905 119911119895) 119889119905 = int

119883119895+1

0119889119895(119905 +119879

119895119871

119899 minus 119895) 119889119905 (21)

Hence

119888 (1199050) = 1198880 + 1198880

119898minus1sum

119895=0int

119883119895+1

0119889119895(119905 +119879

119895119871

119899 minus 119895) 119889119905

+ 1198880 int1199050minus119879119898

0119889119898(119905 +119879

119898119871

119899 minus 119898)119889119905

(22)

where the distribution of 119883119895is 119865119883119895(Δ119905) and the degradation

rate at each phase is 119889119895(119905 119911119895)

According to (22) the distribution of the degradedcomponent performance 119888(1199050) is calculated on basis of thedistribution of 119883

119895denoted by 119865

119883119895(Δ119905) and degradation rate

119889119895(119905 119911119895) Then the discrete probability distribution of (1199050)

could be obtained through

(1199050) = lceil119862

119888 (1199050)rceil = [

[[[

119862

1198880 + 1198880sum119898minus1119895=0 int119883119895+1

0 119889119895(119905 + 119879

119895 119871119899 minus 119895) 119889119905 + 1198880 int

1199050minus119879119898

0 119889119898(119905 + 119879

119898 119871119899 minus 119898) 119889119905

]]]]

(23)

Using the probability distribution of (1199050) estimatedby the performance degradation model and the reliabilitycalculated by the TFRmodel when the system has a specified119870 the reliability of load-sharing 119870-out-of-119873 system withdegrading components is computed in formula as

(1199050) = sum119901(119895) 119877119895 (1199050) (24)

where 119901(119895) is the probability when the (1199050) is equal to

119895and 119877

119895(1199050) is the reliability of 119870-out-of-119873 system when

the (1199050) is equal to 119895 Using the proposed formula to

evaluate the reliability of load-sharing 119870-out-of-119873 systemwith degrading components the load-sharing effect on com-ponent failure rate and the degradation effect coupled withload-sharing effect are all included in the model Thereforethe reliability is calculated more accurately

4 Monte-Carlo Simulation

In the load-sharing system the failure rate and degradationrate are variable during the lifetime and the duration in eachphase is stochastic because a component failure occurs at ran-dom time The analytic expression of degraded performancecould be obtained only if the failure rate or degradation rateis subject to some specified formats such as componentsfollowing the exponential lifetime distribution or degrada-tion rate being constant This paper employs Monte-Carlosimulation to estimate the discrete probability distribution of119870 The simulation procedure is shown in Figure 1

In order to evaluate the discrete probability distributionof 119870 the system configuration must be set firstly as thetotal component number denoted by 119899 the total workloaddenoted by 119871 the system performance requirement denotedby 119862 the initial performance of each component denoted by

6 Mathematical Problems in Engineering

Start

Initialization

Develop FX119894(Δt)

Sample the failure time interval Xi

Calculate Ti

Calculate the probability of each K(t0)

Compute the component performance degradation in [Timinus1 t0]

Compute the component performance

Set i = i + 1

No

No

Yes

Calculate the K(t0)Count the number of different K(t0)

End

Yes

Set j = j + 1

Ti gt t0

j = S

degradation in [Timinus1 Ti]

Set initial value n L C c0 t0 S j

Formulate d(t z) 1205820(t) 120575(z)

Reset T0 = 0 i = 1

Calculate ziminus1 and refresh 120575(ziminus1) d(t ziminus1)

Figure 1 The Monte-Carlo simulation procedure

1198880 and the system required operation time denoted by 1199050Furthermore the baseline failure rate function denoted by1205820(119905) the degradation rate function denoted by 119889(119905 119911) andthe tampered factor function denoted by 120575(119911) all need to beformulated in advance Set the required number of simulationcycle 119878 and initialize cyclic variable 119895 = 1 and then carry outthe Monte-Carlo simulation

At the beginning of a simulation cycle reset the startmoment 1198790 = 0 and sampling variable 119894 = 1 At the 119894th sam-pling in a cycle of the simulation there are (119899minus119894+1) remainingcomponents The workload shared on each component isdenoted by 119911

119894minus1 Then the tampered factor can be calculatedby 120575(119911) and the failure rate under current load is obtainedthrough the baseline failure rate function multiplied by thetampered factor Besides the degradation rate under currentload is computed by 119889(119905 119911) According to the current failurerate and last component failure time 119879

119894minus1 the distributionof 119883119894between (119894 minus 1)th and 119894th component failure denoted

by 119865119883119894(Δ119905) is obtained Using the distribution 119865

119883119894(Δ119905) the

sampling formula of119883119894denoted by 119865minus1

119883119894(120578) is derived where 120578

follows uniform distribution in the interval [0 1] Generatingthe 119883

119894from 119865minus1

119883119894(120578) the 119894th component failure time is 119879

119894=

119879119894minus1 + 119883119894 If 119879119894 is less than 1199050 compute the component

performance degradation in time interval [119879119894minus1 119879119894] and then

start the next sampling When 119879119894reaches system required

operation time 1199050 compute the component performancedegradation in time interval [119879

119894minus1 1199050] and then the degradedcomponent performance at 1199050 is obtained The (1199050) isdetermined by the system performance requirement 119862 andthe degraded component performance 119888(1199050) and then carryout the next cycle of the simulation When the number ofsimulation cycles meets the requirement denoted by 119878 countthe number when (1199050) is equal to different values Then theprobability of each (1199050) is estimated by dividing the numberof different (1199050) by the number of simulation cycles 119878

Mathematical Problems in Engineering 7

5 Case Study

Satellites or space station which needs to work for long peri-ods in aerospace must be equipped with suitable and reliablesolar panels to provide power for the normal operation ofequipment Once a solar panel cannot supply required powerthe spacecraft will lose its function and it may become ldquospacejunkrdquo As the critical module there are strict requirementsabout the performance and reliability of solar panel A solarpanelmodule consists ofmany solar cells Factors of the spaceenvironment like space radiation temperature cycling andcharge-discharge cycles will destroy some solar cells Besidesthe performance of solar cells will degrade in lifetime Thefailure solar cell cannot be repaired in outer space timely sothe workload of solar panel is shared by the remaining solarcells and surviving solar cells should supply the requiredpower of system

A kind of solar panel of geostationary orbit satellite has 20solar cellsThe satellite needs 10 kWoutput power (the systemperformance requirement) from solar panel to ensure normaloperation of other subsystems The initial output power ofeach solar cell is 1 kW (initial performance of component)The solar panel is subject to 10A steady current (the totalworkload) and it is shared by 20 solar cells evenly Thebaseline failure time distribution is Weibull one with shapeparameter 120573 = 2 and location parameter 120578 = 200000 h Thetampered factor is 120575(119911) = 11991115 under the operating current 119911The degradation law of solar cell output power is 119863(119905 119911) =exp(1198861199051199114) where 119886 = minus16000000 According to therequirement the solar panel should supply sufficient powerfor 15 years (consider one year as 365 days and then 15 yearsare equal to 131400 hours) The output power of a solar cellis degrading with service time The random failure of a solarcell will increase the current shared on remaining solar cellsresulting in higher failure rate of solar cells and the degra-dation rate of solar cells output power will increase corre-spondinglyThe value of119870 is determined by the output powerof surviving solar cells and the output power requirement ofsolar panel It is obvious that the solar panels are a typicalload-sharing 119870-out-of-119873 system with component perfor-mance degradationWith the proposedmethod in this paperthe reliability of solar panels could be estimated accurately

Firstly we should calculate the discrete probability of 119870by performance degradation model

According to the Monte-Carlo simulation method thetime interval 119883

119894between 119894th and (119894 minus 1)th solar cell failure

is subject to

119865119883119894(Δ119905)

= 1minus exp [minus (119899 minus 119894 + 1) intΔ119905

0120582119894minus1 (119905 + 119879119894minus1) 119889119905]

(25)

Using the given value 119899 = 20 119871 = 10 1205820(119905) and 120575(119911119894minus1)the distribution of119883

119894is expressed as

Table 1 The count and possibility of119870(131400)

119870(131400) 12 13 14 15Count 29 1710 6749 1512Possibility 029 171 6749 1512

119865119883119894(Δ119905) = 1minus expminus1015 sdot (21minus 119894)minus05

sdot [(119879119894minus1 + Δ119905

200000)

2minus(

119879119894minus1

200000)

2]

(26)

Hence the sampling formula of119883119894is derived

Δ119905 = 200000

sdot [(119879119894minus1

200000)

2minus(21 minus 119894)05

1015ln (1minus 120578)]

12

minus119879119894minus1

(27)

where 120578 is subject to uniform distribution in [0 1]The degradation rate of a solar cell output power is

119889 (119905 119911) =120597119863 (119905 119911)

120597119905=119886

1199114exp(119886119905

1199114) (28)

where 119886 = minus16000000 The required output power of thesolar panel is 119862 = 10 and the initial output power of a solarcell is 1198880 = 1

The output power degradation in time interval [119879119894 119879119894+1]

is

Δ119888119894= 1198880 int

119879119894

119879119894minus1

119889119894minus1 (119905 119911119894minus1) 119889119905

= 1198880 int119879119894

119879119894minus1

119889 (119905119871

119899 + 1 minus 119894) 119889119905

(29)

With the given value

Δ119888119894

= minusint

119879119894

119879119894minus1

16000000 (1021 minus 119894)4

exp[minus 119905

6000000 (1021 minus 119894)4]119889119905

(30)

After a cycle of the simulation the output power of a solarcell 119888(131400) at 119905 = 131400 h can be calculated Hence thevalue of119870 is expressed as

(131400) = lceil 10119888 (131400)

rceil (31)

Set the number of cycle as 119878 = 10000 the count andpossibility of (131400) are shown in Table 1

If the output power of a solar cell does not degrade thevalue of119870 is

119870 = lceil101198880rceil = lceil

101rceil = 10 (32)

8 Mathematical Problems in Engineering

Table 2 The coefficients of 120572119894

1205721 1205722 1205723 1205724 1205725 1205726 1205727 1205728 1205729 12057210

70711 72548 74536 76696 79057 81650 84515 87706 91287 9534612057211 12057212 12057213 12057214 12057215 12057216 12057217 12057218 12057219 12057220

10 105409 111803 119523 129099 141421 158114 182574 223607 316228

Table 3 The coefficients of 119860119894when 119870(131400) = 12

1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609

977231198907 minus578581198908 147681198909 minus211931198909 186671198909 minus103111198909 347871198908 minus653321198907 520861198906

From the above results because the output power of asolar cell is degrading the value of119870 at 119905 = 131400 h is greaterthan the situation ignoring solar cell degradation

Secondly after getting the discrete probability distri-bution of (131400) the reliability when (131400) is aspecified value should be calculated by the TFR model

TFR model of solar panel is shown as follows

Model 119899 = 20 119871 = 10 119905 = 131400 h 120573 = 2 and120578 = 200000Baseline failure rate

1205820 (119905) =120573

120578sdot (119905

120578)

120573minus1=

1100000

sdot119905

200000 (33)

The cumulative failure rate

Λ 0 (119905) = (119905

120578)

120573

= (119905

200000)

2 (34)

The TFR model of failure rate caused by operatingcurrent

120582 (119905) = 120575 (119911) sdot 1205820 (119905) (35)

where the tampered factor is 120575(119911) = 11991115

Solution is as follows

The tampered factor

120575119894= 120575 (119911

119894) = 119911

15119894= (

119871

119899 minus 119894)

15= (

1020 minus 119894

)

15 (36)

The corresponding TFR model with exponentialbaseline failure distribution is as followsTransformed scale

119897 = Λ 0 (119905) = (131400200000

)

2 (37)

Because 120582119894= 120575119894= (10(20 minus 119894))15 120572

119894= (119899 minus 119894 + 1)120582

119894minus1 =

1015(21 minus 119894)minus05 the coefficients of 120572119894are shown in Table 2

When (131400) = 12

119860119894equiv

20minus12+1prod

119895=1119895 =119894

120572119895

120572119895minus 120572119894

=

9prod

119895=1119895 =119894

120572119895

120572119895minus 120572119894

119894 = 1 2 9 (38)

The coefficients of119860119894when (131400) = 12 are shown in

Table 3

=12 (131400) =

20minus12+1sum

119894=1119860119894sdot exp [minus120572

119894Λ 0 (131400)]

=

9sum

119894=1119860119894sdot exp [minus(1314

2000)

2120572119894]

= 09912

(39)

In the same way the reliability when the system has other119870 is calculated as follows

=13 (131400) = 09778

=14 (131400) = 09477

=15 (131400) = 08866

(40)

Therefore the reliability of solar panel that it could supplysufficient power for 15 years is computed as

(131400) = sum119901 (119895) 119877119895 (131400) = 09437 (41)

If the output power of solar cells does not degrade mean-ing that119870 = 10 the coefficients of 119860

119894are shown in Table 4

Hence the reliability is

119877 (131400) =20minus10+1sum

119894=1119860119894sdot exp [minus120572

119894Λ 0 (131400)]

=

11sum

119894=1119860119894sdot exp [minus(1314

2000)

2120572119894] = 09988

(42)

Comparing 119877(131400) and (131400) we can see thatrandom failures of some solar cells make the operatingcurrent of the remaining solar cells increase also leading toan increase of failure rate and degradation rate Due to thedegradation of solar cells output power the reliability of solarpanel (131400) is less than 119877(131400) which ignores thedegradation of solar cell output power

6 Conclusion

In a load-sharing119870-out-of-119873 systemwith degrading compo-nents a component random failure raises the load shared on

Mathematical Problems in Engineering 9

Table 4 The coefficients of 119860119894when 119870(131400) = 10

1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 11986011

129131198909 minus881411198909 2657111989010 minus4649511989010 5217111989010 minus3911511989010 1977711989010 minus663151198909 140421198909 minus168481198908 862831198906

each remaining component resulting in a higher failure rateand increased degradation rate of surviving componentsThispaper proposes amethod combining a TFRmodel with a per-formance degradationmodel to analyze the reliability of load-sharing119870-out-of-119873 systemwith degrading componentsTheTFR model deals with load-sharing effect on failure rateand the reliability when the system has a specified 119870 iscalculated by the TFR model The performance degradationmodel is derived to evaluate degradation effect coupledwith load-sharing effect and then the degraded componentperformance is estimated considering the load-sharing effecton degradation rateThe case of a solar panel is a typical load-sharing119870-out-of-119873 systemwith degrading componentsTheresults calculated by the proposed method show that thereliability considering component degradation is less thanthat ignoring component degradation With utilization ofthe proposed method the degradation effect is quantitativelyevaluated and then the reliability of load-sharing 119870-out-of-119873 system can be calculated more accurately

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was funded by National Natural Science Foun-dation of China (NSFC 61304218) In addition the authorswould like to thank the anonymous reviewers who havehelped to improve the paper

References

[1] S V Amari H Pham and R B Misra ldquoReliability characteris-tics of k-out-of-n warm standby systemsrdquo IEEE Transactions onReliability vol 61 no 4 pp 1007ndash1018 2012

[2] R Moghaddass M J Zuo and W Wang ldquoAvailability of ageneral k-out-of-nG system with non-identical componentsconsidering shut-off rules using quasi-birthdeath processrdquo Reli-ability Engineering and SystemSafety vol 96 no 4 pp 489ndash4962011

[3] P Boddu and L Xing ldquoReliability evaluation and optimizationof series-parallel systems with k-out-of-n G subsystems andmixed redundancy typesrdquo Proceedings of the Institution ofMechanical Engineers Part O Journal of Risk and Reliability vol227 no 2 pp 187ndash198 2013

[4] RMoghaddassM J Zuo and J Qu ldquoReliability and availabilityanalysis of a repairable k-out-of-n G system with R repairmensubject to shut-off rulesrdquo IEEE Transactions on Reliability vol60 no 3 pp 658ndash666 2011

[5] S V Amari K B Misra and H Pham ldquoTampered failure rateload-sharing systems status and perspectivesrdquo in Handbook on

Performability Engineering chapter 20 Springer New York NYUSA 2007

[6] K C Kapur and L R Lambberson Reliability in EngineeringDesign John Wiley amp Sons 1977

[7] R K Iyer and D J Rossetti ldquoA measurement-based model forworkload dependency of CPU errorsrdquo IEEE Transactions onComputers vol C-35 no 6 pp 511ndash519 1986

[8] E M Scheuer ldquoReliability of an m-out-of-n system whencomponent failure induces higher failure rates in survivorsrdquoIEEE Transactions on Reliability vol 37 no 1 pp 73ndash74 1988

[9] H Liu ldquoReliability of a load-sharing fc-out-of-nG system non-iid components with arbitrary distributionsrdquo IEEE Transactionson Reliability vol 47 no 3 pp 279ndash284 1998

[10] S V Amari K B Misra and H Pham ldquoReliability analysisof tampered failure rate load-sharing k-out-of-n G systemsrdquoin Proceedings of the 12th ISSAT International Conference onReliability and Quality in Design pp 30ndash35 August 2006

[11] S V Amari and R Bergman ldquoReliability analysis of k-out-of-n load-sharing systemsrdquo in Proceedings of the 54th AnnualReliability and Maintainability Symposium (RAMS rsquo08) pp440ndash445 IEEE Las Vegas Nev USA January 2008

[12] G Levitin and S V Amari ldquoMulti-state systemswithmulti-faultcoveragerdquo Reliability Engineering and System Safety vol 93 no11 pp 1730ndash1739 2008

[13] Y Ding M J Zuo A Lisnianski and W Li ldquoA frameworkfor reliability approximation of multi-state weighted k-out-of-n systemsrdquo IEEE Transactions on Reliability vol 59 no 2 pp297ndash308 2010

[14] S K Chaturvedi S H Bahsa S V Amari et al ldquoReliability anal-ysis of generalized multi-state k-out-of-n systemsrdquo Proceedingsof the Institution of Mechanical Engineers Part O Journal of Riskand Reliability vol 226 pp 327ndash336 2011

[15] S Eryilmaz ldquoOn reliability analysis of a k-out-of-n system withcomponents having random weightsrdquo Reliability Engineeringand System Safety vol 109 pp 41ndash44 2013

[16] S V Amari M J Zuo and G Dill ldquoA fast and robust reliabilityevaluation algorithm for generalized multi-state k-out-of-nsystemsrdquo IEEE Transactions on Reliability vol 58 no 1 pp 88ndash97 2009

[17] G Levitin ldquoMulti-state vector-k-out-of-n systemsrdquo IEEE Trans-actions on Reliability vol 62 no 3 pp 648ndash657 2013

[18] J Fan K C Yung and M Pecht ldquoPhysics-of-failure-basedprognostics and health management for high-power whitelight-emitting diode lightingrdquo IEEE Transactions on Device andMaterials Reliability vol 11 no 3 pp 407ndash416 2011

[19] R Baillot Y Deshayes L Bechou et al ldquoEffects of siliconecoating degradation on GaN MQW LEDs performances usingphysical and chemical analysesrdquoMicroelectronics Reliability vol50 no 9ndash11 pp 1568ndash1573 2010

[20] Y-F Li E Zio and Y-H Lin ldquoA multistate physics modelof component degradation based on stochastic petri nets andsimulationrdquo IEEE Transactions on Reliability vol 61 no 4 pp921ndash931 2012

[21] L Xing and J B Dugan ldquoAnalysis of generalized phased-mission system reliability performance and sensitivityrdquo IEEETransactions on Reliability vol 51 no 2 pp 199ndash211 2002

10 Mathematical Problems in Engineering

[22] A Shrestha L Xing and Y Dai ldquoReliability analysis ofmultistate phased-mission systemswith unordered and orderedstatesrdquo IEEETransactions on SystemsMan andCybernetics PartASystems and Humans vol 41 no 4 pp 625ndash636 2011

[23] G Levitin L Xing S V Amari and Y Dai ldquoReliability ofnonrepairable phased-mission systems with common causefailuresrdquo IEEE Transactions on Systems Man and CyberneticsPart ASystems and Humans vol 43 no 4 pp 967ndash978 2013

[24] L Xing S V Amari and C Wang ldquoReliability of k-out-of-nsystems with phased-mission requirements and imperfect faultcoveragerdquo Reliability Engineering and System Safety vol 103 pp45ndash50 2012

[25] G K Bhattacharyya and Z Soejoeti ldquoA tampered failure ratemodel for step-stress accelerated life testrdquo Communications inStatisticsTheory andMethods vol 18 no 5 pp 1627ndash1643 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Start

Initialization

Develop FX119894(Δt)

Sample the failure time interval Xi

Calculate Ti

Calculate the probability of each K(t0)

Compute the component performance degradation in [Timinus1 t0]

Compute the component performance

Set i = i + 1

No

No

Yes

Calculate the K(t0)Count the number of different K(t0)

End

Yes

Set j = j + 1

Ti gt t0

j = S

degradation in [Timinus1 Ti]

Set initial value n L C c0 t0 S j

Formulate d(t z) 1205820(t) 120575(z)

Reset T0 = 0 i = 1

Calculate ziminus1 and refresh 120575(ziminus1) d(t ziminus1)

Figure 1 The Monte-Carlo simulation procedure

1198880 and the system required operation time denoted by 1199050Furthermore the baseline failure rate function denoted by1205820(119905) the degradation rate function denoted by 119889(119905 119911) andthe tampered factor function denoted by 120575(119911) all need to beformulated in advance Set the required number of simulationcycle 119878 and initialize cyclic variable 119895 = 1 and then carry outthe Monte-Carlo simulation

At the beginning of a simulation cycle reset the startmoment 1198790 = 0 and sampling variable 119894 = 1 At the 119894th sam-pling in a cycle of the simulation there are (119899minus119894+1) remainingcomponents The workload shared on each component isdenoted by 119911

119894minus1 Then the tampered factor can be calculatedby 120575(119911) and the failure rate under current load is obtainedthrough the baseline failure rate function multiplied by thetampered factor Besides the degradation rate under currentload is computed by 119889(119905 119911) According to the current failurerate and last component failure time 119879

119894minus1 the distributionof 119883119894between (119894 minus 1)th and 119894th component failure denoted

by 119865119883119894(Δ119905) is obtained Using the distribution 119865

119883119894(Δ119905) the

sampling formula of119883119894denoted by 119865minus1

119883119894(120578) is derived where 120578

follows uniform distribution in the interval [0 1] Generatingthe 119883

119894from 119865minus1

119883119894(120578) the 119894th component failure time is 119879

119894=

119879119894minus1 + 119883119894 If 119879119894 is less than 1199050 compute the component

performance degradation in time interval [119879119894minus1 119879119894] and then

start the next sampling When 119879119894reaches system required

operation time 1199050 compute the component performancedegradation in time interval [119879

119894minus1 1199050] and then the degradedcomponent performance at 1199050 is obtained The (1199050) isdetermined by the system performance requirement 119862 andthe degraded component performance 119888(1199050) and then carryout the next cycle of the simulation When the number ofsimulation cycles meets the requirement denoted by 119878 countthe number when (1199050) is equal to different values Then theprobability of each (1199050) is estimated by dividing the numberof different (1199050) by the number of simulation cycles 119878

Mathematical Problems in Engineering 7

5 Case Study

Satellites or space station which needs to work for long peri-ods in aerospace must be equipped with suitable and reliablesolar panels to provide power for the normal operation ofequipment Once a solar panel cannot supply required powerthe spacecraft will lose its function and it may become ldquospacejunkrdquo As the critical module there are strict requirementsabout the performance and reliability of solar panel A solarpanelmodule consists ofmany solar cells Factors of the spaceenvironment like space radiation temperature cycling andcharge-discharge cycles will destroy some solar cells Besidesthe performance of solar cells will degrade in lifetime Thefailure solar cell cannot be repaired in outer space timely sothe workload of solar panel is shared by the remaining solarcells and surviving solar cells should supply the requiredpower of system

A kind of solar panel of geostationary orbit satellite has 20solar cellsThe satellite needs 10 kWoutput power (the systemperformance requirement) from solar panel to ensure normaloperation of other subsystems The initial output power ofeach solar cell is 1 kW (initial performance of component)The solar panel is subject to 10A steady current (the totalworkload) and it is shared by 20 solar cells evenly Thebaseline failure time distribution is Weibull one with shapeparameter 120573 = 2 and location parameter 120578 = 200000 h Thetampered factor is 120575(119911) = 11991115 under the operating current 119911The degradation law of solar cell output power is 119863(119905 119911) =exp(1198861199051199114) where 119886 = minus16000000 According to therequirement the solar panel should supply sufficient powerfor 15 years (consider one year as 365 days and then 15 yearsare equal to 131400 hours) The output power of a solar cellis degrading with service time The random failure of a solarcell will increase the current shared on remaining solar cellsresulting in higher failure rate of solar cells and the degra-dation rate of solar cells output power will increase corre-spondinglyThe value of119870 is determined by the output powerof surviving solar cells and the output power requirement ofsolar panel It is obvious that the solar panels are a typicalload-sharing 119870-out-of-119873 system with component perfor-mance degradationWith the proposedmethod in this paperthe reliability of solar panels could be estimated accurately

Firstly we should calculate the discrete probability of 119870by performance degradation model

According to the Monte-Carlo simulation method thetime interval 119883

119894between 119894th and (119894 minus 1)th solar cell failure

is subject to

119865119883119894(Δ119905)

= 1minus exp [minus (119899 minus 119894 + 1) intΔ119905

0120582119894minus1 (119905 + 119879119894minus1) 119889119905]

(25)

Using the given value 119899 = 20 119871 = 10 1205820(119905) and 120575(119911119894minus1)the distribution of119883

119894is expressed as

Table 1 The count and possibility of119870(131400)

119870(131400) 12 13 14 15Count 29 1710 6749 1512Possibility 029 171 6749 1512

119865119883119894(Δ119905) = 1minus expminus1015 sdot (21minus 119894)minus05

sdot [(119879119894minus1 + Δ119905

200000)

2minus(

119879119894minus1

200000)

2]

(26)

Hence the sampling formula of119883119894is derived

Δ119905 = 200000

sdot [(119879119894minus1

200000)

2minus(21 minus 119894)05

1015ln (1minus 120578)]

12

minus119879119894minus1

(27)

where 120578 is subject to uniform distribution in [0 1]The degradation rate of a solar cell output power is

119889 (119905 119911) =120597119863 (119905 119911)

120597119905=119886

1199114exp(119886119905

1199114) (28)

where 119886 = minus16000000 The required output power of thesolar panel is 119862 = 10 and the initial output power of a solarcell is 1198880 = 1

The output power degradation in time interval [119879119894 119879119894+1]

is

Δ119888119894= 1198880 int

119879119894

119879119894minus1

119889119894minus1 (119905 119911119894minus1) 119889119905

= 1198880 int119879119894

119879119894minus1

119889 (119905119871

119899 + 1 minus 119894) 119889119905

(29)

With the given value

Δ119888119894

= minusint

119879119894

119879119894minus1

16000000 (1021 minus 119894)4

exp[minus 119905

6000000 (1021 minus 119894)4]119889119905

(30)

After a cycle of the simulation the output power of a solarcell 119888(131400) at 119905 = 131400 h can be calculated Hence thevalue of119870 is expressed as

(131400) = lceil 10119888 (131400)

rceil (31)

Set the number of cycle as 119878 = 10000 the count andpossibility of (131400) are shown in Table 1

If the output power of a solar cell does not degrade thevalue of119870 is

119870 = lceil101198880rceil = lceil

101rceil = 10 (32)

8 Mathematical Problems in Engineering

Table 2 The coefficients of 120572119894

1205721 1205722 1205723 1205724 1205725 1205726 1205727 1205728 1205729 12057210

70711 72548 74536 76696 79057 81650 84515 87706 91287 9534612057211 12057212 12057213 12057214 12057215 12057216 12057217 12057218 12057219 12057220

10 105409 111803 119523 129099 141421 158114 182574 223607 316228

Table 3 The coefficients of 119860119894when 119870(131400) = 12

1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609

977231198907 minus578581198908 147681198909 minus211931198909 186671198909 minus103111198909 347871198908 minus653321198907 520861198906

From the above results because the output power of asolar cell is degrading the value of119870 at 119905 = 131400 h is greaterthan the situation ignoring solar cell degradation

Secondly after getting the discrete probability distri-bution of (131400) the reliability when (131400) is aspecified value should be calculated by the TFR model

TFR model of solar panel is shown as follows

Model 119899 = 20 119871 = 10 119905 = 131400 h 120573 = 2 and120578 = 200000Baseline failure rate

1205820 (119905) =120573

120578sdot (119905

120578)

120573minus1=

1100000

sdot119905

200000 (33)

The cumulative failure rate

Λ 0 (119905) = (119905

120578)

120573

= (119905

200000)

2 (34)

The TFR model of failure rate caused by operatingcurrent

120582 (119905) = 120575 (119911) sdot 1205820 (119905) (35)

where the tampered factor is 120575(119911) = 11991115

Solution is as follows

The tampered factor

120575119894= 120575 (119911

119894) = 119911

15119894= (

119871

119899 minus 119894)

15= (

1020 minus 119894

)

15 (36)

The corresponding TFR model with exponentialbaseline failure distribution is as followsTransformed scale

119897 = Λ 0 (119905) = (131400200000

)

2 (37)

Because 120582119894= 120575119894= (10(20 minus 119894))15 120572

119894= (119899 minus 119894 + 1)120582

119894minus1 =

1015(21 minus 119894)minus05 the coefficients of 120572119894are shown in Table 2

When (131400) = 12

119860119894equiv

20minus12+1prod

119895=1119895 =119894

120572119895

120572119895minus 120572119894

=

9prod

119895=1119895 =119894

120572119895

120572119895minus 120572119894

119894 = 1 2 9 (38)

The coefficients of119860119894when (131400) = 12 are shown in

Table 3

=12 (131400) =

20minus12+1sum

119894=1119860119894sdot exp [minus120572

119894Λ 0 (131400)]

=

9sum

119894=1119860119894sdot exp [minus(1314

2000)

2120572119894]

= 09912

(39)

In the same way the reliability when the system has other119870 is calculated as follows

=13 (131400) = 09778

=14 (131400) = 09477

=15 (131400) = 08866

(40)

Therefore the reliability of solar panel that it could supplysufficient power for 15 years is computed as

(131400) = sum119901 (119895) 119877119895 (131400) = 09437 (41)

If the output power of solar cells does not degrade mean-ing that119870 = 10 the coefficients of 119860

119894are shown in Table 4

Hence the reliability is

119877 (131400) =20minus10+1sum

119894=1119860119894sdot exp [minus120572

119894Λ 0 (131400)]

=

11sum

119894=1119860119894sdot exp [minus(1314

2000)

2120572119894] = 09988

(42)

Comparing 119877(131400) and (131400) we can see thatrandom failures of some solar cells make the operatingcurrent of the remaining solar cells increase also leading toan increase of failure rate and degradation rate Due to thedegradation of solar cells output power the reliability of solarpanel (131400) is less than 119877(131400) which ignores thedegradation of solar cell output power

6 Conclusion

In a load-sharing119870-out-of-119873 systemwith degrading compo-nents a component random failure raises the load shared on

Mathematical Problems in Engineering 9

Table 4 The coefficients of 119860119894when 119870(131400) = 10

1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 11986011

129131198909 minus881411198909 2657111989010 minus4649511989010 5217111989010 minus3911511989010 1977711989010 minus663151198909 140421198909 minus168481198908 862831198906

each remaining component resulting in a higher failure rateand increased degradation rate of surviving componentsThispaper proposes amethod combining a TFRmodel with a per-formance degradationmodel to analyze the reliability of load-sharing119870-out-of-119873 systemwith degrading componentsTheTFR model deals with load-sharing effect on failure rateand the reliability when the system has a specified 119870 iscalculated by the TFR model The performance degradationmodel is derived to evaluate degradation effect coupledwith load-sharing effect and then the degraded componentperformance is estimated considering the load-sharing effecton degradation rateThe case of a solar panel is a typical load-sharing119870-out-of-119873 systemwith degrading componentsTheresults calculated by the proposed method show that thereliability considering component degradation is less thanthat ignoring component degradation With utilization ofthe proposed method the degradation effect is quantitativelyevaluated and then the reliability of load-sharing 119870-out-of-119873 system can be calculated more accurately

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was funded by National Natural Science Foun-dation of China (NSFC 61304218) In addition the authorswould like to thank the anonymous reviewers who havehelped to improve the paper

References

[1] S V Amari H Pham and R B Misra ldquoReliability characteris-tics of k-out-of-n warm standby systemsrdquo IEEE Transactions onReliability vol 61 no 4 pp 1007ndash1018 2012

[2] R Moghaddass M J Zuo and W Wang ldquoAvailability of ageneral k-out-of-nG system with non-identical componentsconsidering shut-off rules using quasi-birthdeath processrdquo Reli-ability Engineering and SystemSafety vol 96 no 4 pp 489ndash4962011

[3] P Boddu and L Xing ldquoReliability evaluation and optimizationof series-parallel systems with k-out-of-n G subsystems andmixed redundancy typesrdquo Proceedings of the Institution ofMechanical Engineers Part O Journal of Risk and Reliability vol227 no 2 pp 187ndash198 2013

[4] RMoghaddassM J Zuo and J Qu ldquoReliability and availabilityanalysis of a repairable k-out-of-n G system with R repairmensubject to shut-off rulesrdquo IEEE Transactions on Reliability vol60 no 3 pp 658ndash666 2011

[5] S V Amari K B Misra and H Pham ldquoTampered failure rateload-sharing systems status and perspectivesrdquo in Handbook on

Performability Engineering chapter 20 Springer New York NYUSA 2007

[6] K C Kapur and L R Lambberson Reliability in EngineeringDesign John Wiley amp Sons 1977

[7] R K Iyer and D J Rossetti ldquoA measurement-based model forworkload dependency of CPU errorsrdquo IEEE Transactions onComputers vol C-35 no 6 pp 511ndash519 1986

[8] E M Scheuer ldquoReliability of an m-out-of-n system whencomponent failure induces higher failure rates in survivorsrdquoIEEE Transactions on Reliability vol 37 no 1 pp 73ndash74 1988

[9] H Liu ldquoReliability of a load-sharing fc-out-of-nG system non-iid components with arbitrary distributionsrdquo IEEE Transactionson Reliability vol 47 no 3 pp 279ndash284 1998

[10] S V Amari K B Misra and H Pham ldquoReliability analysisof tampered failure rate load-sharing k-out-of-n G systemsrdquoin Proceedings of the 12th ISSAT International Conference onReliability and Quality in Design pp 30ndash35 August 2006

[11] S V Amari and R Bergman ldquoReliability analysis of k-out-of-n load-sharing systemsrdquo in Proceedings of the 54th AnnualReliability and Maintainability Symposium (RAMS rsquo08) pp440ndash445 IEEE Las Vegas Nev USA January 2008

[12] G Levitin and S V Amari ldquoMulti-state systemswithmulti-faultcoveragerdquo Reliability Engineering and System Safety vol 93 no11 pp 1730ndash1739 2008

[13] Y Ding M J Zuo A Lisnianski and W Li ldquoA frameworkfor reliability approximation of multi-state weighted k-out-of-n systemsrdquo IEEE Transactions on Reliability vol 59 no 2 pp297ndash308 2010

[14] S K Chaturvedi S H Bahsa S V Amari et al ldquoReliability anal-ysis of generalized multi-state k-out-of-n systemsrdquo Proceedingsof the Institution of Mechanical Engineers Part O Journal of Riskand Reliability vol 226 pp 327ndash336 2011

[15] S Eryilmaz ldquoOn reliability analysis of a k-out-of-n system withcomponents having random weightsrdquo Reliability Engineeringand System Safety vol 109 pp 41ndash44 2013

[16] S V Amari M J Zuo and G Dill ldquoA fast and robust reliabilityevaluation algorithm for generalized multi-state k-out-of-nsystemsrdquo IEEE Transactions on Reliability vol 58 no 1 pp 88ndash97 2009

[17] G Levitin ldquoMulti-state vector-k-out-of-n systemsrdquo IEEE Trans-actions on Reliability vol 62 no 3 pp 648ndash657 2013

[18] J Fan K C Yung and M Pecht ldquoPhysics-of-failure-basedprognostics and health management for high-power whitelight-emitting diode lightingrdquo IEEE Transactions on Device andMaterials Reliability vol 11 no 3 pp 407ndash416 2011

[19] R Baillot Y Deshayes L Bechou et al ldquoEffects of siliconecoating degradation on GaN MQW LEDs performances usingphysical and chemical analysesrdquoMicroelectronics Reliability vol50 no 9ndash11 pp 1568ndash1573 2010

[20] Y-F Li E Zio and Y-H Lin ldquoA multistate physics modelof component degradation based on stochastic petri nets andsimulationrdquo IEEE Transactions on Reliability vol 61 no 4 pp921ndash931 2012

[21] L Xing and J B Dugan ldquoAnalysis of generalized phased-mission system reliability performance and sensitivityrdquo IEEETransactions on Reliability vol 51 no 2 pp 199ndash211 2002

10 Mathematical Problems in Engineering

[22] A Shrestha L Xing and Y Dai ldquoReliability analysis ofmultistate phased-mission systemswith unordered and orderedstatesrdquo IEEETransactions on SystemsMan andCybernetics PartASystems and Humans vol 41 no 4 pp 625ndash636 2011

[23] G Levitin L Xing S V Amari and Y Dai ldquoReliability ofnonrepairable phased-mission systems with common causefailuresrdquo IEEE Transactions on Systems Man and CyberneticsPart ASystems and Humans vol 43 no 4 pp 967ndash978 2013

[24] L Xing S V Amari and C Wang ldquoReliability of k-out-of-nsystems with phased-mission requirements and imperfect faultcoveragerdquo Reliability Engineering and System Safety vol 103 pp45ndash50 2012

[25] G K Bhattacharyya and Z Soejoeti ldquoA tampered failure ratemodel for step-stress accelerated life testrdquo Communications inStatisticsTheory andMethods vol 18 no 5 pp 1627ndash1643 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

5 Case Study

Satellites or space station which needs to work for long peri-ods in aerospace must be equipped with suitable and reliablesolar panels to provide power for the normal operation ofequipment Once a solar panel cannot supply required powerthe spacecraft will lose its function and it may become ldquospacejunkrdquo As the critical module there are strict requirementsabout the performance and reliability of solar panel A solarpanelmodule consists ofmany solar cells Factors of the spaceenvironment like space radiation temperature cycling andcharge-discharge cycles will destroy some solar cells Besidesthe performance of solar cells will degrade in lifetime Thefailure solar cell cannot be repaired in outer space timely sothe workload of solar panel is shared by the remaining solarcells and surviving solar cells should supply the requiredpower of system

A kind of solar panel of geostationary orbit satellite has 20solar cellsThe satellite needs 10 kWoutput power (the systemperformance requirement) from solar panel to ensure normaloperation of other subsystems The initial output power ofeach solar cell is 1 kW (initial performance of component)The solar panel is subject to 10A steady current (the totalworkload) and it is shared by 20 solar cells evenly Thebaseline failure time distribution is Weibull one with shapeparameter 120573 = 2 and location parameter 120578 = 200000 h Thetampered factor is 120575(119911) = 11991115 under the operating current 119911The degradation law of solar cell output power is 119863(119905 119911) =exp(1198861199051199114) where 119886 = minus16000000 According to therequirement the solar panel should supply sufficient powerfor 15 years (consider one year as 365 days and then 15 yearsare equal to 131400 hours) The output power of a solar cellis degrading with service time The random failure of a solarcell will increase the current shared on remaining solar cellsresulting in higher failure rate of solar cells and the degra-dation rate of solar cells output power will increase corre-spondinglyThe value of119870 is determined by the output powerof surviving solar cells and the output power requirement ofsolar panel It is obvious that the solar panels are a typicalload-sharing 119870-out-of-119873 system with component perfor-mance degradationWith the proposedmethod in this paperthe reliability of solar panels could be estimated accurately

Firstly we should calculate the discrete probability of 119870by performance degradation model

According to the Monte-Carlo simulation method thetime interval 119883

119894between 119894th and (119894 minus 1)th solar cell failure

is subject to

119865119883119894(Δ119905)

= 1minus exp [minus (119899 minus 119894 + 1) intΔ119905

0120582119894minus1 (119905 + 119879119894minus1) 119889119905]

(25)

Using the given value 119899 = 20 119871 = 10 1205820(119905) and 120575(119911119894minus1)the distribution of119883

119894is expressed as

Table 1 The count and possibility of119870(131400)

119870(131400) 12 13 14 15Count 29 1710 6749 1512Possibility 029 171 6749 1512

119865119883119894(Δ119905) = 1minus expminus1015 sdot (21minus 119894)minus05

sdot [(119879119894minus1 + Δ119905

200000)

2minus(

119879119894minus1

200000)

2]

(26)

Hence the sampling formula of119883119894is derived

Δ119905 = 200000

sdot [(119879119894minus1

200000)

2minus(21 minus 119894)05

1015ln (1minus 120578)]

12

minus119879119894minus1

(27)

where 120578 is subject to uniform distribution in [0 1]The degradation rate of a solar cell output power is

119889 (119905 119911) =120597119863 (119905 119911)

120597119905=119886

1199114exp(119886119905

1199114) (28)

where 119886 = minus16000000 The required output power of thesolar panel is 119862 = 10 and the initial output power of a solarcell is 1198880 = 1

The output power degradation in time interval [119879119894 119879119894+1]

is

Δ119888119894= 1198880 int

119879119894

119879119894minus1

119889119894minus1 (119905 119911119894minus1) 119889119905

= 1198880 int119879119894

119879119894minus1

119889 (119905119871

119899 + 1 minus 119894) 119889119905

(29)

With the given value

Δ119888119894

= minusint

119879119894

119879119894minus1

16000000 (1021 minus 119894)4

exp[minus 119905

6000000 (1021 minus 119894)4]119889119905

(30)

After a cycle of the simulation the output power of a solarcell 119888(131400) at 119905 = 131400 h can be calculated Hence thevalue of119870 is expressed as

(131400) = lceil 10119888 (131400)

rceil (31)

Set the number of cycle as 119878 = 10000 the count andpossibility of (131400) are shown in Table 1

If the output power of a solar cell does not degrade thevalue of119870 is

119870 = lceil101198880rceil = lceil

101rceil = 10 (32)

8 Mathematical Problems in Engineering

Table 2 The coefficients of 120572119894

1205721 1205722 1205723 1205724 1205725 1205726 1205727 1205728 1205729 12057210

70711 72548 74536 76696 79057 81650 84515 87706 91287 9534612057211 12057212 12057213 12057214 12057215 12057216 12057217 12057218 12057219 12057220

10 105409 111803 119523 129099 141421 158114 182574 223607 316228

Table 3 The coefficients of 119860119894when 119870(131400) = 12

1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609

977231198907 minus578581198908 147681198909 minus211931198909 186671198909 minus103111198909 347871198908 minus653321198907 520861198906

From the above results because the output power of asolar cell is degrading the value of119870 at 119905 = 131400 h is greaterthan the situation ignoring solar cell degradation

Secondly after getting the discrete probability distri-bution of (131400) the reliability when (131400) is aspecified value should be calculated by the TFR model

TFR model of solar panel is shown as follows

Model 119899 = 20 119871 = 10 119905 = 131400 h 120573 = 2 and120578 = 200000Baseline failure rate

1205820 (119905) =120573

120578sdot (119905

120578)

120573minus1=

1100000

sdot119905

200000 (33)

The cumulative failure rate

Λ 0 (119905) = (119905

120578)

120573

= (119905

200000)

2 (34)

The TFR model of failure rate caused by operatingcurrent

120582 (119905) = 120575 (119911) sdot 1205820 (119905) (35)

where the tampered factor is 120575(119911) = 11991115

Solution is as follows

The tampered factor

120575119894= 120575 (119911

119894) = 119911

15119894= (

119871

119899 minus 119894)

15= (

1020 minus 119894

)

15 (36)

The corresponding TFR model with exponentialbaseline failure distribution is as followsTransformed scale

119897 = Λ 0 (119905) = (131400200000

)

2 (37)

Because 120582119894= 120575119894= (10(20 minus 119894))15 120572

119894= (119899 minus 119894 + 1)120582

119894minus1 =

1015(21 minus 119894)minus05 the coefficients of 120572119894are shown in Table 2

When (131400) = 12

119860119894equiv

20minus12+1prod

119895=1119895 =119894

120572119895

120572119895minus 120572119894

=

9prod

119895=1119895 =119894

120572119895

120572119895minus 120572119894

119894 = 1 2 9 (38)

The coefficients of119860119894when (131400) = 12 are shown in

Table 3

=12 (131400) =

20minus12+1sum

119894=1119860119894sdot exp [minus120572

119894Λ 0 (131400)]

=

9sum

119894=1119860119894sdot exp [minus(1314

2000)

2120572119894]

= 09912

(39)

In the same way the reliability when the system has other119870 is calculated as follows

=13 (131400) = 09778

=14 (131400) = 09477

=15 (131400) = 08866

(40)

Therefore the reliability of solar panel that it could supplysufficient power for 15 years is computed as

(131400) = sum119901 (119895) 119877119895 (131400) = 09437 (41)

If the output power of solar cells does not degrade mean-ing that119870 = 10 the coefficients of 119860

119894are shown in Table 4

Hence the reliability is

119877 (131400) =20minus10+1sum

119894=1119860119894sdot exp [minus120572

119894Λ 0 (131400)]

=

11sum

119894=1119860119894sdot exp [minus(1314

2000)

2120572119894] = 09988

(42)

Comparing 119877(131400) and (131400) we can see thatrandom failures of some solar cells make the operatingcurrent of the remaining solar cells increase also leading toan increase of failure rate and degradation rate Due to thedegradation of solar cells output power the reliability of solarpanel (131400) is less than 119877(131400) which ignores thedegradation of solar cell output power

6 Conclusion

In a load-sharing119870-out-of-119873 systemwith degrading compo-nents a component random failure raises the load shared on

Mathematical Problems in Engineering 9

Table 4 The coefficients of 119860119894when 119870(131400) = 10

1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 11986011

129131198909 minus881411198909 2657111989010 minus4649511989010 5217111989010 minus3911511989010 1977711989010 minus663151198909 140421198909 minus168481198908 862831198906

each remaining component resulting in a higher failure rateand increased degradation rate of surviving componentsThispaper proposes amethod combining a TFRmodel with a per-formance degradationmodel to analyze the reliability of load-sharing119870-out-of-119873 systemwith degrading componentsTheTFR model deals with load-sharing effect on failure rateand the reliability when the system has a specified 119870 iscalculated by the TFR model The performance degradationmodel is derived to evaluate degradation effect coupledwith load-sharing effect and then the degraded componentperformance is estimated considering the load-sharing effecton degradation rateThe case of a solar panel is a typical load-sharing119870-out-of-119873 systemwith degrading componentsTheresults calculated by the proposed method show that thereliability considering component degradation is less thanthat ignoring component degradation With utilization ofthe proposed method the degradation effect is quantitativelyevaluated and then the reliability of load-sharing 119870-out-of-119873 system can be calculated more accurately

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was funded by National Natural Science Foun-dation of China (NSFC 61304218) In addition the authorswould like to thank the anonymous reviewers who havehelped to improve the paper

References

[1] S V Amari H Pham and R B Misra ldquoReliability characteris-tics of k-out-of-n warm standby systemsrdquo IEEE Transactions onReliability vol 61 no 4 pp 1007ndash1018 2012

[2] R Moghaddass M J Zuo and W Wang ldquoAvailability of ageneral k-out-of-nG system with non-identical componentsconsidering shut-off rules using quasi-birthdeath processrdquo Reli-ability Engineering and SystemSafety vol 96 no 4 pp 489ndash4962011

[3] P Boddu and L Xing ldquoReliability evaluation and optimizationof series-parallel systems with k-out-of-n G subsystems andmixed redundancy typesrdquo Proceedings of the Institution ofMechanical Engineers Part O Journal of Risk and Reliability vol227 no 2 pp 187ndash198 2013

[4] RMoghaddassM J Zuo and J Qu ldquoReliability and availabilityanalysis of a repairable k-out-of-n G system with R repairmensubject to shut-off rulesrdquo IEEE Transactions on Reliability vol60 no 3 pp 658ndash666 2011

[5] S V Amari K B Misra and H Pham ldquoTampered failure rateload-sharing systems status and perspectivesrdquo in Handbook on

Performability Engineering chapter 20 Springer New York NYUSA 2007

[6] K C Kapur and L R Lambberson Reliability in EngineeringDesign John Wiley amp Sons 1977

[7] R K Iyer and D J Rossetti ldquoA measurement-based model forworkload dependency of CPU errorsrdquo IEEE Transactions onComputers vol C-35 no 6 pp 511ndash519 1986

[8] E M Scheuer ldquoReliability of an m-out-of-n system whencomponent failure induces higher failure rates in survivorsrdquoIEEE Transactions on Reliability vol 37 no 1 pp 73ndash74 1988

[9] H Liu ldquoReliability of a load-sharing fc-out-of-nG system non-iid components with arbitrary distributionsrdquo IEEE Transactionson Reliability vol 47 no 3 pp 279ndash284 1998

[10] S V Amari K B Misra and H Pham ldquoReliability analysisof tampered failure rate load-sharing k-out-of-n G systemsrdquoin Proceedings of the 12th ISSAT International Conference onReliability and Quality in Design pp 30ndash35 August 2006

[11] S V Amari and R Bergman ldquoReliability analysis of k-out-of-n load-sharing systemsrdquo in Proceedings of the 54th AnnualReliability and Maintainability Symposium (RAMS rsquo08) pp440ndash445 IEEE Las Vegas Nev USA January 2008

[12] G Levitin and S V Amari ldquoMulti-state systemswithmulti-faultcoveragerdquo Reliability Engineering and System Safety vol 93 no11 pp 1730ndash1739 2008

[13] Y Ding M J Zuo A Lisnianski and W Li ldquoA frameworkfor reliability approximation of multi-state weighted k-out-of-n systemsrdquo IEEE Transactions on Reliability vol 59 no 2 pp297ndash308 2010

[14] S K Chaturvedi S H Bahsa S V Amari et al ldquoReliability anal-ysis of generalized multi-state k-out-of-n systemsrdquo Proceedingsof the Institution of Mechanical Engineers Part O Journal of Riskand Reliability vol 226 pp 327ndash336 2011

[15] S Eryilmaz ldquoOn reliability analysis of a k-out-of-n system withcomponents having random weightsrdquo Reliability Engineeringand System Safety vol 109 pp 41ndash44 2013

[16] S V Amari M J Zuo and G Dill ldquoA fast and robust reliabilityevaluation algorithm for generalized multi-state k-out-of-nsystemsrdquo IEEE Transactions on Reliability vol 58 no 1 pp 88ndash97 2009

[17] G Levitin ldquoMulti-state vector-k-out-of-n systemsrdquo IEEE Trans-actions on Reliability vol 62 no 3 pp 648ndash657 2013

[18] J Fan K C Yung and M Pecht ldquoPhysics-of-failure-basedprognostics and health management for high-power whitelight-emitting diode lightingrdquo IEEE Transactions on Device andMaterials Reliability vol 11 no 3 pp 407ndash416 2011

[19] R Baillot Y Deshayes L Bechou et al ldquoEffects of siliconecoating degradation on GaN MQW LEDs performances usingphysical and chemical analysesrdquoMicroelectronics Reliability vol50 no 9ndash11 pp 1568ndash1573 2010

[20] Y-F Li E Zio and Y-H Lin ldquoA multistate physics modelof component degradation based on stochastic petri nets andsimulationrdquo IEEE Transactions on Reliability vol 61 no 4 pp921ndash931 2012

[21] L Xing and J B Dugan ldquoAnalysis of generalized phased-mission system reliability performance and sensitivityrdquo IEEETransactions on Reliability vol 51 no 2 pp 199ndash211 2002

10 Mathematical Problems in Engineering

[22] A Shrestha L Xing and Y Dai ldquoReliability analysis ofmultistate phased-mission systemswith unordered and orderedstatesrdquo IEEETransactions on SystemsMan andCybernetics PartASystems and Humans vol 41 no 4 pp 625ndash636 2011

[23] G Levitin L Xing S V Amari and Y Dai ldquoReliability ofnonrepairable phased-mission systems with common causefailuresrdquo IEEE Transactions on Systems Man and CyberneticsPart ASystems and Humans vol 43 no 4 pp 967ndash978 2013

[24] L Xing S V Amari and C Wang ldquoReliability of k-out-of-nsystems with phased-mission requirements and imperfect faultcoveragerdquo Reliability Engineering and System Safety vol 103 pp45ndash50 2012

[25] G K Bhattacharyya and Z Soejoeti ldquoA tampered failure ratemodel for step-stress accelerated life testrdquo Communications inStatisticsTheory andMethods vol 18 no 5 pp 1627ndash1643 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 2 The coefficients of 120572119894

1205721 1205722 1205723 1205724 1205725 1205726 1205727 1205728 1205729 12057210

70711 72548 74536 76696 79057 81650 84515 87706 91287 9534612057211 12057212 12057213 12057214 12057215 12057216 12057217 12057218 12057219 12057220

10 105409 111803 119523 129099 141421 158114 182574 223607 316228

Table 3 The coefficients of 119860119894when 119870(131400) = 12

1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609

977231198907 minus578581198908 147681198909 minus211931198909 186671198909 minus103111198909 347871198908 minus653321198907 520861198906

From the above results because the output power of asolar cell is degrading the value of119870 at 119905 = 131400 h is greaterthan the situation ignoring solar cell degradation

Secondly after getting the discrete probability distri-bution of (131400) the reliability when (131400) is aspecified value should be calculated by the TFR model

TFR model of solar panel is shown as follows

Model 119899 = 20 119871 = 10 119905 = 131400 h 120573 = 2 and120578 = 200000Baseline failure rate

1205820 (119905) =120573

120578sdot (119905

120578)

120573minus1=

1100000

sdot119905

200000 (33)

The cumulative failure rate

Λ 0 (119905) = (119905

120578)

120573

= (119905

200000)

2 (34)

The TFR model of failure rate caused by operatingcurrent

120582 (119905) = 120575 (119911) sdot 1205820 (119905) (35)

where the tampered factor is 120575(119911) = 11991115

Solution is as follows

The tampered factor

120575119894= 120575 (119911

119894) = 119911

15119894= (

119871

119899 minus 119894)

15= (

1020 minus 119894

)

15 (36)

The corresponding TFR model with exponentialbaseline failure distribution is as followsTransformed scale

119897 = Λ 0 (119905) = (131400200000

)

2 (37)

Because 120582119894= 120575119894= (10(20 minus 119894))15 120572

119894= (119899 minus 119894 + 1)120582

119894minus1 =

1015(21 minus 119894)minus05 the coefficients of 120572119894are shown in Table 2

When (131400) = 12

119860119894equiv

20minus12+1prod

119895=1119895 =119894

120572119895

120572119895minus 120572119894

=

9prod

119895=1119895 =119894

120572119895

120572119895minus 120572119894

119894 = 1 2 9 (38)

The coefficients of119860119894when (131400) = 12 are shown in

Table 3

=12 (131400) =

20minus12+1sum

119894=1119860119894sdot exp [minus120572

119894Λ 0 (131400)]

=

9sum

119894=1119860119894sdot exp [minus(1314

2000)

2120572119894]

= 09912

(39)

In the same way the reliability when the system has other119870 is calculated as follows

=13 (131400) = 09778

=14 (131400) = 09477

=15 (131400) = 08866

(40)

Therefore the reliability of solar panel that it could supplysufficient power for 15 years is computed as

(131400) = sum119901 (119895) 119877119895 (131400) = 09437 (41)

If the output power of solar cells does not degrade mean-ing that119870 = 10 the coefficients of 119860

119894are shown in Table 4

Hence the reliability is

119877 (131400) =20minus10+1sum

119894=1119860119894sdot exp [minus120572

119894Λ 0 (131400)]

=

11sum

119894=1119860119894sdot exp [minus(1314

2000)

2120572119894] = 09988

(42)

Comparing 119877(131400) and (131400) we can see thatrandom failures of some solar cells make the operatingcurrent of the remaining solar cells increase also leading toan increase of failure rate and degradation rate Due to thedegradation of solar cells output power the reliability of solarpanel (131400) is less than 119877(131400) which ignores thedegradation of solar cell output power

6 Conclusion

In a load-sharing119870-out-of-119873 systemwith degrading compo-nents a component random failure raises the load shared on

Mathematical Problems in Engineering 9

Table 4 The coefficients of 119860119894when 119870(131400) = 10

1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 11986011

129131198909 minus881411198909 2657111989010 minus4649511989010 5217111989010 minus3911511989010 1977711989010 minus663151198909 140421198909 minus168481198908 862831198906

each remaining component resulting in a higher failure rateand increased degradation rate of surviving componentsThispaper proposes amethod combining a TFRmodel with a per-formance degradationmodel to analyze the reliability of load-sharing119870-out-of-119873 systemwith degrading componentsTheTFR model deals with load-sharing effect on failure rateand the reliability when the system has a specified 119870 iscalculated by the TFR model The performance degradationmodel is derived to evaluate degradation effect coupledwith load-sharing effect and then the degraded componentperformance is estimated considering the load-sharing effecton degradation rateThe case of a solar panel is a typical load-sharing119870-out-of-119873 systemwith degrading componentsTheresults calculated by the proposed method show that thereliability considering component degradation is less thanthat ignoring component degradation With utilization ofthe proposed method the degradation effect is quantitativelyevaluated and then the reliability of load-sharing 119870-out-of-119873 system can be calculated more accurately

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was funded by National Natural Science Foun-dation of China (NSFC 61304218) In addition the authorswould like to thank the anonymous reviewers who havehelped to improve the paper

References

[1] S V Amari H Pham and R B Misra ldquoReliability characteris-tics of k-out-of-n warm standby systemsrdquo IEEE Transactions onReliability vol 61 no 4 pp 1007ndash1018 2012

[2] R Moghaddass M J Zuo and W Wang ldquoAvailability of ageneral k-out-of-nG system with non-identical componentsconsidering shut-off rules using quasi-birthdeath processrdquo Reli-ability Engineering and SystemSafety vol 96 no 4 pp 489ndash4962011

[3] P Boddu and L Xing ldquoReliability evaluation and optimizationof series-parallel systems with k-out-of-n G subsystems andmixed redundancy typesrdquo Proceedings of the Institution ofMechanical Engineers Part O Journal of Risk and Reliability vol227 no 2 pp 187ndash198 2013

[4] RMoghaddassM J Zuo and J Qu ldquoReliability and availabilityanalysis of a repairable k-out-of-n G system with R repairmensubject to shut-off rulesrdquo IEEE Transactions on Reliability vol60 no 3 pp 658ndash666 2011

[5] S V Amari K B Misra and H Pham ldquoTampered failure rateload-sharing systems status and perspectivesrdquo in Handbook on

Performability Engineering chapter 20 Springer New York NYUSA 2007

[6] K C Kapur and L R Lambberson Reliability in EngineeringDesign John Wiley amp Sons 1977

[7] R K Iyer and D J Rossetti ldquoA measurement-based model forworkload dependency of CPU errorsrdquo IEEE Transactions onComputers vol C-35 no 6 pp 511ndash519 1986

[8] E M Scheuer ldquoReliability of an m-out-of-n system whencomponent failure induces higher failure rates in survivorsrdquoIEEE Transactions on Reliability vol 37 no 1 pp 73ndash74 1988

[9] H Liu ldquoReliability of a load-sharing fc-out-of-nG system non-iid components with arbitrary distributionsrdquo IEEE Transactionson Reliability vol 47 no 3 pp 279ndash284 1998

[10] S V Amari K B Misra and H Pham ldquoReliability analysisof tampered failure rate load-sharing k-out-of-n G systemsrdquoin Proceedings of the 12th ISSAT International Conference onReliability and Quality in Design pp 30ndash35 August 2006

[11] S V Amari and R Bergman ldquoReliability analysis of k-out-of-n load-sharing systemsrdquo in Proceedings of the 54th AnnualReliability and Maintainability Symposium (RAMS rsquo08) pp440ndash445 IEEE Las Vegas Nev USA January 2008

[12] G Levitin and S V Amari ldquoMulti-state systemswithmulti-faultcoveragerdquo Reliability Engineering and System Safety vol 93 no11 pp 1730ndash1739 2008

[13] Y Ding M J Zuo A Lisnianski and W Li ldquoA frameworkfor reliability approximation of multi-state weighted k-out-of-n systemsrdquo IEEE Transactions on Reliability vol 59 no 2 pp297ndash308 2010

[14] S K Chaturvedi S H Bahsa S V Amari et al ldquoReliability anal-ysis of generalized multi-state k-out-of-n systemsrdquo Proceedingsof the Institution of Mechanical Engineers Part O Journal of Riskand Reliability vol 226 pp 327ndash336 2011

[15] S Eryilmaz ldquoOn reliability analysis of a k-out-of-n system withcomponents having random weightsrdquo Reliability Engineeringand System Safety vol 109 pp 41ndash44 2013

[16] S V Amari M J Zuo and G Dill ldquoA fast and robust reliabilityevaluation algorithm for generalized multi-state k-out-of-nsystemsrdquo IEEE Transactions on Reliability vol 58 no 1 pp 88ndash97 2009

[17] G Levitin ldquoMulti-state vector-k-out-of-n systemsrdquo IEEE Trans-actions on Reliability vol 62 no 3 pp 648ndash657 2013

[18] J Fan K C Yung and M Pecht ldquoPhysics-of-failure-basedprognostics and health management for high-power whitelight-emitting diode lightingrdquo IEEE Transactions on Device andMaterials Reliability vol 11 no 3 pp 407ndash416 2011

[19] R Baillot Y Deshayes L Bechou et al ldquoEffects of siliconecoating degradation on GaN MQW LEDs performances usingphysical and chemical analysesrdquoMicroelectronics Reliability vol50 no 9ndash11 pp 1568ndash1573 2010

[20] Y-F Li E Zio and Y-H Lin ldquoA multistate physics modelof component degradation based on stochastic petri nets andsimulationrdquo IEEE Transactions on Reliability vol 61 no 4 pp921ndash931 2012

[21] L Xing and J B Dugan ldquoAnalysis of generalized phased-mission system reliability performance and sensitivityrdquo IEEETransactions on Reliability vol 51 no 2 pp 199ndash211 2002

10 Mathematical Problems in Engineering

[22] A Shrestha L Xing and Y Dai ldquoReliability analysis ofmultistate phased-mission systemswith unordered and orderedstatesrdquo IEEETransactions on SystemsMan andCybernetics PartASystems and Humans vol 41 no 4 pp 625ndash636 2011

[23] G Levitin L Xing S V Amari and Y Dai ldquoReliability ofnonrepairable phased-mission systems with common causefailuresrdquo IEEE Transactions on Systems Man and CyberneticsPart ASystems and Humans vol 43 no 4 pp 967ndash978 2013

[24] L Xing S V Amari and C Wang ldquoReliability of k-out-of-nsystems with phased-mission requirements and imperfect faultcoveragerdquo Reliability Engineering and System Safety vol 103 pp45ndash50 2012

[25] G K Bhattacharyya and Z Soejoeti ldquoA tampered failure ratemodel for step-stress accelerated life testrdquo Communications inStatisticsTheory andMethods vol 18 no 5 pp 1627ndash1643 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Table 4 The coefficients of 119860119894when 119870(131400) = 10

1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 11986011

129131198909 minus881411198909 2657111989010 minus4649511989010 5217111989010 minus3911511989010 1977711989010 minus663151198909 140421198909 minus168481198908 862831198906

each remaining component resulting in a higher failure rateand increased degradation rate of surviving componentsThispaper proposes amethod combining a TFRmodel with a per-formance degradationmodel to analyze the reliability of load-sharing119870-out-of-119873 systemwith degrading componentsTheTFR model deals with load-sharing effect on failure rateand the reliability when the system has a specified 119870 iscalculated by the TFR model The performance degradationmodel is derived to evaluate degradation effect coupledwith load-sharing effect and then the degraded componentperformance is estimated considering the load-sharing effecton degradation rateThe case of a solar panel is a typical load-sharing119870-out-of-119873 systemwith degrading componentsTheresults calculated by the proposed method show that thereliability considering component degradation is less thanthat ignoring component degradation With utilization ofthe proposed method the degradation effect is quantitativelyevaluated and then the reliability of load-sharing 119870-out-of-119873 system can be calculated more accurately

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was funded by National Natural Science Foun-dation of China (NSFC 61304218) In addition the authorswould like to thank the anonymous reviewers who havehelped to improve the paper

References

[1] S V Amari H Pham and R B Misra ldquoReliability characteris-tics of k-out-of-n warm standby systemsrdquo IEEE Transactions onReliability vol 61 no 4 pp 1007ndash1018 2012

[2] R Moghaddass M J Zuo and W Wang ldquoAvailability of ageneral k-out-of-nG system with non-identical componentsconsidering shut-off rules using quasi-birthdeath processrdquo Reli-ability Engineering and SystemSafety vol 96 no 4 pp 489ndash4962011

[3] P Boddu and L Xing ldquoReliability evaluation and optimizationof series-parallel systems with k-out-of-n G subsystems andmixed redundancy typesrdquo Proceedings of the Institution ofMechanical Engineers Part O Journal of Risk and Reliability vol227 no 2 pp 187ndash198 2013

[4] RMoghaddassM J Zuo and J Qu ldquoReliability and availabilityanalysis of a repairable k-out-of-n G system with R repairmensubject to shut-off rulesrdquo IEEE Transactions on Reliability vol60 no 3 pp 658ndash666 2011

[5] S V Amari K B Misra and H Pham ldquoTampered failure rateload-sharing systems status and perspectivesrdquo in Handbook on

Performability Engineering chapter 20 Springer New York NYUSA 2007

[6] K C Kapur and L R Lambberson Reliability in EngineeringDesign John Wiley amp Sons 1977

[7] R K Iyer and D J Rossetti ldquoA measurement-based model forworkload dependency of CPU errorsrdquo IEEE Transactions onComputers vol C-35 no 6 pp 511ndash519 1986

[8] E M Scheuer ldquoReliability of an m-out-of-n system whencomponent failure induces higher failure rates in survivorsrdquoIEEE Transactions on Reliability vol 37 no 1 pp 73ndash74 1988

[9] H Liu ldquoReliability of a load-sharing fc-out-of-nG system non-iid components with arbitrary distributionsrdquo IEEE Transactionson Reliability vol 47 no 3 pp 279ndash284 1998

[10] S V Amari K B Misra and H Pham ldquoReliability analysisof tampered failure rate load-sharing k-out-of-n G systemsrdquoin Proceedings of the 12th ISSAT International Conference onReliability and Quality in Design pp 30ndash35 August 2006

[11] S V Amari and R Bergman ldquoReliability analysis of k-out-of-n load-sharing systemsrdquo in Proceedings of the 54th AnnualReliability and Maintainability Symposium (RAMS rsquo08) pp440ndash445 IEEE Las Vegas Nev USA January 2008

[12] G Levitin and S V Amari ldquoMulti-state systemswithmulti-faultcoveragerdquo Reliability Engineering and System Safety vol 93 no11 pp 1730ndash1739 2008

[13] Y Ding M J Zuo A Lisnianski and W Li ldquoA frameworkfor reliability approximation of multi-state weighted k-out-of-n systemsrdquo IEEE Transactions on Reliability vol 59 no 2 pp297ndash308 2010

[14] S K Chaturvedi S H Bahsa S V Amari et al ldquoReliability anal-ysis of generalized multi-state k-out-of-n systemsrdquo Proceedingsof the Institution of Mechanical Engineers Part O Journal of Riskand Reliability vol 226 pp 327ndash336 2011

[15] S Eryilmaz ldquoOn reliability analysis of a k-out-of-n system withcomponents having random weightsrdquo Reliability Engineeringand System Safety vol 109 pp 41ndash44 2013

[16] S V Amari M J Zuo and G Dill ldquoA fast and robust reliabilityevaluation algorithm for generalized multi-state k-out-of-nsystemsrdquo IEEE Transactions on Reliability vol 58 no 1 pp 88ndash97 2009

[17] G Levitin ldquoMulti-state vector-k-out-of-n systemsrdquo IEEE Trans-actions on Reliability vol 62 no 3 pp 648ndash657 2013

[18] J Fan K C Yung and M Pecht ldquoPhysics-of-failure-basedprognostics and health management for high-power whitelight-emitting diode lightingrdquo IEEE Transactions on Device andMaterials Reliability vol 11 no 3 pp 407ndash416 2011

[19] R Baillot Y Deshayes L Bechou et al ldquoEffects of siliconecoating degradation on GaN MQW LEDs performances usingphysical and chemical analysesrdquoMicroelectronics Reliability vol50 no 9ndash11 pp 1568ndash1573 2010

[20] Y-F Li E Zio and Y-H Lin ldquoA multistate physics modelof component degradation based on stochastic petri nets andsimulationrdquo IEEE Transactions on Reliability vol 61 no 4 pp921ndash931 2012

[21] L Xing and J B Dugan ldquoAnalysis of generalized phased-mission system reliability performance and sensitivityrdquo IEEETransactions on Reliability vol 51 no 2 pp 199ndash211 2002

10 Mathematical Problems in Engineering

[22] A Shrestha L Xing and Y Dai ldquoReliability analysis ofmultistate phased-mission systemswith unordered and orderedstatesrdquo IEEETransactions on SystemsMan andCybernetics PartASystems and Humans vol 41 no 4 pp 625ndash636 2011

[23] G Levitin L Xing S V Amari and Y Dai ldquoReliability ofnonrepairable phased-mission systems with common causefailuresrdquo IEEE Transactions on Systems Man and CyberneticsPart ASystems and Humans vol 43 no 4 pp 967ndash978 2013

[24] L Xing S V Amari and C Wang ldquoReliability of k-out-of-nsystems with phased-mission requirements and imperfect faultcoveragerdquo Reliability Engineering and System Safety vol 103 pp45ndash50 2012

[25] G K Bhattacharyya and Z Soejoeti ldquoA tampered failure ratemodel for step-stress accelerated life testrdquo Communications inStatisticsTheory andMethods vol 18 no 5 pp 1627ndash1643 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

[22] A Shrestha L Xing and Y Dai ldquoReliability analysis ofmultistate phased-mission systemswith unordered and orderedstatesrdquo IEEETransactions on SystemsMan andCybernetics PartASystems and Humans vol 41 no 4 pp 625ndash636 2011

[23] G Levitin L Xing S V Amari and Y Dai ldquoReliability ofnonrepairable phased-mission systems with common causefailuresrdquo IEEE Transactions on Systems Man and CyberneticsPart ASystems and Humans vol 43 no 4 pp 967ndash978 2013

[24] L Xing S V Amari and C Wang ldquoReliability of k-out-of-nsystems with phased-mission requirements and imperfect faultcoveragerdquo Reliability Engineering and System Safety vol 103 pp45ndash50 2012

[25] G K Bhattacharyya and Z Soejoeti ldquoA tampered failure ratemodel for step-stress accelerated life testrdquo Communications inStatisticsTheory andMethods vol 18 no 5 pp 1627ndash1643 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of