Consider a system of components whose lifetimes are governed by a probability distribution. Load sharing refers to a model of stochastic interdependency between components that operate within a system. If components are set up in a parallel system (see Parallel, Series, and Series–Parallel Systems) for example, the system survives as long as at least one component is operating. In a typical load-sharing system, once a component fails, the remaining components suffer an increase in failure rate due to the extra “load” they must encumber due to the failed component

Kvam, Paul H.

Lu, Jye-Chyi

English

University of Richmond

University of RichmondUR Scholarship RepositoryMath and Computer Science Faculty Publications Math and Computer Science2008Load Sharing ModelsPaul H. KvamUniversity of Richmond, pkvam@richmond.eduJye-Chyi LuFollow this and additional works at: https://scholarship.richmond.edu/mathcs-faculty-publicationsPart of the Applied Statistics Commons, and the Mathematics CommonsThis Post-print Chapter is brought to you for free and open access by the Math and Computer Science at UR Scholarship Repository. It has beenaccepted for inclusion in Math and Computer Science Faculty Publications by an authorized administrator of UR Scholarship Repository. For moreinformation, please contact scholarshiprepository@richmond.edu.Recommended CitationKvam, Paul H. and Lu, Jye-Chyi, "Load Sharing Models" (2008). Math and Computer Science Faculty Publications. 212.https://scholarship.richmond.edu/mathcs-faculty-publications/212Load-Sharing ModelsIntroductionConsider a system of components whose lifetimesare governed by a probability distribution. Load shar-ing refers to a model of stochastic interdependencybetween components that operate within a system.If components are set up in a parallel system (seeParallel, Series, and Series–Parallel Systems) forexample, the system survives as long as at leastone component is operating. In a typical load-sharingsystem, once a component fails, the remaining com-ponents suffer an increase in failure rate due to theextra “load” they must encumber due to the failedcomponent.Component interdependencies are discussed inDependence; Aging and Positive Dependence;Measures of Association. Another model for compo-nent interdependency is called a shock model, such asMarshall and Olkin [1] bivariate exponential model,which enables the user to model stochastic dependen-cies between component lifetimes by incorporatinglatent variables to allow simultaneous componentfailures. For example, suppose two components in asystem are characterized by failure processes 1 and2. In a shock model, we can add a new process 3that characterizes events causing both components tofail, hence the overall failure process for the com-ponents become 1 + 3 and 2 + 3. This is thebasis for most dependent failure models in probabilis-tic risk assessment, including Common Cause Failuremodels used in the nuclear industry; see Chapter 8 ofBedford and Cooke [2].In contrast to the shock model, the load shar-ing creates a dynamic reliability model in whichcomponent lifetime distributions may or may notchange throughout the course of a system lifetime.Daniels [3] originally adopted the load-share modelto describe how the strain on yarn fibers increases asindividual fibers within a bundle break. In this case,reliability is measured from strength instead of timeuntil failure. Freund [4] formalized the probabilitytheory for a bivariate exponential load-share model.In general, the shock model provides an easieravenue for multivariate modeling of system compo-nent lifetimes. However, dynamic models such asthe load-share model are deemed more realistic inenvironments where a component’s performance canchange once another component in the system failsor degrades. In typical applications of the load-sharemodel, the group of operating components share aload (e.g., current, stress, weight) and once a com-ponent within the system fails, the failure rate ofthe surviving components increase because there arefewer components to share the constant load.The load-sharing framework can apply to generalproblems of detecting members of a finite population.If resources allocated for finding a finite set of itemsare defined globally, rather than assigned individu-ally, then once items are detected, resources can beredistributed for the problem of detecting the remain-ing items. This action gives rise to a load-sharingmodel. For bridge construction, some beams con-nected to the girder are further supported by a set ofwelded joints. The failure of one or two welded jointscan cause the increase of stress on the remainingjoints, inducing a load-share model.Load-Share RulesPerhaps the most important element of the load-share model is the rule that governs how failure rateschange after some components in the system fail. Thisrule depends on the reliability application and howthe components within the system interact that is,through the structure function. For example, an equalload-sharing rule, illustrated in Figure 1, implies thatthe extra load caused by the failed component isshared equally among the surviving ones. A local0 X1:5 X2:5 X3:5 X4:5Figure 1 Illustrative load transfer with an equal load-sharing rule2 Load-Sharing Models0 X1:5 X2:5 X3:5 X4:5Figure 2 Illustrative load transfer with an local load-sharing rule, where the load from a failed component istransferred to its closest neighborsload-sharing rule, illustrated in Figure 2, dictates thata failed component’s load is transferred to adjacentcomponents; the proportion of the load the survivingcomponents inherit depends on their distance to thefailed component.More generally, a monotone load-sharing rulestates that the load on the surviving components isnondecreasing with respect to the failure of othercomponents in the system. Not all load-share rulesneed to be monotone. In finite population sampling,the event of finding one of the n observations mightdecrease the rate for finding the others. To modelthe detection rate in software debugging for example,the detection time for existing faults can dependon the number of other faults in the software thathave already been found (see Software ReliabilityModeling and Analysis). The discovery of a criticalfault in the software might help reveal or concealother yet undetected bugs. If other faults are moreconcealed, they have a decreased rate of discovery inthe debugging process.For assessing an unknown load-sharing rule,Kim and Kvam [5] model the dynamic nature ofcomponent reliability through proportional hazards.The baseline component failure time distribution isdenoted by F . The hazard function (or cumulativehazard rate) corresponding to F is R(x) = − log(1 −F(x)), and the hazard rate is r(x) = f (x)[1 −F(x)]−1, where f (x) is the density of F . Until thefirst component failure, the failure rate of each ofk components in the system equals the baseline rater(x). Upon the first failure within a system, the failurerates of the k − 1 remaining components jump toγ1r(x), and remain at that rate until the next com-ponent failure. After this failure, the failure rates ofthe k − 2 surviving components jump to γ2r(x), andso on. The failure rate of the last remaining compo-nent is γk−1r(x). An equal load-share rule is impliedby the k − 1 parameters γ1, γ2, . . . , γk−1.To illustrate the relationship of the load-share ruleto the system failure process, we consider Daniels’original application for modeling fiber strength.Stress-Strength ModelsFor researchers in the textile industry who investigatethe reliability of composite materials, a bundle offibers can be considered as a parallel system subjectto a steady tensile load. The rate of failure forindividual fibers depends on how the unbroken fiberswithin the bundle share the load of this overall stress.The load-share rule of such a system depends onthe physical properties of the fiber composite. Yarnbundles or untwisted cables tend to spread the stressload uniformly after individual failures. This leads toan equal load-share rule, which implies the existenceof a constant system load that is distributed equallyamong the working components.In more complex settings, a bonding matrix joinsthe individual fibers as a composite material, andan individual fiber failure affects the load of certainsurviving fibers (e.g., neighbors) more than others,dictated by a local load-sharing rule. If individualfiber strengths are independently and identically dis-tributed with distribution function F , the (parallel)system strength isYn = maxn{n − k + 1nXk:n, k = 1, . . . , n}(1)Unfortunately, the distribution FYn for Yn becomesintractable quickly as n increases. Suh et al. [6]developed a recursive formula for FnFYn =n∑i=1(ni)(−1)i+1 (F (x))i Fn−i(nxn − i)(2)that allows algorithmic computation, but even thissolution becomes computationally infeasible withlarge n.As an alternative approach, consider a system witheach fiber undergoing a positive stress level x. AsLoad-Sharing Models 3x increases, more and more individual fibers break.In this scenario, however, the system load actuallydecreases at the instance of failure because here thesystem load is measured as the total stress divided bythe number of surviving fibers. Now the actual loadin the system (per fiber) at stress level x isL(x) = xn−1n∑i=1I (Xi > x) (3)Because x−1nL(x) is binomially distributed, L(x)has an asymptotic normal distribution with meanx(1 − F(x)) and variance x2F(x)(1 − F(x)). Themean is unimodal in x, so there is a stress valuex0 such that L(x0) represents the maximum loadthe system can endure. For example, if F(x) = x,0 ≤ x ≤ 1 (strength is uniformly distributed between0 and 1) then x0 = 0.5, the maximum value ofx(1 − x).Various techniques have been used to approx-imate Pn(x) = n−1∑ I (Xi ≤ n) in L(x) = x(1 −Pn(x)). These are summarized in Crowder et al. [7];Barbour [8] approximation shows a slight improve-ment over the others based on Pn(x) ≈ (zx), wherezx =√n(x − L0 − λ1/3n−2/3)σ 2 + γ n−1/22/3 (4)L0 = L(x0), σ 2 = x02F(x0)(1 − F(x0)), and = x40F′(x0)22F ′(x0) + x0F ′′(x0) (5)(λ, γ ) are constant with γ ≈ −0.317 and λ ≈ 0.996.McCartney and Smith [9] carefully evaluated severalapproximation techniques, noting that the Barbourtechnique outperform other methods in the lower tailof the distribution.Time-to-Failure ModelsThe load-share model has found broad applica-tion in life testing and dependent systems analy-sis since first being developed for material strengthmodels. In most examples, load-sharing modelsserve as a detriment to systems, especially parallelsystems. Most research has emphasized paramet-ric lifetime distributions for the dependent compo-nents (e.g., Exponential, Weibull) with an assignedload-sharing rule to characterize the dependence.Ryden´ [10] extended the load-sharing frameworkto nonparametric estimation of component andsystem lifetime using the same load-sharing rules ofDaniels [3].As a simple example, if the components of a paral-lel system have independent exponential distributionswith failure rate λ, it is easy to show through spacings(the time between successive failures in the system)that the expected system lifetime is(n+12)λ−1. How-ever, if there is a load-sharing system in which theload of the failed component is added to the othercomponents, the expected system lifetime is reducedto λ−1. This is another consequence of the memory-less property that is, for 0 < a < t , if X ∼ Exp(λ),P(X > t |X > a) = P(X > t − a). In this case, itdoes not even matter if the load is shared equallyby all the surviving components or whether the loadis put only on one or two survivors.Figure 3 shows the expected lifetime for paral-lel system of Weibull(a, b)-lifetime components. Thescale parameter (a) equals one and the shape parame-ter (b) changes from one to five, so the left-most caserefers to the exponential model, where the expectedlifetime is 1.0. For the components, the failure rate isr(t) = tb. Here, the load from a failed component isassumed to be distributed equally among the surviv-ing components via the scale parameter as describedby Kvam and Pen˜a [11] and Liu [12].With complex load-sharing rules, the numberof parameters can escalate dramatically. Lynch [13]showed that complex systems of Weibull-distributedcomponents, in a load-sharing model, can be modeledusing a mixture of gamma-type distributions.2 3 4 50.950.960.970.980.99Figure 3 Expected parallel system lifetime versus shapeparameter of Weibull-distributed components. The shapeparameter changes from 1 (Exponential case) to 5. Curvesare for cases n = 20 (highest), n = 10 (middle) and n = 5(lowest)4 Load-Sharing ModelsEstimating the Load-Sharing RuleUsing nonparametric maximum likelihood estima-tion (MLE), Kvam and Pen˜a [11] estimated a simpleload-sharing rule based on independent identicallydistributed (i.i.d). parallel systems for which the fail-ure rates of all components were equal, but the changein rate after a component failure depends on the setof functioning components in the system. The pro-portional hazard model by Kim and Kvam [5] isapplied to estimate the load-sharing rule; that is, fail-ure rate changes are characterized by k − 1 unknownparameters γ1, γ2, . . . , γk−1 and the unknown base-line distribution or hazard function. In this case,estimating the underlying baseline functions F wasof primary interest.If Xij represents the lifetime of the j th componentin the ith parallel system, then the random spacingTij is the time between j th failure and (j − 1)thfailure for the ith system. Kim and Kvam [5] con-sidered the exponential model. For this special casewhere i = 1, . . . , n and j = 1, . . . , k, the likelihoodfunction for the ith system isLi(θ, γ ; ti1, ti2, · · · , tik)= k!θkk−1∏j=1γjexp(−θk∑j=1(k − j + 1)γj−1tij)(6)Although the likelihood provides no closed formsolution to obtaining MLEs, a Gauss-Seidel algorith-mic method can be used to solve them. Kim andKvam [5] also considered order restricted inferencecaused by monotone load sharing where 1 ≤ γ1 ≤γ2 ≤ · · · ≤ γk−1. In this case, a component failurecauses the increase in the work-load of the other com-ponents, which can equate to an increase of failurerate.Kvam and Pen˜a [11] generalized the special caseof Kim and Kvam [5] in which the components aredistributed exponential. Si,1 < Si,2 < . . . representsthe successive component failure times for the ithsystem. With the counting processesNi(t) =k∑j=1I (Si,j ≤ t), i = 1, 2, . . . , n (7)Ni(t) represents the number of component failuresfor the ith system that occurred by time t . Wecan express γ in terms of the counting processesas γ [Ni(w)] = ∑k−1j=0 γj I (Ni(w) = j). To formulatethe nonparametric likelihood in terms of stochasticprocesses, letYi(w) = (k − Ni(w−)) I (τ ≥ w) (8)Define Fit = σ {(Ni(w), Yi(w+)); w ≤ t} to be thefiltration generated by the ith system up to time t ,and let Ft =∨ni=1 Fit . The load-share model can bedescribed by specifying the failure intensities asPr{dNi(t) = 1|Fit−}= r(t)Yi(t)γ [Ni(t−)] dt, i = 1, . . . , n(9)Note thatMi(t) = Ni(t) −∫ t0γ [Ni(u−)]r(u)Yi(u) du (10)is a vector of orthogonal square-integrable zero-meanmartingales. See, for example, Andersen et al. [14].In terms of {(Ni(w), Yi(w)), 0 ≤ w ≤ τ) :i = 1, 2, . . . , n}, the likelihood isL(R(·), γ )={n∏i=1π0≤w≤τ[Yi(w)γ [Ni(w−)] dR(w)]dNi(w)}× exp{−∫ τ0Yi(w)γ [Ni(w−)] dR(w)}(11)We can estimate R(·) from the equation (10) by fixingγ , in Rˆ(·; γ ). By plugging Rˆ(·; γ ) into the equa-tion (10), we have the profile likelihood Lp(γ ) forγ , which is then maximized in γ to obtain the esti-mator γˆ . The semiparametric estimator of R(·) isRˆ(·) = Rˆ(·; γˆ ). To estimate R this way, we defineJ (w) = I (∑ni=1 Yi(w) > 0), where J (w) = 0 indi-cates all nk components have already failed at timew−. If γ is known, by using the zero-mean propertyof the martingale∑ni=1 Mi(·), we haveRˆ(s; γ ) =∫ s0J (w) dN(w)n∑i=1Yi(w)γ [Ni(w−)](12)This is analogous to the derivation of the Nelson-Aalen estimator in Aalen [15].Load-Sharing Models 5The estimator in the equation (12) is a generalizedNelson-Aalen estimator, and is similar in structureto the hazard function estimator for tensile strengthsderived by Ryden´ [10]. To obtain the estimator ofR(·) for the more general case where γ is unknown,we first obtain the profile likelihood for γ by pluggingin Rˆ(·; γ ) given in equation (12) into the likelihoodfunction in equation (10). From equation (11) andequation (12) we obtain the profile likelihood to beLp(s; γ ) =n∏i=1π0≤w≤sYi(w)γ [Ni(w−)]n∑l=1Yl(w)γ [Nl(w−)]dNi(w)(13)This profile likelihood is maximized with respect toγ to obtain γˆ , which is then plugged in into Rˆ(·; γ )to obtain the semiparametric estimator of R given byRˆ(s) = Rˆ(s; γˆ ) (14)By virtue of the product representation ofF¯ = 1 − F given by F¯ (s) = π 0≤w≤s[1 − R(dw)],we can estimate F¯ withˆ¯F(s) = π0≤w≤s[1 − Rˆ(dw)] (15)The solution to the nonparametric MLE is detailed inKvam and Pen˜a [11], and asymptotic properties forRˆ and γ are discussed in the section titled “Time-to-Failure Models” of that paper.References[1] Marshall, A.W. & Olkin, I. (1967). A multivariate expo-nential distribution, Journal of the American StatisticalAssociation 62, 30–44.[2] Bedford, T. & Cooke, R. (2001). Probalistic Risk Anal-ysis: Foundations and Methods, Cambridge UniversityPress.[3] Daniels, H.E. (1945). The statistical theory of thestrength of bundles of threads, Proceedings of the RoyalSociety of London. Series A 183, 405–435.[4] Freund, J. (1961). A bivariate extension of the expo-nential distribution, Journal of the American StatisticalAssociation 56, 971–977.[5] Kim, H.T. & Kvam, P.H. (2004). Reliability estimationbased on system data with an unknown load share rule,Lifetime Data Analysis 10, 83–94.[6] Suh, M.W., Bhattacharyya, B.B. & Grandage, A. (1970).On the distribution and moments of the strength ofa bundle of fibers, Journal of Applied Probability 7,712–720.[7] Crowder, M.J., Kimber, A.C., Smith, R.L. & Sweeting,T.J. (1991). Statistical Analysis of Reliability Data,Chapman & Hall.[8] Barbour, A.D. (1981). Brownian motion and a sharplycurved boundary, Advances in Applied Probability 13,736–750.[9] McCartney, L.N. & Smith, R.L. (1983). Statistical theoryof the strength of fiber bundles, ASME Journal of AppliedMechanics 105, 601–608.[10] Ryden´, P. (1999). Estimation of the reliability of systemsdescribed by the Daniels load-sharing model, Licen-tiate Thesis, Department of Mathematical Statistics,Umea˚University, Sweden.[11] Kvam, P.H. & Pen˜a, E.A. (2005). Estimating load-sharing properties in a dynamic reliability system,Journal of the American Statistical Association 100,262–272.[12] Liu, H. (1999). Reliability of a load-sharing k-out-of-n:Gsystem: non-iid components with arbitrary distributions,IEEE Transactions on Reliability 47, 279–284.[13] Lynch, J.D. (1999). On the joint distribution of compo-nent failures for monotone load sharing systems, Journalof Statistical Planning and Inference 78, 13–21.[14] Andersen, P.K., Borgan, O., Gill, R.D. & Keiding, N.(1993). Statistical Models Based on Counting Processes,Springer-Verlag, New York.[15] Aalen, O.O. (1978). Nonparametric estimation for afamily of counting processes, Annals of Statistics 6,701–726.Further ReadingGu¨cer, D.E. & Gurland, J. (1962). Comparison of statistics oftwo fracture models, Journal of the Mechanics and Physicsof Solids 10, 365–373.Smith, R.L. (1982). The asymptotic distribution of the strengthof a series-parallel system with equal load-sharing, Annalsof Probability 10, 137–171.Related ArticlesCompeting Risks; Cumulative Distribution Func-tion (CDF); Dependence; Markov Processes; Mea-sures of Association; Parallel, Series, and Ser-ies–Parallel Systems; Renewal Theory; SoftwareFailure Data Analysis; Software Reliability Mod-eling and Analysis; Stochastic Deterioration;Stochastic Orders and Aging; Stress–StrengthModel.PAUL H. KVAM AND JYE-CHYI LU

Load Sharing Models

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