International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

A unified formalism for modeling and reliability estimation of degrading systems Riascos-Ochoa, Javier; Sánchez-Silva, Mauricio; Klutke, Georgia-Ann Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015A Unified Formalism For Modeling And Reliability Estimation OfDegrading SystemsJavier Riascos-OchoaGraduate Student, Dept. of Civil and Environmental Engineering, Universidad de losAndes, Bogotá, ColombiaMauricio Sánchez-SilvaProfessor, Dept. of Civil and Environmental Engineering, Universidad de los Andes,Bogotá, ColombiaGeorgia-Ann KlutkeProfessor, Dept. of Industrial and Systems Engineering, Texas A&M University, CollegeStation, USAABSTRACT: We present a framework to model degradation on deteriorating systems, and to find easy-to-evaluate expressions for important reliability quantities. The model allows to combine multiple degra-dation mechanisms in the form of shock-based degradation with arbitrary shock sizes and progressivedegradation (as deterministic drift plus a gamma process, among others). In addition, the proposed ap-proach can be used to obtain expressions for the reliability function, the mean and n-moments of thedeterioration process Xt, and the probability density and moments of the lifetime L. In this paper we alsopresent a novel numerical method to compute these expressions, which is highly efficient and accurate.Furthermore, several deterioration models are compared in terms of the calculated reliability quantities,and the feasible moments of the deterioration Xt. The results demonstrate the generality, versatility, effi-ciency and accuracy of the proposed framework, which can open a new productive research field in thearea of probabilistic degradation models.1. INTRODUCTIONThe evaluation and prediction of the performance ofengineered systems and components over time is ofparticular importance, especially to support opera-tional decisions. Evaluating the infrastructure sys-tem performance requires that we address two im-portant aspects: the identification and modeling ofthe degradation mechanisms; and the evaluation ofthe system’s lifetime distribution, from which im-portant quantities such as the mean and the reliabil-ity function can be obtained.A flexible and general framework to model dete-rioration should include the combined effect of twobasic degradation mechanisms Sánchez-Silva et al.(2011): (1) shock-based degradation, which is theeffect of extreme events (such as earthquakes orhurricanes) over the system’s condition at discretetimes, and (2) progressive degradation, due to pro-cesses such as corrosion or wear, and consists inthe continuous removal of the system conditionover time. In addition, this framework should pro-vide easy-to-evaluate expressions for the reliabilityquantities that solves the numerical issues usuallypresented in their computation.This paper proposes a general type of deteriora-tion Xt as a stochastic process with independent, in-creasing and (possibly) non-stationary increments,i.e., as a non-homogeneous increasing Lévy processor increasing additive Process. Particular cases ofthis process are: the non-stationary gamma process(van Noortwijk (2009), Iervolino et al. (2013)),the Compound Poisson Process (CPP) Nakagawa(2007), Riascos-Ochoa et al. (2014), and determin-istic drift (Sánchez-Silva et al. (2011)), that have112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015been used extensively in the last years in a widerange of applications. With the proposed frame-work it is possible to combine any of these pro-cesses to model combined degradation mechanismswith no additional difficulty.Some previous works have used Lévy pro-cesses in deterioration (e.g., Abdel-Hameed (1984),Abdel-Hameed (2014), Yang and Klutke (2000)).However, the present paper explicitly addresses theproblem of multiple degradation mechanisms, pro-vides a clear definition of each mechanism based onthe Lévy measure, and gives easy-to-evaluate ex-pressions for the reliability function, and the life-time density and moments. In addition, it pro-vides analytical expressions for the mean and n-central moments of the deterioration Xt over time.With these results it is possible to compare severalsystems under different additive degradation pro-cesses, and to show that our model increases thefeasible region of deterioration moments that canbe reproduced. The present work is mostly basedon the paper of Riascos-Ochoa et al. (2015), but themodel presented here is more general, as the as-sumption of stationary increments is relaxed. Asa consequence, non-linear deterioration trends canbe obtained with the new model.2. THE STOCHASTIC DETERIORATION AP-PROACHIn this paper, the condition of an infrastructure sys-tem is modeled as a non-negative stochastic processVt, with V0 = 0 the deterministic value when thesystem is new. The condition decreases over timedue to degradation, and the accumulated deteriora-tion until time t is defined by the random variableXt. Hence, the condition Vt is related with the dete-rioration Xt by Sánchez-Silva et al. (2011):Vt =max {V0−Xt,0}, (1)if there is no maintenance, and the system is aban-doned at first failure.The system fails when the condition falls bellowa predefined threshold k∗ (with 0 ≤ k∗ ≤ V0) thatrepresents a safety threshold or limit state. The sys-tem lifetime L is a r.v.given by:L = inf {t ≥ 0 : Vt ≤ k∗} = inf {t ≥ 0 : Xt ≥ V0− k∗}.(2)The reliability is defined as the probability thatthe system accomplishes its intended function ina designated period of time tm (system’s missiontime), i.e., it is the probability that the condition Vtmat tm is greater than the limit threshold k∗.R(tm) = P(Vtm ≥ k∗)= P(Xtm ≤ V0− k∗). (3)Setting the mission time tm as variable, the previ-ous expression defines the reliability function R(t)for all t ≥ 0. The probability density function (pdf )f (t) of the lifetime can be defined, which is givenby:f (t) = − ddtR(t) = − ddtP(Xt ≤ V0− k∗), (4)if such derivative exists. From this expression, anyof the n-moments (n ≥ 1) of the lifetime can be ob-tained by:E[Ln] =∫ ∞0f (t)tndt, (5)where E[·] is the expectation operator.3. BASIC NOTIONS OF ADDITIVE PRO-CESSES3.1. DefinitionGiven a filtered probability space (Ω,F ,F,P), anadapted process {Xt}t≥0 that takes values on Rd (d =1,2, . . .), with X0 = 0 almost surely (a.s.) is a Lévyprocess if Protter (2004):1. {Xt}t≥0 has increments independent of the past;that is, Xt−Xs is independent of Fs, 0 ≤ s < t <∞;2. {Xt}t≥0 has stationary increments; that is, Xt −Xs has the same distribution as Xt−s, 0 ≤ s <t < ∞; and3. Xt is continuous in probability; that is,limt→s P(Xt ∈ ·) = P(Xs ∈ ·).An additive process {Xt}t≥0 relaxes the condition2 for a Lévy process. This means that, althoughthe process still has independent increments, thedistribution of Xt − Xs not only depends on thetime difference t − s but also on s Sato (1999),Cont and Tankov (2004).212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20153.2. Characteristic function and CharacteristicexponentThe characteristic function and exponent of an ad-ditive process Xt plays an important role in the for-malism proposed in this paper. The characteristicfunction φXt(ω) of the r.v. Xt is defined from theprobability density P (Xt ∈ dx) by Resnick (1999) :φXt(ω) := E[ei〈ω,Xt〉]=∫Rdei〈ω,x〉P(Xt ∈ dx), (6)where i =√−1 is the imaginary unit, and 〈·, ·〉 is theinner product in Rd.Now, because of the independence property ofthe increments of an additive process Xt, the dis-tribution of Xt is infinitely divisible, which meansthat the characteristic function can be expressed asCont and Tankov (2004), Sato (1999):φXt(ω) = e−Ψt(ω), (7)where Ψt is a function from Rd to C (the set ofcomplex numbers), and it is called the character-istic exponent of the process Xt. If Xt is a Lévyprocess the characteristic exponent is linear in timeRiascos-Ochoa et al. (2015), Bertoin (1996), i.e.,φXt(ω) = e−tΨ(ω).3.3. Decomposition of an additive process and theLévy measureThe characteristic exponent Ψt of an additive pro-cess satisfies the renowned Lévy-Kintchine formulafor any t ≥ 0 Cont and Tankov (2004), Sato (1999):Ψt(ω) = −i〈Γ(t),ω〉+12Qt(ω) (8)+∫Rd(1− ei〈ω,x〉+ i〈ω, x〉1|x|<1)Πt(dx),where 1|x|<1 is the indicator function; Γ(t) is a deter-ministic and continuous function that takes valueson Rd and Γ(0) = 0; Qt is a semi-definite quadraticform on Rd for each t ≥ 0, varies continuously witht, and Q0 = 0; and Πt is a measure defined on Rdfor each t ≥ 0 called the Lévy measure of the pro-cess Xt, it varies continuously with t, and satisfiesΠt(B) ≥Πs(B) for all s, t such that t ≥ s and all mea-surable sets B in Rd, and Π0 = 0.A useful interpretation of the Lévy measure is asfollows: denote Nt(B) the random variable of thenumber of jumps of Xt with sizes in a measurableset B in Rd (∆X ∈ B) that occur until time t. Its ex-pected value is equal to the Lévy measure evaluatedat the set B:Πt(B) = E [Nt(B)] . (9)The Lévy measure is finite if Πt(Rd) < ∞, mean-ing that the number of jumps occurring in any timeinterval [0, t] is finite. If Πt is infinite there are in-finitely many jumps in any time interval. In orderto have a convergent process, the sizes of these in-finitely jumps tend to 0.A consequence of the formula (8) is that any ad-ditive process Xt can be written as the sum of threeindependent processes:Xt = X{1}t +X{2}t +X{3}t , (10)where X{1}t = Γ(t) is the deterministic contribution,X{2}t is a Gaussian process (e.g., a Brownian mo-tion but in multiple dimensions) described by thequadratic form Qt, and X{3}t is a pure-jump processdetermined by the Lévy measureΠt. Then, an addi-tive process Xt is uniquely determined by the char-acteristic triplet (Γ(t),Qt,Πt).3.4. Mean and central moments of an additiveprocessIn the same way as in Riascos-Ochoa et al. (2015)for Lévy processes, the moments of the additiveprocess Xt evaluated at any t ≥ 0 can be obtainedfrom its characteristic function φXt = e−Ψt(ω) by:E[Xnt]= (−i)n dndωne−Ψt(ω)∣∣∣∣∣ω=0. (11)The mean E [Xt] and the n-central momentsµn(t)= E [(Xt −E(Xt))n] (with n= 2,3) are given by:E[Xt] = i e−Ψt(0)Ψ′t(0) = i Ψ′t(0) (12)µ2(t) = Ψ(2)t (0) (13)µ3(t) = −i Ψ(3)t (0), (14)where Ψ(n)t denotes the n derivative of Ψt with re-spect to ω. Note that if Xt is a Lévy process theprevious expressions vary linearly with time.312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20154. MODELING DEGRADATION VIA ADDI-TIVE PROCESSESThe framework proposed in this paper to modeldegradation and combined degradation mecha-nisms satisfies the following assumptions:1. Increasing and independent increments: In ab-sence of maintenance, deterioration Xt is a 1-dimensional increasing additive process (alsoknown as a nonhomogeneous subordinator inAbdel-Hameed (2014)). In other words, Xttakes values in R+ := [0,∞) and has increas-ing sample paths Bertoin (1996). Therefore,the Gaussian component X{2}t in equation (10)must be zero; i.e., Qt = 0 for all t ≥ 0; and theLévy measure Πt has support on [0,∞) for allt ≥ 0 (i.e., the process has no negative jumps).2. Shock-based deterioration Wt is modeled as apure jump process X{3}t with finite and positiveLévy measure ΠWt . This means that there is afinite number of shocks during any finite timeinterval. Hence, Wt is a Non-HomogeneousCompound Poisson Process (NHCPP):Wt = X{3}t =∑Nti=1Yi, (15)where Nt is a Non-Homogeneous Poisson Pro-cess with rate λ(t); the sequence {Yi}i≥1 cor-responds to the shock sizes of each ith shock,which are independent and identically dis-tributed with distribution H(·) supported on[0,∞).3. Progressive deterioration Zt is modeled as thesum of two independent processes: a deter-ministic process X{1}t = Γ(t) (with Γ(t) ≥ 0 forall t ≥ 0) and a jump process X{3}t with infi-nite and positive Lévy measure. Hence, onlyshocks with sizes tending to 0 happen in-finitely often, a property that is adequate tomodel the ‘continuous’ removal of the systemcondition in progressive degradation. There-fore,Zt = Γ(t)+X{3}t . (16)Examples of the process X{3}t are: the (non-stationary) gamma process van Noortwijk(2009) and the inverse gaussian processYe and Chen (2013).4. The combined effect of shocks Wt and pro-gressive deterioration Zt is modeled by assum-ing that these processes are independent. Al-though this condition is not general and mightnot be satisfied in some cases, it is a conve-nient approximation to the general problemKlutke and Yang (2002), Sánchez-Silva et al.(2011), Iervolino et al. (2013). Then, the com-bined degradation process Kt is obtained bythe superposition, which is also an additiveprocess:Kt = αsWt +αpZt= αs∑Nti=1Yi+αp(Γ(t)+X{3}t), (17)where the parameters αs ≥ 0 and αp ≥ 0 areweights that control the contribution of theshock-based and progressive mechanisms, re-spectively, to the combined degradation.5. EXPRESSIONS FOR SPECIFIC MODELS5.1. Shock-based deteriorationFor the shock-based process Wt, define Λ(t) =∫ t0 λ(s)ds which represents the expected number ofshocks by time t. It can be shown that the expres-sions for its Lévy measure ΠWt , characteristic ex-ponent ΨWt (λ), mean E[Xt] and n-central momentsµn,W(t) (with n = 2,3) are given by:ΠWt (dx) = H(dx)Λ(t), ΨWt (ω) = (1−φY(ω))Λ(t)(18)E[Wt] = E[Y]Λ(t), µn,W(t) = E[Yn]Λ(t), (19)where φY is the characteristic function of the dis-tribution H(·) of the shock sizes Yi. Table 1 showsexpressions for the moments of Wt for several Yidistributions. See Riascos-Ochoa et al. (2015) forthe explicit expressions of φY for these same distri-butions.5.2. Progressive deteriorationTwo progressive deterioration models are studied:the deterministic process (DP) and the gamma pro-cess (GP).Recall that the deterioration for a DP is givenby Zt = Γ(t). Hence, its characteristic exponent is412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Table 1: Examples of shock-based (NHCPP) addi-tive models Wt with common Poisson rate λ(t), alongwith the expressions for its mean E[Wt] and n-centralmoments µn,W(t) (n = 2,3) for several distributions ofshock sizes Yi with common mean y = E[Y]. (Notation:Λ(t) =∫ t0 λ(s)ds, the exponential distribution has rateβ = 1/y, and the Log-normal has coefficient of variationcov(Y) =√E[Y2]− y2/y).SHOCK-BASED (NHCPP) MODELSDeteriorationmomentsDeltaYi ∼ δ(y)Uniform Yi ∼U(y−a,y+a)ExponentialYi ∼ Exp(β)E[Wt] yΛ(t) yΛ(t) yΛ(t)µ2,W (t) y2Λ(t)(y2+a2/3)Λ(t)2y2Λ(t)µ3,W (t) y3Λ(t) (ya2+ y3)Λ(t) 6y3Λ(t)DeteriorationmomentsLog-normalYi ∼ LN(µ,σ)PH-typeYi ∼ PH(τ,T)E[Wt] yΛ(t) yΛ(t)µ2,W (t) y2(cov(Y)2+1)Λ(t) 2τT−21Λ(t)µ3,W (t) y3(cov(Y)2+1)3Λ(t) −6τT−31Λ(t)ΨZt (ω) = −i Γ(t)ω. On the other hand, the (non-stationary) GP, Zt, with shape function ΛZ(t) > 0and scale parameter βZ > 0 is usually defined fromthe following assumptions van Noortwijk (2009):(1) Z0 = 0 with probability one; (2) Zt − Zs ∼Ga(ΛZ(t)−ΛZ(s),βZ) for all t > s ≥ 0, with Ga thegamma distribution; and (3) Zt has independent in-crements. Properties (1) and (3) make the GP pro-cess an additive process. It can be shown that it is apure jump process with Lévy measure ΠZt (dx) andcharacteristic exponent ΨZt (ω) given by:ΠZt (dx) = ΛZ(t)x−1e−βZ xdxΨZt (ω) = ΛZ(t) ln (1− iω/βZ) . (20)Table 2 shows the deterioration moments of boththe DP and GP processes, obtained by applying theformulas (13).5.3. Combined deteriorationThe Lévy measure and characteristic exponent as-sociated to the combined deterioration process KtTable 2: Examples of two progressive additive modelsZt: Deterministic process (DP) following the determin-istic function Γ(t) and the gamma process (GP) withshape function ΛZ(t) and scale parameter βZ , alongwith their mean E[Zt] and n-central moments µn,Z(t)(n = 2,3).PROGRESSIVE MODELSDeteriorationmoments DPGPE[Xt] Γ(t) ΛZ(t)/βZµ2,Z(t) 0 ΛZ(t)/β2Zµ3,Z(t) 0 2ΛZ(t)/β3Zgiven by (17) are the weighted sum of the Lévymeasures and characteristic exponents of the shock-based process Wt and the progressive process Zt(Equations (18) and (20)), i.e.,:ΠKt (dx) = αsΠWt (dx)+αpΠZt (dx) (21)= αsΛ(t)H(dx)+αpΠZt (dx),ΨKt (ω) = αsΨWt (ω)+αpΨZt (ω). (22)Table 3 shows the deterioration moments of thecombination of the DP and GP with any of theshock models in Table 1.5.4. Feasible moments of Lévy deterioration pro-cessesThe objective of this section is to show that thecombination of different additive models can in-crease the region of feasible second and third cen-tral moments, compared with each model taken in-dependently. For comparison purposes the meanis set equal to E[Xt] = yΛ(t) for all models, withy = E[Y] the mean of shock sizes Yi, and Λ(t) theintegral of the Poisson rate λ(t) of the NHCPP mod-els.Note that in the previous examples, the depen-dency on t of the moments of Xt comes fromΛ(t) for the NHCPP models and the shape func-tion ΛZ(t) for the GP. Then, the axis in Figure 1represent the quantities µ2(t)/Λ(t) and µ3(t)/Λ(t),which have dimensions of y2 and y3, respectively.512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Table 3: Examples of combined additive models Ktby the independent superposition of the shock-basedmodels in Table 1 with the progressive models in Table2, and expressions for the mean E[Kt] and n-centralmoments µn,K(t) (n = 2,3). (Weights: αs and αp, respec-tively.)COMBINED MODELSDeteriorationmomentsShock-based +DPShock-based +GPE[Kt] αsΛ(t)y+αpΓ(t)αsΛ(t)y+αpΛZ(t)/βZµ2,K(t) αsΛ(t)y2αsΛ(t)y2+αpΛZ(t)/β2Zµ3,K(t) αsΛ(t)y3αsΛ(t)y3+2αpΛZ(t)/β3ZThe models in Tables 1 and 2 are sketched as dotsor filled lines, depending on the case. The (con-vex) combination of any two models (with super-position coefficients αi summing 1) is representedas dashed lines joining the corresponding points ofeach model. The shadowed region at the right ofy2 is the feasible region of any shock-based modelwith sizes distributed Phase-type (i.e., NHCPP-PH), while the region at left can be obtained bycombining the Deterministic Process (DP) or theGP model with the shock NHCPP models. Theseresults can be useful to fit combined additive degra-dation processes to degradation data from its dete-rioration moments.6. RELIABILITY ESTIMATION AND SAM-PLE PATHS OF ADDITIVE DETERIORA-TION PROCESSES6.1. Reliability function and lifetime densityEquation (6) can be inverted to obtain the probabil-ity distribution P(Xt ∈ ·) of the additive process Xtfrom its characteristic function φXt (see, e.g, Durret(2010)). From this expression, the reliability func-tion Rx(t) in (3) can be deduced:Rx(t) =12− 12pii∫ ∞−∞e−iωxωe−Ψt(ω)dω. (23)Figure 1: Feasible second and third central mo-ments (in terms of their temporal rates µ2(t)/Λ(t) andµ3(t)/Λ(t)) for several Lévy deterioration models andtheir combination with common mean deteriorationE[Xt] = yΛ(t). The axis are in natural units of y2 andy3, with y the mean shock size.From equations (4) and (23) we obtain the life-time density:fx(t)=−12pii∫ ∞−∞e−iωxω(ddtΨt(ω))e−Ψt(ω)dω. (24)6.2. Numerical solutionFormulas (23) and (24) can be solved numericallyby discretizing and truncating the improper inte-grals on the variable ω. The results are:Rx(t) ≈ Rx(t;h,M) (25)=12− 12piiM∑m=−Me−ix(m−1/2)h(m−1/2) e−Ψt((m−1/2)h).fx(t) ≈ fx(t;h,M) (26)= − 12piiM∑m=−Me−ix(m−1/2)h(m−1/2)(ddtΨt((m−1/2)h))e−Ψt((m−1/2)h),where h and M are the discretization step and trun-cation level, respectively, of the integrals in (23)and (24). Good results are obtained by taking:M ≈ 105, h = h(x, t;r) = r 2pix+E[Xt]+E[X1],(27)with r = 1/20.612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20156.3. Lifetime momentsThe moments of the lifetime E[Ln] can be approx-imated by numerically integrating equation (5) by,say, the trapezoidal rule. The procedure consists oftwo steps:1. Define a time increment ∆t > 0 and the set oftimes t1, t2, ..., tN with ti = ti−1+∆t and t0 = 0 atwhich the pdf fx(t) of the lifetime L is eval-uated by using the approximation fx(t;h,M)from (26). The final time tN and the increment∆t are set in order to have the trapezoidal ap-proximation∫ ∞0fx(t)dt ≈ Fx(∆t, tN) (28)=tN − t02Nfx(t0)+2N−1∑i=1fx(ti)+ fx(tN)close to 1 with an absolute error|1−Fx(∆t, tN)| less than a predefined er-ror . The examples in Section 7 show goodresults with  = 10−4.2. Approximate the moments E[Ln] by applyingthe trapezoidal rule to the integral (5):E[Ln] ≈ E[Ln;∆t, tN] (29)=tN − t02Ntn0 fx(t0)+2N−1∑i=1tni fx(ti)+ tnN fx(tN).7. ILLUSTRATIVE EXAMPLES AND COM-PARISONThis section shows the results of the lifetime den-sity and moments of several additive deteriorationmodels, by applying the algorithms proposed inSection 6.Consider a system that degrades with failurethreshold x = 99. Fig. 2(a) shows the lifetimedensity fx(t) (using formula (26)) of such sys-tem with different Lévy deterioration processes(i.e., with stationary increments): a stationaryGP model and shock-based (CPP) models withshock sizes distributed: Delta (CPP-Delta), ex-ponential (CPP-Exp), Uniform (CPP-U) and Log-Normal (CPP-LN). The curves obtained for thecases GP, CPP-Delta and CPP-LN match exactlywith those obtained from available expressions (seeRiascos-Ochoa et al. (2015)) for the lifetime den-sity. Furthermore, the mean, second and thirdmoments of the lifetime were calculated with themethod in Section 6.3 and compared with the com-puted from the available expressions. The resultsshow errors less than 10−4%. This demonstrates theaccuracy of the numerical method proposed in Sec-tions 6.2 and 6.3.Finally, Fig.2(b) shows the pdf ’s for the com-bined cases; thus each CPP model was combinedwith the GP using superposition coefficients αs =αp = 1. It can be observed that the combined mod-els lead to smaller failure times, which is expectedsince we have added two sources of degradation.8. CONCLUSIONSThe formalism presented in this paper provides amathematical and numerical framework to modeldeterioration. The proposed methodology has sev-eral advantages over other deterioration models: (1)It provides a conceptual definition of each degrada-tion mechanism (shocks, progressive and combineddegradation) by means of the Lévy measure and thedeterministic function (if present); (2) it includesprocesses widely used in deterioration modeling(gamma process, CPP, deterministic deterioration);(3) it can accommodate in the same mathematicalframework the combination of several sources ofdegradation; in this sense, several cases reported inthe literature can be reproduced: shock-based CPPplus linear drift Klutke and Yang (2002) or shock-based CPP with shock sizes distributed gamma plusa stationary gamma process Iervolino et al. (2013));(4) it provides analytical expressions for the mo-ments of deterioration over time, and reproducenon-linear deterioration trends; and (5) it providesaccurate numerical solutions to the main reliabilityquantities.Further research in this topic should be directedtowards the inclusion of other degradation mod-els into this formalism (e.g., stable processes, in-verse gaussian processes). Also, it is necessaryto extend the present formalism to model depen-dency between the degradation mechanisms, andto state-dependent degradation processes. Finally,it is important to formalize the statistical fitting ofan additive degradation model to deterioration data.712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015(a)0 50 100 150 20000.010.020.03f x(t)Time(t)  GPCPP-DeltaCPP-UCPP-ExpCPP-LN(b)0 50 100 15000.010.020.030.04  f x(t)Time(t)GP + CPP-DeltaGP + CPP-UGP + CPP-ExpGP + CPP-LNFigure 2: pdf’s fx(t) of the lifetime L of a system withthreshold level x = 99 for several Lévy deteriorationmodels: (a) Not combined GP with shape functionΛZ(t) = 0.1t and shape parameter βZ = 1/20 and CPP’smodels with shock sizes Yi distributed: δ(y = 20);Exp(β= 1/20); U(0,40); and LN(µ,σ) with parameters(µ,σ) such that the mean of shock sizes is E[Y] = 20with a coefficient of variation COV(Y) = 2, accordingto Table 1. (b) Combined degradation of the previousGP model with each of the CPP’s models with weightsαs = αp = 1. The CPP Poisson rate is λ = 0.1 in allcases.The analysis done in this paper about the feasiblemoments of Lévy deterioration processes could behelpful in this aspect.9. REFERENCESAbdel-Hameed, M. (1984). “Life distribution propertiesof devices subject to a lévy wear process..” Mathe-matics of Operations Research, 9(4), 606–614.Abdel-Hameed, M. (2014). Lévy Processes and TheirApplications in Reliability and Storage. Springer,New York.Bertoin, J. (1996). Lévy Processes. Cambridge Univer-sity Press, Cambridge, U.K.Cont, R. and Tankov, P. (2004). Financial ModellingWith Jump Processes. Chapman & Hall/CR, UnitedStates of America.Durret, R. (2010). Probability: Theory and Examples.Cambridge University Press, USA.Iervolino, I., Giorgio, M., and Chioccarelli, E. (2013).“Gamma degradation models for earthquake-resistantstructures..” Structural Safety, 45, 48–58.Klutke, G.-A. and Yang, Y. (2002). “The availabilityof inspected systems subject to shocks and grace-ful degradation..” IEEE Transactions on Reliability,51(3), 371–374.Nakagawa, T. (2007). Shock and Damage Models in Re-liability. Springer, London.Protter, P. E. (2004). Stochastic Integration and Differ-ential Equations. Springer, Germany.Resnick, S. (1999). A Probability Path. Birkh auser,Boston.Riascos-Ochoa, J., Sanchez-Silva, M., and Akhavan-Tabatabaei, R. (2014). “Reliability analysis of shock-based deterioration using phase-type distributions..”Probabilistic Engineering Mechanics, 38, 88–101.Riascos-Ochoa, J., Sanchez-Silva, M., and Klutke, G.-A. (2015). “Modeling and reliability analysis of sys-tems subject to combined degradation mechanismsbased on lévy processes.” Probabilistic EngineeringMechanics. [Under Review].Sánchez-Silva, M., Klutke, G.-A., and Rosowsky, D. V.(2011). “Life-cycle performance of structures sub-ject to multiple deterioration mechanisms..” Struc-tural Safety, 33, 206–217.Sato, K.-I. (1999). Lévy Processes and Infinitely Divisi-ble Distributions. Cambridge University Press, Cam-bridge.van Noortwijk, J. (2009). “A survey of the applicationof gamma processes in maintenance..” Reliability En-gineering and System Safety., 94, 2–21.Yang, Y. and Klutke, G.-A. (2000). “Lifetime-characteristics and inspection-schemes for lévydegradation processes..” IEEE Transactions on Reli-ability, 49(4), 377–382.Ye, Z.-S. and Chen, N. (2013). “The inverse gaussianprocess as a degradation model..” Technometrics.8

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