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

Stochastic renewal process models for life cycle cost and utility analysis Pandey, Mahesh D.; Wang, Z.; Cheng, T. 2015-07

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Stochastic Renewal Process Models for Life Cycle Cost and UtilityAnalysisMahesh D. Pandey and Z. WangDepartment of Civil and Environmental Engineering, University of Waterloo, Waterloo,Ontario N2L 3G1, CanadaT. ChengTechnology and Engineering Center for Space Utilization, Chinese Academy of Science,Beijing, ChinaABSTRACT: The paper presents a systematic formulation of the life cycle cost and utility analysisbased on the theory of stochastic renewal processes. The paper derives integral equations for expectedcost and net present value, variance and expected utility over a given period. The proposed formulationcan be used to optimize the design and rehabilitation activities to improve the life cycle performance ofstructures that are vulnerable to external hazards, such as earthquakes, winds storms and floods.1. INTRODUCTIONThe life-cycle cost analysis focuses on the estima-tion of the total cost of construction, operation,maintenance, decommissioning, and many otheractivities, over a given time horizon or service pe-riod of a structure or facility. In this analysis, oneof the most uncertain elements is the cost of repair-ing and restoring the structural damage caused byexternal hazards, such as earthquakes, wind stormsand floods. Uncertainty in the estimation of re-pair cost arises from intrinsic uncertainty associ-ated with the occurrence frequency and intensity ofa given type of hazard.As the focus of the performance-based designhas shifted to life-cycle cost analysis (LCCA), thistopic has become an important area of research(Koduru and Haukaas, 2010). LCCA is also in-tended to support the decision making regardingimprovements in design and retrofitting of struc-ture. For this purpose, a suitable metric of cost isused, such as the expected cost and expected dis-counted cost (or Net Present Value - NPV). The ex-pected utility approach is also finding applicationwith the purpose of incorporating the risk aversionof the decision maker (Cha and Ellingwood, 2013).In a technical sense, LCC estimation problemshould be analyzed using the theory of compoundrenewal processes. In structural engineering, aspecial case of renewal process, the homogeneousPoisson process (HPP), has been traditionally usedLCCA. A most notable example is the seismic riskanalysis. Although the HPP model greatly simpli-fies the analytical formulation, it has a downsidethat it completely masks the real probabilistic struc-ture of the solution. What it means is that the resultsobtained under the HPP assumption cannot be read-ily extended to other situations without understand-ing the theory of the stochastic renewal processes.The main objective of this paper is to providea clear exposition of key ideas of the theory ofstochastic renewal processes and illustrate their ap-plications to the life-cycle cost analysis. In partic-ular, derivations of the expected value and the vari-ance of the cost, expected NPV, and expected util-ity are presented for a stochastic renewal processmodel. Results for HPP model are obtained as aspecial case of the renewal process. The example112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015of seismic risk analysis is discussed in more detail.An ulterior motive of this study is to help new gen-eration of engineers understand the key concepts ofstochastic models for life-cycle cost analysis.2. STOCHASTIC RENEWAL PROCESS2.1. Stochastic Point Process0 tS1 S2 Sn−1 Sn Sn+1T1 T2 Tn Tn+1Figure 1: A schematic of the renewal process.Mathematically, a point process is a strictly increas-ing sequence of real numbers, S1 < S2 < ∙ ∙ ∙ , with-out a finite limit point, i.e., as i→ ∞, limSi → ∞and S0 = 0. In an engineering sense however, Si de-notes the time of arrival of an ith event (or hazard),as shown in Figure 1.A point process can be equivalently representedby a sequence of inter-arrival times, T1,T2, ∙ ∙ ∙ , withTn = Sn+1 − Sn. An ordinary renewal process isdefined as a sequence of non-negative, indepen-dent and identically distributed random variablesT1,T2, ∙ ∙ ∙ ,Tn with a distribution FT (t).The arrival time of an nth event, Sn, can be writtenas a partial sum, Sn = T1 + T2 ∙ ∙ ∙+ Tn. The proba-bility distribution of Sn can be derived in principlefrom an n-fold convolution of distribution, FT (t), asFSn(x) = P [T1 + T2 ∙ ∙ ∙+ Tn ≤ x] = F(n)T (x) (1)This convolution can be evaluated in a sequentialmanner asF(n)T (x) =∫ t0F(n−1)T (x− y)dFT (y), (n≥ 2) (2)Note that dFT (y) = fT (y) dy when the probabilitydensity of T , fT (y), exists.The number of events, N(t), in the time interval(0, t] is referred to as a counting (or renewal) pro-cess associated with the partial sums Sn, n≥ 1, andformally defined asN(t) = max{n,Sn ≤ t} (3)Note that N(t) = n is equivalent to the eventSn ≤ t < Sn+1. This analogy is very useful in an-alyzing functions of the renewal process, as shownlater in the paper.The probability distribution of N(t) can be writ-ten asP [N(t) = n] = P [Sn ≤ t < Sn+1] = FSn(t)−FSn+1(t)(4)The distribution of N(t) is not easy to derive in ageneral setting. Instead, the expected number offailures is used to characterize the renewal process.2.1.1. Renewal FunctionThe renewal function, Λ(t), is defined as the ex-pected number of renewals in (0, t], which is com-puted using an integral equation (Tijms, 2003). Thederivation of the integral equation is based on therenewal argument that is extremely useful in ana-lyzing a variety of problems.The use of a binary indicator function makes iteasier to write concise mathematical statements. Itis used to test a logical condition in the followingway:1{A} ={1 only if A is true0 otherwise. (5)From probability theory, it is known that E[1{A}]=P [A]. Using the indicator function, the countingprocess can be written asN(t) =∞∑i=11{Si≤t} andE [N(t)] = Λ(t) =∞∑i=1P [Si ≤ t] (6)Eq (6) and Eq. (1) implies that the renewal functioncan be written as a sum of convolutions:Λ(t) =∞∑i=1F(n)T (t) (7)However, this approach is not fruitful, as the com-putation of higher order convolutions is not an easytask. Therefore, the computation relies on the re-newal property of the process.212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015The analysis starts with breaking the sum givenin Eq (6) asΛ(t) = P [S1 ≤ t]+∞∑i=2P [Si ≤ t] or= P [S1 ≤ t]+∞∑i=1P [Si+1 ≤ t] (8)The essence of the renewal argument is that aftera renewal a (probabilistic) replica of the original re-newal process starts again. Noting that S1 = T1 andSi+1 = T1 + Si, Eq. (8) can be rewritten asΛ(t) = P [T1 ≤ t]+∞∑i=1P [T1 + Si ≤ t] (9)Any ith term in the sum in Eq. (9) can be simplifiedasP [T1 + Si ≤ t] =∫ t0P [Si ≤ t− x]dFT (x) (10)Substituting this in Eq. (9) and inter-changing theorder of summation and integration leads toΛ(t) = P [T1 ≤ t]+∫ t0(∞∑i=1P [Si ≤ t− x])dFT (x)(11)Comparing the summation term inside the paren-theses with Eq. (6), it is nothing but the expectednumber of renewals in time interval (0, t−x], whichis equal to Λ(t − x) by virtue of the renewal argu-ment. This leads to the final result known as therenewal equation (Tijms, 2003):Λ(t) == FT (t)+∫ t0Λ(t− x)dFT (x). (12)The renewal rate is defined as the expected num-ber of renewals per unit time given as:λ (t) = dΛ(t)dt (13)2.1.2. Marked and Compound Renewal ProcessesIn addition to the time of occurrence of a hazard,its severity (or intensity) tends to be highly uncer-tain. Variability in the intensity can be modelledby a random variable, which is referred to as the0 tX1X2Xn−1XnS1 S2 Sn−1 SnT1 T2 TnFigure 2: An example of a marked renewal process.mark of the renewal process. If a random markXi is assigned to the arrival time Si, then the se-quence {(S1,X1),(S2,X2), ∙ ∙ ∙} is called a markedpoint process, as shown in Figure 2. This modelis useful to evaluate the probability of intensity ex-ceeding a critical limit in a time interval (0, t).The compound process refers to the cumulativeeffect of a renewal process. For example, each oc-currence of a hazard results in damage costing $ Cto repair the structure. Suppose the repair cost ismodelled as a random variable, then total (cumula-tive) repair cost in the interval (0, t) is given as arandom sum:K(t) =N(t)∑i=1Ci (14)K(t) is technically referred to as the compound pro-cess. Note that K(t) = 0 for t < S1. The mean andvariance of the compound process are useful in thelife cycle cost analysis, as shown later in the paper.2.2. Homogeneous Poisson ProcessThis is the simplest and most widely used renewalprocess model in which the time between the occur-rence of events is an exponentially distributed ran-dom variable with the distribution FT (x) = 1−e−λxand the mean of μT = 1/λ . The distribution of N(t)is explicitly given by the Poisson probability massfunction asP [N(t)) = k] = pk =(λ t)k e−λ tk! (15)The probability of no renewal in (0, t] (e.g., no oc-currence of failure) is synonymous with the relia-bility in time interval, (0, t] asP [N(t)) = 0] = P [T1 > t] = R(t) = e−λ t , (16)312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015which is essentially the complementary distributionof T . The renewal function of HPP is given asΛ(t) = λ t (17)Here, λ is referred to as the failure rate. In sum-mary, the probabilistic structure of the homoge-neous Poisson process is completely defined bysimple analytical formulas.2.2.1. Decomposition and SuperpositionThe two most useful properties of the Poisson pro-cess are related to the decomposition and superpo-sition defined as follows.Suppose the seismic events occur with a rate λ ,but the probability of structural failure given anevent is p, then the structural failure is also a Pois-son process with the rate λ p. This assumes that thestructure is renewed to the original state after eachfailure, and the repair time is negligible. In fact, aPoisson process can be randomly decomposed inton sub-processes, each with the rate parameters asλ pi, i = 1, ∙ ∙ ∙ ,n and p1 + ∙ ∙ ∙+ pn = 1.The superposition is related to the merger ofPoisson processes. Suppose there are n seismicsources (faults) and λi is earthquake occurrence ratefrom an ith source. The occurrence of earthquakesfrom any of n sources is also a Poisson processwith rate being the sum of rates of all the seismicsources, i.e.,λ = λ1 + λ2 + ∙ ∙ ∙+ λn (18)This property is extensively used in probabilisticseismic hazard assessment (PSHA).2.2.2. Marked and Compound Poisson ProcessesRandomness in the intensity and occurrence of anexternal hazard can be modelled as a marked Pois-son process. For example the peak ground acceler-ation (pga) associated with an earthquake event ismodelled as a random variable. The marked pro-cess is useful in structural reliability analysis over atime interval (or life-cycle).Given the distribution of the pga, FS(s), earth-quake events exceeding certain level, say s1, alsoconstitute a Poisson process with the rate λ p1,where p1 = 1−FS(s1) This is based on the decom-position property of the Poisson process. Based onEq. (16), the structural reliability is given as e−tλ p1 .The compound Poisson process also has a simpleanalytical structures, which will be discussed in thenext Section.3. LIFE-CYCLE COST ANALYSIS (LCCA)3.1. Expected Life-Cycle CostSuppose the occurrence of a hazard costs the ownerof a facility C $, in repairing and restoring the struc-ture. The repair cost is uncertain due to uncertaintyassociated with the intensity of hazard and otherdesign features. The repair cost has a mean μCand standard deviation σC. The total cost, K(t),in a time interval (0, t] is a random sum given byEq. (14). The evaluation of expected cost beginswith rewriting it asK(t) =∞∑i=1Ci1{Si≤t} (19)If the cost is assumed to be independent of inter-occurrence time (T ) of hazards, then the evaluationof expected cost becomes rather simple as shownbelow.E [K(t)] =∞∑i=1E [Ci]E[1{Si≤t}]= E [C]∞)∑i=1P [Si ≤ t]E [K(t)] = μC Λ(t) (from Eq. (6)) (20)Eq. (20) is a standard result of the compound re-newal process (Gallager, 2013).3.2. Variance of Life-Cycle CostThe evaluation of the variance of LCC begins ina much more systematic way (Cheng and Pandey,2012). Firstly, second moment of the cost is writtenusing the law of total expectation asE[K2(t)]= E[K2(t)1{T1≤t}]+E[K2(t)1{T1>t}](21)In case of T1 > t, i.e., the first occurrence of hazardis beyond the time interval of interest, no cost willaccrue, i.e., K(t) = K2(t) = 0.412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015On the other hand T1 ≤ t means that at least onehazard would occur in the time interval. Therefore,the total cost will be the sum of damage cost due tothe first hazard, C1 at time T1, and the cost, K(T1, t),in the remaining time interval. Thus, K(t) = C1 +K(T1, t). Using these result,E[K2(t)]= E[(C1 + K(T1, t))2 1{T1≤t}](22)In the spirit of the renewal argument that the re-newal process starts again after the first renewal attime T1, the damage cost in the remaining intervalcan be written as K(t−T1) which has the same dis-tribution as K(t). Using this argument and expand-ing the square term in Eq. (22) lead toE[K2(t)]= E[C211{T1≤t}]+E[2C1K(t−T1)1{T1≤t}]+E[K2(t−T1))21{T1≤t}] (23)Assuming that the C is independent of T , these ex-pectations can be evaluated in the following man-ner.E[C211{T1≤t}]= E[C2]FT (t), (24)E[2C1K(t−T1)1{T1≤t}]= 2 μC∫ t0E [K(t− x)]dFT (x) (25)and lastlyE[K2(t−T1))21{T1≤t}]=∫ t0E[K2(t− x)]dFT (x)(26)Substituting all these simplified terms in Eq.(23)leads to the following renewal equation:E[K2(t)]= g(t)+∫ t0E[K2(t− x)]dFT (x) (27)whereg(t) = μ2CFT (t)+ 2μC∫ t0E [K(t− x)]dFT (x)(28)Note that μ2C is the second moment of the damagecost, and E [K(t− x)] is already given by Eq.(20).In fact, the expected cost can also be derived us-ing the renewal argument in form of the followingintegral equation:E [K(t)] = μCFT (t)+∫ t0E [K(t− x)]dFT (x) (29)Using this result, a simpler expression for g(t) isobtained asg(t) = (μ2C−2μ2C) FT (t)+ 2μC E [K(t)] (30)Since g(t) is a bounded and integrable function, therenewal equation, Eq.(27) has the following solu-tion:E[K2(t)]= g(t)+∫ t0g(t− x)dΛ(x)which can also be written asE[K2(t)]= g(t)+∫ t0g(t− x)λ (x)dx (31)In summary, given the renewal function andthe expected life cycle cost, the second mo-ment of the life-cycle cost can be calculated us-ing the above equation. Then the variance canbe calculated using the standard relationship as[E[K2(t)]− (E [K(t)])2].3.3. Discounted Life Cycle CostThe expected value of the discounted life-cycle costor (NPV) can be obtained using the renewal argu-ment. Here, the damage cost Ci incurring at timeSi is discounted back to present time, S0 = 0, asCie−ρSi , where ρ is the interest rate. Thus,the totaldiscounted life-cycle cost can be written asKD(t) =∞∑i=1Ci e−ρSi 1{Si≤t} (32)Its expected value is then written asE [KD(t)] = E [Ci]∞∑i=1E[e−ρSi 1{Si≤t}]= μC∞∑i=1∫ t0e−ρxdFSi(x) (33)Recall that FSi(x) = F(i)T (x) and substituting it inabove equation leads toE [KD(t)] = μC∞∑i=1∫ t0e−ρxdF(i)T (x)= μC∫ t0e−ρx ∑dF(i)T (x) (34)512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Based on Eq. (7), the final result is obtained asE [KD(t)] = μC∫ t0e−ρx dΛ(x) (35)It is interesting to note that even for a general re-newal process the expected discounted cost has afairly simple analytical form.3.4. Expected Utility AnalysisA renewal type integral equation can be derived fora restricted family of the utility function, such asan exponential function of the following form (Chaand Ellingwood, 2013):U(K(t)) =−a e−αK(t) (36)For sake of notational simplicity, define the ex-pected utility without any multiplicative constantsasE [U(K(t)))] = φ(t) = E[e−αK(t)](37)The formulation proceeds in the same way as thatfor the second moment in Section 3.2.E [U(K(t)] = E[U(K(t)1{T1≤t}]+ E[U(K(t)1{T1>t}] (38)The first term can be simplified by invoking the re-newal argument as (Cheng et al., 2012)E[U(K(t)1{T1≤t}]= E[e−α(C1+K(t−T1))1{T1≤t}]=∫ t0E[e−αK(t−x)]E[e−αC1]dFT (x) (39)The second expectation term in Eq.(38) simplyturns out to be FT (t), since K(t) = 0 when T1 > t.The final solution can be written as an integralequationφ(t) = FT (t)+∫ t0φ(t− x) fφ (x) d(x) (40)wherefφ (x) = E[e−αC1]fT (x) (41)is a defective density.Eq.(40) is also referred to as a defective renewalequation, and its solution will depend on the proba-bility density of the renewal interval, fT (x), as wellas the density of the cost, C, which is needed toevaluate E[e−αC1].Since the solution of a defective renewal equa-tion is a more involved task, further analysis of thisproblem is pursued in a separate investigation.3.5. Life-Cycle Cost Analysis: HPP ModelIf the occurrence of a hazard is modelled as an HPP,then the life-cycle cost is equivalent to a compoundPoisson process. Using Eq.(20) and noting that therenewal function of HPP is simply Λ(t) = λ t, theexpected cost can be easily obtained asE [K(t)] = μC λ t (42)The second moment (or mean square) of the costcan be obtained from Eq.(31) with λ (x) = λ andFT (t) = 1− e−λ t :E[K2(t)]= μ2C λ t +(μCλ t)2 (43)The variance of cost is simplyσ2K(t) = μ2C λ t (44)The expected discounted cost can be obtainedfrom Eq.(34) asE [KD(t)] = μC∫ t0e−ρx λdx = μCλρ (1− e−ρt)(45)This standard formula is commonly used in seismicrisk analysis (Liu et al., 2004; Porter et al., 2004).4. SEISMIC RISK ANALYSIS: DISCUS-SION4.1. Seismic Hazard AnalysisConsider a hypothetical site that is vulnerable todamage from earthquakes originating from threeseismic sources shown in Figure 3. This Figure alsoprovides occurrence rates, magnitude and depth ofseismic sources. Here, the earthquake intensity isquantified as the spectral acceleration Sa at 1 sec-ond period of the structure situated at the site.For details of the probabilistic seismic hazardanalysis (PSHA), readers are referred to McGuire(2004). The ground motion prediction equation, afunction of earthquake magnitude (M) and distance(R), given by Abrahamson and Silva (1997) is used.612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Figure 3: Seismic sources near the site.PSHA leads to the complementary distribution ofSa from an ith source asFi(si) = P [Sai > si]=∫M∫RP [Sai > si|m,r] fR(r) fM(m)drdm (46)Since the earthquake occurrence is modelled asan HPP with rate λi per year, earthquakes exceedingan intensity si is also an HPP with rate λiFi(si). Thisresult is based on the decomposition property of thePoisson process.From the superposition principle, the combinedearthquake hazard from all the three sources is alsoa Poisson process with an overall rate of λ = λ1 +λ2 +λ3 = 0.1 per year. In particular, the occurrencerate of earthquakes exceeding a magnitude s is alsogiven by the superposition principle asλ (s) = λ1F1(s)+ λ2F2(s)+ λ3F3(s) (47)A plot of λ (s) versus s, referred to as the hazardcurve, is shown in Figure 4.The probability of no occurrence of earthquakeexceeding the intensity s in (0, t] is equivalent tothe cumulative probability that the earthquake in-tensity does not exceed s in (0, t], which is given asFSa,t(s) = e−λ (s)t . On an annual basis, i.e., t = 1,this distribution is given asFSa(s) = e−λ (s) (48)×××××××Figure 4: Seismic hazard curves resulting from threesources.Figure 5: Probability distribution (CDF) of the spectralacceleration (Sa at 1 sec period).In summary, the earthquake hazard at the site inquestion is described as a marked Poisson processwith the rate of λ = 0.1 per year and the distributionof mark (Sa) is shown in Figure 5.4.2. Life-Cycle Damage CostThe damage cost, C, given the occurrence of anearthquake tends to be a function of the extent ofstructural damage, which in turn depends on theearthquake intensity, Sa. The seismic damage tostructure is typically quantified in terms of drift (D),which is related with Sa by a simple empirical rela-tionship, such as the following logarithmic relation(Cornell et al., 2002):lnD = a + b lnSa + ε (49)712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015where ε is a normally distributed random variablewith mean 0 and standard deviation of βD.The damage cost is then related to drift by a sim-ilar simple, empirical relation. The overall aim isto derive a conditional distribution of damage costgiven Sa, i.e., FC|Sa(c). However, the estimation ofthe moments of life-cycle cost, K(t), requires onlythe first two moments of C, as shown by the resultsgiven in Section 3.5. This simplification is a resultof two assumptions, namely, (1) damage cost is in-dependent of inter-occurrence time, and (2) earth-quake occurrences follow the homogeneous Pois-son process. What it means is that conditional re-lationships between the cost, damage and intensityare needed to estimate only the conditional mo-ments of the cost, E [Cn|s]. Given these conditionalmoments, an nth order moment can be calculated as(Porter et al., 2004):E [Cn] =∫SaE [Cn|s] dFSa(s)ds (50)where FSa(s) is given by Eq.(48). Using the mo-ments of the cost and occurrence rate, formulasgiven in Section 3.5 can be used to calculate var-ious estimates related to the expected cost5. CONCLUSIONSIn the life-cycle cost estimation, damage causedby external hazards introduces a great deal of un-certainty due to random nature of the occurrenceand intensity of hazards. The marked renewal pro-cess serves as a conceptual model of occurrenceand intensity of a hazard, whereas the cumulativecost of repairing and restoring the structure can betreated as a compound renewal process. This paperpresents systematic derivations of various measuresof the life-cycle cost, such as the expected value,variance, expected NPV and expected utility.In structural engineering, the homogeneous Pois-son process model is widely used which has sucha simple probabilistic structure that it completelyby passes the use of formal theory of renewal pro-cess. The paper emphasizes that the understandingof the renewal theory is necessary to analyze life-cycle cost measures in a general setting.6. REFERENCESAbrahamson, N. A. and Silva, W. J. (1997). “Empiricalresponse spectral attenuation relations.” Seismologi-cal Research Letters, 68(1), 94–127.Cha, E. J. and Ellingwood, B. R. (2013). “Seismic riskmitigation of building structures: The role of riskaversion.” Structural Safety, 40, 11–19.Cheng, T. and Pandey, M. D. (2012). “An accurate anal-ysis of maintenance cost of structures experiencingstochastic degradation.” Structure and InfrastructureEngineering, 8(4), 329–339.Cheng, T., Pandey, M. D., and van der Weide, J. (2012).“The probability distribution of maintenance cost ofa system affected by the gamma process of degra-dation: Finite time solution.” Reliability Engineeringand System Safety, 108, 65–76.Cornell, C. A., Jalayer, F., Hamburger, R. O., andFoutch, D. A. (2002). “Probabilistic basis for 2000sac federal emergency management agency steel mo-ment frame guidelines.” ASCE J. Structural Engineer-ing, 128(4), 526–533.Gallager, R. (2013). Stochastic Processes: Theory andApplications. Cambridge University Press, Cam-bridge, U.K.Koduru, S. D. and Haukaas, T. (2010). “probabilis-tic seismic loss assessment of a vancouver high-risebuilding.” Journal of Structural Engineering, 136(3),235–245.Liu, M., Wen, Y. K., and Burns, S. A. (2004). “Life cyclecost oriented seismic design optimization of steel mo-ment frame structures with risk-taking preference.”Engineering Structures, 26, 1407–1421.McGuire, R. (2004). Seismic Hazard and Risk Analysis.Earthquake Engineering Research Institute, Oakland,CA.Porter, K. A., Beck, J. L., Shaikhutdinov, R., Au, S.,Mizukoshi, K., Miyamura, M., Ishida, H., Moroi,T., Tsukada, Y., and Masuda, M. (2004). “Effect ofseismic risk on lifetime property value.” EarthquakeSpectra, 20(4), 1211–1237.Tijms, H. C. (2003). A First Course in Stochastic Mod-els. John Wiley & Sons, New York.8


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