Probabilistic Engineering Mechanics 38 (2014) 88101Contents lists available at ScienceDirectProbabilistic Engineering Mechanicshttp://d0266-89

n CorrE-m

msanchr.akhavajournal homepage: www.elsevier.com/locate/probengmechReliability analysis of shock-based deterioration using phase-typedistributions

Javier Riascos-Ochoa a, Mauricio Sanchez-Silva b,n, Raha Akhavan-Tabatabaei c

a Department of Civil and Environmental Engineering, Universidad de Los Andes, Carrera 1E 19A-40, Ed. Mario Laserna, Bogot 110111, Colombiab Department of Civil and Environmental Engineering, Universidad de Los Andes, Carrera 1E 19A-40, Ed. Mario Laserna (ML 630), Bogot 110111, Colombiac Department of Industrial Engineering, Universidad de Los Andes., Carrera 1E 19A-40, Ed. Mario Laserna, Bogot 110111, Colombiaa r t i c l e i n f o

Article history:Received 14 August 2014Accepted 18 September 2014

Keywords:DegradationShocksStructural lifetimeConvolutionMatrix-analytic methodsPhase-type distributionsx.doi.org/10.1016/j.probengmech.2014.09.00420/& Elsevier Ltd. All rights reserved.

esponding author.ail addresses: j.riascos96@uniandes.edu.co (J.ez@uniandes.edu.co (M. Sanchez-Silva),n@uniandes.edu.co (R. Akhavan-Tabatabaei).a b s t r a c t

This paper presents a model to estimate the lifetime of degrading infrastructure systems subject toshocks based on the family of Phase-type (PH) distributions. In particular, the paper focuses on damageaccumulation when both the inter-arrival time of shocks and their sizes are random. PH distributions areapplied to approximate any probability distribution with positive support; furthermore, their matrix-geometric properties allow to handle problems involving the calculation of convolutions (e.g., sum ofshock sizes). The proposed PH shock model relaxes the identically distributed assumption for the inter-arrival times and/or shock sizes. Besides, the model provides easy-to-evaluate expressions for importantreliability quantities such as the density function and the moments of the lifetime, and the mean andmoments of the cumulative shock deterioration at any time. In order to fit data or theoreticaldistributions to PH, the paper compares and discusses two PH fitting algorithms: the Moment Matching(MM) and the Expectation Maximization (EM) methods in terms of accuracy, computational efficiencyand the available information of the random variables to fit. Then, it provides an algorithm for thereliability estimation of infrastructures along with a study of its accuracy and efficiency; the results showacceptable execution times for most practical applications. Finally, the use of PH to handle degradation isillustrated with several examples of engineering interest; i.e., deterioration due to crack growth,corrosion, aftershocks sequences, among others.

& Elsevier Ltd. All rights reserved.1. Introduction

Life-cycle analysis and other time-dependent reliability pro-blems in structures and infrastructures require the evaluation ofits uncertain future performance [1]. For this purpose, manystudies focus on evaluating only the properties of the lifetimedistribution (e.g., mean time to failure) regardless of the processleading to failure. However, understanding and modeling thisprocess, described commonly as degradation, is an essential partof design and analysis of modern large infrastructure systems (e.g.,bridges) [2,3]; it is also of great value for lifecycle analysis and todefine optimum maintenance or monitoring programs.

Degradation models describe the damage accumulation processover time in any system as a result of its dynamic interaction withexternal (environmental) demands [4]. Frequently, degradationRiascos-Ochoa),models are divided into two basic mechanisms [57]: (1) progres-sive degradation, in which the system condition (system proper-ties) decreases continuously over the time, e.g., aging, wear orcorrosion; and (2) shock-based degradation, where the systemcondition is subject to sudden changes (decays) at specific pointsin time e.g., structures subjected to earthquakes or blasts [8]. Thispaper focuses only on modeling degradation as a result of shocks.This problem can be observed in a wide range of engineeringproblems; for example, the deterioration of structures (buildingsand bridges) due to earthquakes [6,7] and the damage in nuclearreactor components [9].

There are two main challenges that arise in the time-depen-dent reliability assessment of systems subjected to shock-baseddegradation. First, there is not enough and dependable data to fitthe distributions of the time between shocks, Xi, and of shocksizes, Yi. Secondly, even if this information is available, in mostcases the reliability estimation is numerically intractable. How-ever, the main difficulty arises in the evaluation of the distributionof the sum of these random variables, e.g., the time until theoccurrence of k consecutive shocks and the magnitude of

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J. Riascos-Ochoa et al. / Probabilistic Engineering Mechanics 38 (2014) 88101 89accumulated damage caused by them; i.e., total damage after agiven number of events of variable sizes. This evaluation requirescomputing convolutions, which in most cases does not have aneasy-to-evaluate expression and are difficult to approximatenumerically. This problem has been tackled in the literature byassuming that the inter-arrival times Xi are independent andidentically distributed, i.e., iid, as well as the shock sizes Yi, andthat these two processes are mutually independent [5,6,8,10].Moreover, the random variables are typically assumed to followeasy to handle probabilistic models such as exponential, gamma,or normal [7,9] which clearly diminish the actual description ofthe phenomena.

This paper presents a model for the reliability estimation ofsystems subject to cumulative shock degradation based on theformalism of Phase-type (PH) distributions. Thanks to the matrix-geometric properties of PH distributions [11], the model providesan alternative to compute the convolutions, and gives easy-to-evaluate expressions for the main reliability quantities: the relia-bility function, the probability density function and the momentsof the lifetime. Moreover, as PH distributions can fit any prob-ability distribution with positive support, with arbitrary accuracy,they can approximate general datasets of Xi and Yi, allowing theefficient reliability computation for general distributions of theserandom variables (under their PH approximation). Additionally,the PH shock model allows the evaluation of other importantquantities which in general are computationally expensive: therenewal function of the shock arrival process (i.e. the expectednumber of shocks by time t), their time-derivative or renewaldensity, and the moments of the structure's deterioration D(t).

Phase-type distributions have been used extensively in otherengineering areas such as queueing problems [12,13], finance [14]and reliability analysis [15,16]. Although PH distributions have alsobeen used to model systems subject to shocks [1720], the failuremechanisms considered in these studies are not cumulative asproposed in this paper, and they do not mention real applicationsfrom the analysis of the deterioration trends (i.e. mean of D(t)).

This paper is structured as follows: in Section 2 we review thedifferent cumulative-shock models and present the typical ap-proach for the time-dependent reliability estimation. The theory ofPH distributions is summarized in Section 3, while in Section 4 wepresent the specific details of using PH distributions for fittingexperimental data by means of the MM and EM algorithms;herein, we illustrate their differences with several examples. InSection 5 the assumptions of the PH shock model are providedalong with the derivations of the expressions for reliability. InSection 6 we present the numerical algorithm of the PH shockmodel and its efficiency. Finally, in Section 7 we provide twoapplications of PH distributions for modeling degradation. Section8 concludes the paper and gives directions for future research.2. Time-dependent reliability of cumulative shock degradation

2.1. Time-dependent reliability problem

Consider the case of deteriorating systems without any main-tenance or intervention actions to provide the basis for morecomplex cases. In lifecycle analysis, finding the distribution of thetime to failure of such systems is of particular importance [9,21].Two valuable reliability quantities can be obtained from thisdistribution: the mean time to failure (MTTF) and the probabilitythat the system meets its mission time (i.e., designed lifetime)[1,22]. The first result is frequently used in cost-based decisions,while the second in defining intervention (retrofitting) strategies.

Assume now that at time t0 the structure's initial condition isz. Since the condition may decay over time, at time t it can bedescribed by a random quantity V(t) that takes values in the set ofpositive real numbers (including zero), as in [6,7]. Such quantitycan be related with some structural property (e.g., capacity) or anyother performance indicator (e.g., resistance, stiffness, etc.). With-out loss of generality, we denote V(t) the structural or systemcapacity. The structure's lifetime, L, can be thought of as a randomvariable describing the length of time required for the structuralcapacity V(t) to reach a predefined performance threshold; e.g., an,with * a z . In practice, an is a prescribed design or operationthreshold that can be used to define limit states (e.g., service-ability, ultimate) or intervention policies; it is the conditionbeyond which the system's operation is deficient or deemedunsafe. Note that frequently * =a 0 but in some cases it might bereasonable to assume * >a 0.

Deterioration D(t) is defined as the loss of value of thestructural capacity V(t) at time t. Suppose that at t0, =D t( ) 0with probability 1. Then, the capacity at time t can be computed as

= *V t z D t a( ) max ( ( ), ), (1)

which takes into account the no-maintenance assumption. Then,the structural lifetime can be written as

= *L t V t ainf { 0: ( ) }, (2)

i.e., L is the infimum time t such that the capacity V(t) is below thethreshold an. Consequently, the (time-dependent) structure's re-liability can be expressed as: = >R t P L t( ) ( )m m , where tm is thelength of the system's mission; i.e., the lifetime for which thesystem is intended. Alternatively, at a specific point in time t, thereliability can be written in terms of the capacity V(t), or thedeterioration D(t) as

= > * = < *R t P V t a P D t z a( ) ( ( ) ) ( ( ) ). (3)

Note that in this case, computing the reliability or the struc-ture's lifetime requires keeping track of the system performanceover time, which is a complicated task due to the complexity anduncertainty of the change in the structural properties as damageaccumulates.

2.2. Cumulative shock degradation

As mentioned before, degradation is usually divided intoprogressive and shock-based. Progressive degradation can bemodeled with a (deterministic or random) deterioration rate

( , ), i.e., =D t d( ) ( , )t0

, where is a vector of (determi-nistic or random) parameters; as a stochastic process with in-dependent increments, like the gamma process, inverse gammaprocess or inverse gaussian process [7,2325]; or as a state-dependent stochastic process for which the increments dD(t) inthe deterioration depends on the accumulated damage D(t) at timet [26,27].

In shock-based degradation, the occurrence of the externalevents is modeled by shocks that occur at discrete points in time.Then, let us define the random variable Xi of the inter-arrival timebetween the i( 1)th and ith shocks, for = i 1, 2, . Additionally,each ith shock deteriorates the system by an amount or size Yi,which depends on the relationship between the external eventand the structural response. For notational convenience, supposethat =X 00 and =Y 00 with probability 1. This means that afictitious 0th shock with size zero occurs at time t0. Thus, thedeterioration in a cumulative shock-model can be expressed as [58,28,10], (see Fig. 1):

==

D t Y( ) ,(4)i

N t

i0

( )

Fig. 1. Cumulative degradation caused by shocks.

J. Riascos-Ochoa et al. / Probabilistic Engineering Mechanics 38 (2014) 8810190where N(t) is the counting process for the shock arrivals (i.e.,number of shocks by time t). Then, the structural capacity at time t(Eq. (1)) is given by

= = =

V t z D t z Y( ) ( ) .(5)i

N t

i0

( )

A general cumulative shock model assumes dependency amongthe processes X{ }i i 0 and Y{ }i i 0. Some examples of these processesare: the arrival of aftershocks after a major earthquake whichdepends on the magnitude of the main shock, i.e., Y1, [29,30]; allstate-dependent shock models for which the damage of subse-quent shocks depends upon the cumulated damage, i.e., Ykdepends on + + Y Yk1 1; or the case where the damage Ykdepends on the previous inter-arrival time Xk 1, like in the -shock model [31]. The reliability estimation of these models isnumerically expensive, and requires the evaluation of expressionsin terms of infinite sums and convolution integrals [28].

A common assumption is that the processes X{ }i i 0 and Y{ }i i 0are mutually independent, and the random variables Xi are iid, aswell as the Yi, i 0 [6]. In other words, both X{ }i i 0 and Y{ }i i 0constitute independent renewal processes (RP) [8]. This can be thecase of earthquakes with large inter-arrival times, whose times ofoccurrence and magnitudes are not correlated. Also, a large set ofmodels are based on the assumption that X{ }i i 0 are iid andexponentially distributed; and that Y{ }i i 0 has an arbitrary dis-tribution. This defines a Poisson Process (PP) for N(t) and aCompound Poisson Process (CPP) for the deterioration D(t). Thesemodels are more tractable mathematically due to their Markovianproperty. However, when Yi possesses a general distribution, thereliability computation is still difficult because of the convolutions.This problem can be solved by assuming Yi distributed exponen-tial, gamma or normal, for which the convolutions take simplerforms [7]. A more general case is a Non-Homogeneous PoissonProcess (NHPP) for the arrival of shocks, which allows deteriora-tion trends different from linear, like in the CPP model. This casehas been applied to model aftershock sequences [7,29] and crackgrowth [32].

2.3. Reliability of cumulative shock degradation

In order to assess the evolution of the system capacity overtime it is necessary to define both the cumulated deterioration Dkand the total time Sk until the kth shock (Fig. 1); i.e.,

= == =

D Y S Xand .(6)

ki

k

i ki

k

i0 0

We define FX and fX as the Cumulative Density Function (CDF)and probability density function pdf of the random variable X.Now, recall that =X 00 and =Y 00 , then, = =F t F t( ) ( ) 1S X0 0 and

= =F z F t( ) ( ) 1D Y0 0 , for >t 0. According to Eqs. (3) and (4) thestructural reliability at time t can be expressed as

= > = < * = < *

=

R t P L t P D t z a P Y z a( ) ( ) ( ( ) ) ,(7)i

N t

i0

( )

If it is assumed that the processes X{ }i i 0 and Y{ }i i 0 areindependent and conditioning out the number of shocks, Eq. (7)can be rewritten as

= < * =

= * =

=

=

=

R t P Y z a P N t k

P z a P N k

( ) ( ( ) ),

( ) ( ),(8)

k i

k

i

kk t

0 0

0

where *P z a( )k is the probability of survival until the kth shock,given an initial capacity z and failure threshold an. Without loss ofgenerality we can assume * =a 0 and, then, P z( )k is given by theCDF of the cumulated damage Dk (Eq. (6)), which depends only onz:

= < = < =

=

P z P Y z P D z F z( ) ( ) ( ),(9)

ki

k

i k D0

k

Note also that the term =P N k( )t is the probability of havingexactly k shocks by time t, which leads to the following expres-sion:

= = < < =

= =

+

+P N k P X t P X t F t F t( ) ( ) ( ).(10)

ti

k

ii

k

i S S0 0

1

k k 1

Furthermore, the pdf f(t) of the time to failure can be computedas [22]

= = =

= =

=

=

f tddt

R tddt

P z P N k

P zddt

P N k

( ) ( ) ( ) ( )

( ) ( ),(11)

kk t

kk t

0

0

with

= = +ddt

P N k f t f t( ) ( ) ( ). (12)t S Sk k 1

Recall that, for k0 and >t 0, =F 1S0 , then =f t( ) 0S0 . Finally,the raw moments of the lifetime distribution L, i.e., L[ ]m , are givenby

=

=

=

=

=

+

+

L t f t dt

P z t f t f dt

P z S S

[ ] ( )

( ) [ ( ) ]

( )[ [ ] [ ]].(13)

m m

kk

mS S

kk k

mkm

0

0 0

01

k k 1

J. Riascos-Ochoa et al. / Probabilistic Engineering Mechanics 38 (2014) 88101 912.4. Mean and moments of the deterioration, and mean and arrivalrate of shocks

Since most degradation mechanisms obey specific deteriora-tion trends, the mean of deterioration =D t D t( ) [ ( )] and its mth-moment D t[ ( )]m ( = m 1, 2, ) are also quantities of interest. Forinstance, deterioration due to corrosion of reinforced concreterebars follows a linear law, i.e., =D t ct( ) , [23] (see Section 7.2 formore examples). In order to obtain expressions for the mean andmoments of D(t) note first that Eq. (3) corresponds to the CDF of

the deterioration D(t) evaluated at z (with * =a 0), i.e.,=F z R t( ) ( )D t( ) . The pdf of D(t) is obtained by differentiating R(t)

in (8) with respect to z, and using (9):

= = = ==

=

f zddz

P z P N k f z P N k( ) ( ) ( ) ( ) ( ),(14)

D tk

k tk

D t( )0 0

k

and therefore, the moments of D(t) can be computed from:

= =

= =

=

=

D t z f z dzP N k

D P N k

[ ( )] ( ) ( )

[ ] ( ).(15)

m

k

mD t

kkm

t

0 0

0

k

Similar expressions have been reported in [9,8]. However, Eq.(15) is more general for it only requires independency in bothprocesses.

Another important quantity is the expected number of shocksby time t, =N t N t( ) [ ( )], and the rate of arrival of shocks

=n t d dt N t( ) / [ ( )]. When the process X{ }i i 0 is a renewal process,these quantities are known as the renewal function and renewaldensity [8], respectively. In the shock-model studied here they aregiven by

= =

= =

=

=

N t N t F t

n tddt

N t f t

( ) [ ( )] ( ) and

( ) [ ( )] ( ),(16)

kS

kS

1

1

k

k

which have the same form as those reported in [8].

2.5. Challenges in the reliability estimation of cumulative shockdegradation

Two main problems arise in the reliability analysis of theprevious shock-based model:1. find the appropriate probabilistic model for the random vari-ables: inter-arrival times Xi and shock sizes Yi; and2. numerically compute the reliability quantities: R(t), f(t), L[ ]m , D t[ ( )]m , N t( ) and n t( ).

The first problem can be addressed by identifying and model-ing the physical phenomena behind the process. This informationcombined with field data can be used to fit probabilistic models. Ingeneral, in most large civil infrastructure systems there is asignificant lack of dependable degradation data. The secondproblem is the complexity associated with the numerical compu-tations of the distributions (and moments) of the variables Sk andDk. These computations involve convolutions in the case that inter-arrival times X{ }i i 0 are independent, as well as shock sizes Y{ }i i 0.These convolutions are difficult to evaluate numerically and only afew cases have analytical or easy-to-compute solutions (e.g., [7,9]).

One of the objectives of this paper is to overcome these issuesby using Phase-type (PH) distributions. The basic notions of thesedistributions are explained in the next section, where weemphasize and discuss the properties that are particularly im-portant for estimating the reliability quantities; further informa-tion on PH distributions can be found, for instance, in [11,15].3. Phase-type distributions: basic concepts

3.1. Definition and examples

Phase-type (PH) distributions are based on the method ofstages, first introduced by A. K. Erlang and later formalized byNeuts [15]. This family of distributions model random times T (orother discrete or continuous quantities) as being made up of anumber of exponentially distributed segments and exploit theresulting Markovian structure to simplify the analysis. Then, theexponential distribution is used as the building block to constructmore complex distributions.

A PH distribution is the distribution of the time until absorp-tion X in a Continuous Time Markov Chain (CTMC) [11]. Thus,consider a CTMC defined on the finite state-space= S n{0, 1, 2, , }, where state 0 is an absorbing state and the last

n states are transient. Define the initial probability vector of theCTMC as [ , ]0 , where is a row vector of size n, and theinfinitesimal generator matrix as:

=Q

0t T0 ,

(17)

where T is an nn matrix, and t is a column vector of size n. Thematrix T has the exponential transition rates between the tran-sient states. Then, a random variable X is said to be distributed

PH T( , ) of order n, denoted X PH T( , ), if X is the time untilabsorption in the CTMC previously defined. Note that since Q isthe generator of a CTMC and 0 is an absorbing state, the elementsof the matrix T and vector t satisfy [11]:

< + = i j nT T t T 1 t 00, 0, 0, and for 1 ,ii ij i

where 1 is a column vector of 1's and is the matrix product. Wealso have:

+ =1 1.0

Note that the simplest PH distribution is the exponentialdistribution with only one phase. Other examples are presentedbelow, for clarity:

Example 1: Erlang distribution Consider the case of two iidexponential random variables X1 and X2 with X exp ( )i , i1,2.Then = +S X X1 2 is the 2-phase Erlang distribution with mean 2/ .Fig. 2(a) shows the transition rate diagram of this PH distribution.At t0 the system is in state 1 and stays there for a randomamount of time, which is exponentially distributed with rate .Then, it moves into state 2 and remains there for another amountof time that is also exponentially distributed with rate . Finally, inthe next step it is absorbed into state 0. The total time since themovement begins until it is absorbed in 0 is distributed PH T( , ),with parameters and associated generator matrix given by

= =

=

T Q[1 0],

0and

0 0 00

0.

Example 2: Hyper-exponential distribution A n-state Hyper-exponential distribution results from the convex mixture of nexponential distributions, each with rate k, = k n1, , . Fig. 2(b) shows a 2-phase hyper-exponential distribution. Then, thesystem might initially move to state 1 with probability 1 andspend an exponential amount of time 1 in that state; or it canmove directly to state 2 with probability = 12 1 and spend

Fig. 2. Particular cases of PH distributions: (a) 2-phase Erlang distribution and(b) 2-phase hyper-exponential distribution.

J. Riascos-Ochoa et al. / Probabilistic Engineering Mechanics 38 (2014) 8810192some time in state 2 with rate 2. Thus, the total time spent in thissystem is distributed PH T( , ) with

= =

=

T Q,

0

0and

0 0 00

0.1 2

1

21 1

2 2

Other PH distributions such as Coxian [33], hyper-Erlang [34]and acyclic PH distributions [35] have more complex structuresand have been described elsewhere.3.2. Properties of PH distributions

There are three properties of PH distributions that are ofparticular use for modeling degradation and help to solve thechallenges presented in Section 2.5:1.Fig.algorDenseness property: PH distributions are dense in the set ofcontinuous density functions with support on [0, ) (Latoucheet al., Theorem 2.6.5 [11]). The term dense refers to thecomplete coverage of the continuous density functions (inthe sense of weak convergence of distribution), and meansthat there is a PH distribution arbitrarily close to any contin-uous distribution. A number of efficient algorithms have beenproposed in the literature to fit a PH distribution to arbitrary(positive) datasets (numerically generated from any continu-ous distribution or from field measurements) [35,33,34,36].This denseness property and the available fitting algorithmscan solve the first challenge presented in Section 2.5, aboutfinding the appropriate probabilistic model for the r.v's Xi andYi.2. Closure under convolutions: The convolution of PH distribu-tions is also a PH distribution. In other words, the sum of twoor more independent PH distributed random variables is alsoPH distributed. It has been proved (Theorem 2.6.1 in [11]) thatif X1 and X2 are two independent random variables withdistributions PH T( , )1 1 of order n1 and PH T( , )2 2 of order n2,3. Structures of the PH distributions used. For the MM method in [36]: n-phase generithm in [34], hyper-Erlang distribution.the sum +X X1 2 is distributed PH s S( , ) of order +n n1 2 with

= =

s S

T t

Tand

0,

(18)1 1

02

1 1 2

2

where + =T 1 t 01 1 . Note that the sum of k independent PHdistributed random variables is also PH distributed, and its PHrepresentation can be obtained by applying Eq. (18) succes-sively. Section 5.2 provides a simple deduction of this propertyapplied to the PH representation of the cumulated damage Dk.3. Specific form of the distributions and moments: For a PH randomvariable X PH T( , ) the CDF, pdf and the mth moment( = m 1, 2, ) are as follows [11]:

= = F x P X x xT 1( ) ( ) 1 exp ( ) , (19)

= f x xT t( ) exp ( ) , (20)

= ! X m T 1[ ] ( 1) , (21)m m m

where exp ( ) is the matrix exponential operator, defined as !=

x x kT Texp ( ) ( ) /kk

1 when applied on xT [37]. There areefficient algorithms to compute exponential matrices, whichare discussed briefly in Section 6.3.In the next section, we will discuss the problem of fitting anarbitrary density function (or dataset) to a PH distribution.4. Fitting PH distributions to shock data

The objective of fitting is to obtain a PH representation of arandom variable X, which will in turn allow the efficient computa-tion of the reliability quantities. In general, PH fitting algorithmsare classified into two major groups: Moment Matching (MM)algorithms [33,35,36,38] and Maximum Likelihood or ExpectationMaximization (EM) algorithms [34]. MM algorithms use somemoments of the distribution or the dataset in order to find theparameters of the PH representation. EM algorithms use the wholedataset which can be empirically generated (e.g., from fieldmeasurements), or artificially generated from a known distribu-tion (e.g., lognormal, gamma, Weibull). The selection of a parti-cular algorithm depends on three important aspects: the availableinformation, e.g., the number of data points or moments; the levelof accuracy needed for the PH fitting; and the computational effortfor the fitting and later for the evaluation of reliability. Theseaspects are evaluated for the MM and EM algorithms in thefollowing sections.alized Erlang ( < 0 COV 0.52 ) and 2-phase Coxian ( >COV 0.52 ). For the EM

Table 1Accuracy and efficiency of the MM and EM algorithms applied to

=Y LN ( 20, COV )i Y Y for different values of COVY; the term n is number of statesof the PH representation and SKY is the skewness. The results are presented interms of the percentage of error of the PH representations compared with those ofthe generated dataset.

MM algorithm =COV 0.2Y =COV 0.5Y =COV 0.8Y =COV 1.0Y =COV 2.0Y

n: Number ofstates

25 4 2 2 2

ET: Executiontime (s)

0.01 0.01 0.01 0.07 0.07

%Error Y 0.0% 0.0% 0.0% 0.0% 0.0%%Error COVY 0.0% 0.0% 0.0% 0.0% 0.0%%Error SKY 38% 38% 49% 46% 0.0%

EM algorithm

=N 10data 5 =COV 0.2Y =COV 0.5Y =COV 0.8Y =COV 1.0Y =COV 2.0Y

n: Number ofstates

25 10 10 10 10

M: Number ofbranchesanalyzed

1, 2, 3 1, 2, 3 1, 2, 3 1, 2, , 10 1, 2, , 10

ET: Executiontime (s)

6.9 5.1 6.5 40 30

%Error Y 0.0% 0.0% 0.0% 0.0% 0.0%%Error COVY 0.1% 0.7% 0.4% 2.0% 6.6%%Error SKY 34% 10% 8.6% 15% 53%

EM algorithm=N 50data =COV 0.2Y =COV 0.5Y =COV 0.8Y =COV 1.0Y =COV 2.0Y

n: Number ofstates

25 10 10 10 10

M: Number ofbranchesanalyzed

1, 2, 3 1, 2, 3 1, 2, 3 1, 2, , 10 1, 2, , 10

ET: Executiontime (s)

2.2 1.4 1.5 12 27

%Error Y 0.0% 0.0% 0.0% 0.0% 0.0%%Error COVY 0.1% 0.7% 4.5% 2.1% 5.0%%Error SKY 57% 27% 30% 20% 49%

J. Riascos-Ochoa et al. / Probabilistic Engineering Mechanics 38 (2014) 88101 934.1. MM algorithm

The MM algorithm presented here was initially proposed byKharoufeh et al. [36]. It uses different PH structures according tothe Coefficient Of Variation (COV) of the dataset (Fig. 3). Thus, forthe range < 0 COV 0.52 the algorithm uses only the first twomoments, because for low COV's the lower moments influence thedistributions more than the higher moments. For this range, thenumber of PH phases n is given by n n1/ COV 1/( 1)2 . For theRange < 0.5 COV 12 the algorithm uses the first two moments,and the PH representation involves only 2 states, which isconvenient in terms of computational efficiency. Finally, for therange >COV 12 the fitting uses the first three moments. This isbecause in this case the third moment is necessary to reduce themaximum relative error [36].

4.2. EM algorithm

Within this context, the EM algorithmwas initially proposed byThmmler et al. [34]. This algorithm is based on the hyper-Erlangdistribution (HErD), which is a mixture of Erlang distributions(Fig. 3). It consists of = m M1, 2, , Erlang-branches, each con-sisting of rm phases (states) with scale parameter m. In the HErD,the process starts with the system in one of the M first states ofeach Erlang-branch according to the initial probability vector = [ , , ]M1 with 0m and == 1m

Mm1 . Given the number

of phases n in the HErD, the EM algorithm maximizes the log-likelihood to find the best n-state HErD structure (i.e. the numberof Erlang branches M and the vector = r r r r[ , , , ]M1 2 of thenumber of states in each Erlang branch such that == r nm

Mm1 ),

the initial probability vector and the rates = [ , , ]M1 .

4.3. Examples of the EM and MM algorithms

The objective of the following example is to illustrate thedifferences between the EM and MM algorithms in terms of thethree aspects mentioned above: information available for thefitting, fitting accuracy and execution times.

We first consider fitting a log-normal (LN) distribution, whichis a convenient choice when modeling shock sizes in structuraldegradation as a result of earthquakes [6,39]. For this example, fivedifferent models of shock sizes, Y, were studied; in all cases themean value is the same i.e., = 20Y , but the COV varies taken thefollowing values: =COV 0.2Y , 0.5, 0.8, 1.0, 2.0. A dataset of

=N 10data 5 samples using Monte Carlo simulation was generatedfor Y, and used for the EM fitting; furthermore, based on themoments of this dataset the MM algorithm was also performed.

In addition, another dataset with =N 50data was generated andfitted with the EM algorithm in order to test it with sparse dataand compared with the results for =N 10data 5. The results of thefittings are shown in Table 1. Also, for the EM fitting, this tableshows the number of different branches M used to estimate thebest HErD [34]. The pdf's of the PH fittings and the empiricaldatasets are shown in Fig. 4.

PH distributions are very versatile and can be used also to fitdata traces with irregular shapes. In this example a dataset for amultimodal random variable Y was generated and fitted with theMM algorithm and the EM algorithm varying the number ofphases: n 10, 20, 30, 50. Fig. 5 shows the distribution of themultimodal data and the pdf's of the PH representations.

The results of the experiment and previous studies show thatthe MM algorithm is appropriate when there is not enough datasamples but there is information of some of the moments of therandom variable. However, even if data is available, in some casesan exact fitting is not required and the first moments are enoughto approximate the solution. On the other hand, the EM algorithmis based only on the dataset available and results are in generalacceptable even in the case where data is scarce (see Table 1 andFig. 4).

The numerical experiments show that the MM algorithm per-forms better for fitting the moments. The EM algorithm also fitsthe mean Y exactly and the COVY with low errors. In terms of thepdf's of the original and PH-fitted, the EM algorithm is the bestoption (Figs. 4 and 5), with similar results in the case of sparsedatasets. Better results are obtained with greater number of statesn as it is clear with the multimodal distribution.

Finally, in terms of the computational efficiency (executiontimes), the MM algorithm gives better results than the EMalgorithm by a factor of approximately 100 (Table 1). Sparse data

=N 50data is fitted faster (around 4 times) than complete data=N 10data 5 with the EM algorithm. The EM algorithm performs

slower for greater number of states n and number of analyzedbranches M.5. Reliability estimation of cumulative shock degradationusing PH distributions

The problem of evaluating the convolution, which is the maincomputational difficulty when modeling shock-based degradation,can be solved by fitting the distributions of inter-arrival times Xiand shock sizes Yi to PH distributions. Thus, in this section we

Fig. 4. Distribution of the datasets from Y LN ( ,Y COV )Y and pdf's of the PH fitting (EM and MM algorithms).

Fig. 5. Distribution of the multimodal data Y and pdf's from the MM and EM fittingfor different number of PH phases n.

J. Riascos-Ochoa et al. / Probabilistic Engineering Mechanics 38 (2014) 8810194present easy to evaluate expressions for the reliability quantitiespresented in Section 2 based on the properties of PH distributionspresented in Section 3.2.

5.1. Description of the PH-shock model

In order to build the PH-shock model some assumptionsconcerning the random variables Xi and Yi and their PH represen-tations are required; these are

Assumption 1. The processes of inter-arrival times X{ }i i 0 andshock sizes Y{ }i i 0 are independent.

Assumption 2. The inter-arrival times X{ }i i 1 are independent butnot necessarily identically distributed and follow PH distributionswith representation and generator matrix:

=X PH nT Q

0t T

( , ) of order ,0

.(22)

i i i X Xi i

i i

Assumption 3. The probability that the Markov chain associatedwith the PH distribution of Xi starts in its absorbent state is zero;i.e., = i0, 1i0 . Therefore, Xi is always strictly greater than zero, i 1 (two shocks cannot arrive at the same time).

Assumption 3. Shock sizes Y{ }i i 1 are independent but not neces-sarily identically distributed and follow PH distributions withrepresentation and generator matrix:

=Y PH nY Q

0y Y( , ) of order ,0

,(23)

i i i Y Yi i

i i

The probability i0 that the Markov chain associated with thePH distribution of Yi starts in its absorbent state can be differentfrom zero, i 1. Therefore, each ith shock may not produce anydamage (i.e., Yi0) with probability given by i0.

Assumption 4. The deterioration is given by the cumulative shockmodel described in Section 2.2.

As defined, the PH-shock model relaxes the condition ofidentical distributions (i.e., inter-arrival times and shock sizes asRP) which is assumed in most of the shock models (e.g., [6,7,9,8])as was explained in Section 2.2. Also, this assumption canapproximate the general (and more complex) dependency case,by defining different distributions for the sequences X{ }k k 1 and

J. Riascos-Ochoa et al. / Probabilistic Engineering Mechanics 38 (2014) 88101 95Y{ }k 1. Several examples in Section 7.2 demonstrate that typicaldependent processes can be modeled with this assumption, whichcannot be addressed with the simpler iid case. The PH-shockmodel facilitates the reliability estimation in these cases, as itprovides easy-to-evaluate expressions for the convolutionsinvolved.5.2. Distribution of the accumulated damage Dk

The total damage after k shocks, i.e., Dk, is the sum of the shocksizes Y{ }i i k. As discussed in Section 3.2, the closure property of PHguarantees that if each shock is PH distributed, Dk is also PHdistributed. Suppose D PH d D( , )k i i . To find this representation,consider first the simple case of D2, that is, the distribution of theaccumulated damage after two successive shocks; thus, by usingEq. (18):

= =

d D

Y y

Y[ , ],

0,2 1 1

02 2

1 1 2

2

of order = +n n nD Y Y2 1 2. This means that the accumulated damageuntil the second shock is described by joining the two Markovchains associated with the first two shock sizes Y1 and Y2 (Fig. 6).The vector of initial probabilities d2 indicates that the Markovchain starts with probabilities 1 in the first nY1 states (correspond-ing to the states of Y1) or with probabilities 1

02 in the last nY2

states (which corresponds to the Y2 states). In the first case thesystem evolves according to the matrix Y1. The transition from thestates of Y1 to Y2 is obtained when absorption to the state 0 of Y1happens. Remember that the absorption rates from each state of Y1are described by the vector y1. Then, the transition rate from thefirst nY1 states to the next nY2 states of the second shock can beobtained by multiplying (element by element) the initial vector 2by the Markov chain associated with the second shock (Fig. 6).

In general, for Dk the representation is given by

=

=

D

Y y

Y y

Y y

Y

d

0

0 0

0 ,

[ , , , ( ) ], (24)

k

k k k

k

k k k

1 1 2

2 2 3

1 1

1 10

2 10

10

of order = + + +n n n nD Y Y Yk k1 2 . This means that the PH repre-sentation of Dk has nDk number of phases which results fromjoining the PH representations of each Y{ }i i k (Fig. 6).Fig. 6. Description of the PH structure asThen, according to Eq. (19) the expression for =P z F z( ) ( )k Dk ,>z 0, is given by

= =

=P z F z

z k

k

d D 1( ) ( )

1 exp ( ) , 1

1, 0 (25)k D

k kk

Remember that the case k0 corresponds to no shocks, for whichthe survival probability P z( )k is equal to 1.

5.3. Distribution of the time until the kth shock (FSk)

As Sk is given by the sum of the inter-arrival times X{ }i i k whichare PH distributed, then Sk also follows a PH distribution given byPH s S( , )k k . The same procedure used to compute the distributionof the accumulated damage Dk can be applied to find PH s S( , )k k , bymaking the following substitutions: Y Ti i, i i, n nY Xi i,

D Si i, y ti i and d si i in Fig. 6 and Eq. (24); i.e.,

=

=

S

T t

T t

T t

T

s

0

0 0

0

, [ , 0, , 0]

(26)

k

k k k

k

k

1 1 2

2 2 3

1 1

1

of order = + + +n n n nS X X Xk k1 2 . Then, F t( )Sk and f t( )Sk , >t 0,are given by Eqs. (19) and (20):

=

=F t

t k

k

s S 1( )

1 exp( ) , 1

1, 0 (27)S

k kk

=

=f t

t k

k

s S S 1( )

exp( ) ( ), 1

0, 0, (28)S

k k kk

Then, by substituting Eq. (27) into Eqs. (10) and (28) into Eq.(12) expressions for the distribution of the number of shocks andits time-derivative are obtained:

= =

=+ +

P N kt t k

t k

s S 1 s S 1

s S 1( )

exp( ) exp( ) , 1

exp( ) , 0, (29)t

k k k k1 1

1 1

= =

=+ + +

ddt

P N k

t

t

k

t k

s S S 1

s S S 1

s S S 1

( )

exp( ) ( )

exp( ) ( ),

1

exp( ) ( ), 0. (30)

t

k k k

k k k1 1 1

1 1 1

5.4. Evaluation of the reliability quantities

The results found in Sections 5.2 and 5.3 can be used to find theexpressions for the reliability function, R(t), the lifetime density,f(t), and its moments L[ ]m (i.e., Eqs. (8), (11) and (13)). Thesesociated with damage accumulation.

J. Riascos-Ochoa et al. / Probabilistic Engineering Mechanics 38 (2014) 8810196expressions are based on computing P z( )k , =P N k( )t and=d dtP N k/ ( )t . The quantities of interest are

= =

=

=

=

+

R t P z P N k

z F t F td D 1

( ) ( ) ( )

[1 exp( ) ][ ( ) ( )],(31)

kk t

kk k S S

0

0k k 1

= =

=

=

=

+

f t P zddt

P N k

z f t f td D 1

( ) ( ) ( )

[1 exp( ) ][ ( ) ( )],(32)

kk t

kk k S S

0

0k k1

=

= ! !

=

+

=

+ +

L P z S S

P z m ms S 1 s S 1

[ ] ( )[ ( ) ( )]

( )[ ( ) ( ) ],(33)

m

kk k

mkm

kk k k

mk k

m

01

01 1

1 1

In particular, the Mean Time To Failure (MTTF) is given bytaking m1 in Eq. (33); i.e.,

= ==

+=

+MTTF P z S S P z X( )( [ ] [ ]) ( ) [ ],(34)k

k k kk

k k0

10

1

= =

P z T 1( ) ( ) .

(35)kk k k

0

1

Note that when the Xi's are identically distributed withPH T( , ):

= =

MTTF P zT 1( ) ( ).(36)k

k1

0

In addition, the moments D t[ ( )]m and the mean D t( ) of thedeterioration at any time t, in Eq. (15), are given by

= =

= ! =

=

=

D t D P N k

m P N kd D 1

[ ( )] [ ] ( )

( ) ( ),(37)

m

kkm

t

kk k

mt

1

1

1

= ==

D t P N kd D 1( ) ( ) ( ).

(38)kk k t

0

1

Finally, the expected number of shocks by time t, =N t( ) N t[ ( )], and the rate of arrival of shocks =n t d dt N t( ) / [ ( )] (Eq. (16)) is

= = =

=

N t F t ts S 1( ) ( ) (1 exp( ) ),(39)k

Sk

k k1 1

k

= = =

=

n t f t ts S S 1( ) ( ) exp( ) ( ).(40)k

Sk

k k k1 1

k

6. Numerical computation of the PH-based shock model

6.1. Approximation to the infinite sums of the reliability quantities

The expressions obtained in Eqs. (31)(33) imply the evaluationof an infinite sum of terms, each one associated with one shock k,and the evaluation of an infinite number of matrix exponentials

tSexp( )k , zDexp( )k and vectors sk and dk. In order to overcome thisissue, the infinite sums are truncated in a Kth term, obtainingapproximations for the quantities R(t), f(t), and L[ ]m given by= ==

R t P z P N k( ) ( ) ( ),(41)

Kk

K

k t0

= ==

f t P zddt

P N k( ) ( ) ( ),(42)

Kk

K

k t0

= =

+L P z S S[ ] ( )( [ ] [ ]).(43)

Km

k

K

k km

km

01

It can be shown that the correct value R(t) is bounded frombelow by R t( )K 1 and from above by +R t( )K 1 for all >t 0, with Ksuch that the probability of survival 0 and >z 0 (see Theorem A.1 in Appendix A). Inother words, for the numerical computation it is sufficient to takeinto account the first K( 1) shocks in order to have a probabilityof failure R t(1 ( )) greater than 1 . Values of error within theorder of = 10 6 can be considered acceptable. For the quantities D t N t[ ( )], ( )m , and n t( ) the truncation depends on the time ofevaluation t. In these cases it is reasonable to carry out the analysisuntil the shock that leads to a relative error smaller than 108.6.2. Description of the numerical computation of the reliabilityquantities

The numerical computation of RK(t), fK(t) and L[ ]K m consists offour stages that are described in Table 2. A similar algorithm isfollowed for the quantities D t N t[ ( )], ( )m , and n t( ).

6.3. Efficiency of the PH shock algorithm

The numerical solution described above requires the evaluationof Dexp ( )k and Sexp( )k , which can be computed numerically usingexisting software (e.g., function expm in flMATLAB ). According to[40], the efficiency of this function is O n( )3 , where n is the size ofthe matrices (Dk or Sk). Since the size of these matrices grows asshocks accumulate (Eq. (6)), the algorithm efficiency also dependson the truncation; i.e., the total number of evaluated shocks K.Fig. 7 shows that the efficiency decreases with K and the size n ofthe matrices. However, the execution times ET for the PH algo-rithm are reasonable for failures with number of shocks K100 orless (

Table 2Pseudocode for the reliability estimation (Reliability function, pdf of lifetime andmoments of lifetime) with the proposed PH-shock model.

Stage 1: Definition of the system (fitting)(1) PH-representation: Define the PH-representation of the inter-arrival

times and shock sizes: T{ , }k k , Y{ , }k k

(2) Define , z and t{ }i , with >t 0i , as the times of evaluationStage 2: Computation of P z( )k(3) Set k0, =P z( ) 10(4) WHILE >P z( )k(5) Increase k in 1(6) Compute the PH-representation of the cumulative damage Dk until the

kth shock: d D{ , }k k(7) Compute the exponential matrix zDexp ( )k(8) Compute = P z zd D 1( ) 1 exp( )k k k(9) Set the number of computed shocks KkStage 3: Computation of =P N k( )t AND =P N k( )

ddt t

(10) Set =F t( ) 1S0 , =f t( ) 0S0 , t t{ }i(11) FOR k1 to = +k K 1(12) Compute the PH-representation of the time Sk until the kth shock:

s S{ , }k k(13) FOR all t t{ }i(14) Compute the exponential matrix tSexp ( )k(15) Compute = F t ts S 1( ) 1 exp( )Sk k k and

= f t ts S S 1( ) exp( ) ( )Sk k k k

(16) Compute = = P N k F t F t( 1) ( ) ( )t Sk Sk1 and

= = P N k f t f t( 1) ( ) ( )ddt t Sk Sk1

(17) END(18) ENDStage 4: Computation of the reliability quantities(19) FOR all t t{ }i(20) Compute = ==R t P z P N k( ) ( ) ( )K k

Kk t0

(21) = ==f t P z P N k( ) ( ) ( )K kK

kddt t0

(22) END(23) Compute = = +L P z S S[ ] ( ) [ ] [ ]K m kK k kn kn0 1

Fig. 7. Execution times ET (seconds) of the PH-shock algorithm in Table 2 as afunction of K for several sizes n of the PH matrices of the inter-arrival time Xi andshock sizes Yi (supposing they have the same size: = = =n n n 2, 4, 16Xi Yi , for alli 1).

J. Riascos-Ochoa et al. / Probabilistic Engineering Mechanics 38 (2014) 88101 977.1. Example 1: reliability assessment of a structure subjected toearthquakes

Consider a structural system that deteriorates as a result ofearthquakes; with inter-arrival times Xi distributed exponentialwith mean = 10yearsX and shock sizes Yi, lognormal (LN)distributed with mean = 20Y (in appropriate capacity units)and coefficient of variation =COV 0.2Y , 0.5, 0.8, 1.0, 2.0 (as inSection 4.3). The initial system capacity is z100 and the failurethreshold * =a 0; the purpose is to evaluate the reliability function.

Note first that Xi is already PH distributed since the exponentialdistribution is the simplest form of PH. For Yi the PH approxima-tions obtained in Section 4.3 were used (i.e., both the MM and EMfitting algorithms with =N 10data 5). The results of the analysis aresummarized in Table 3 and the pdf's f(t) of the lifetime in Fig. 8.They show very close fits of both PH approximations with MonteCarlo simulations. In particular, the EM fitting shows relativeerrors of around 1% in MTTF and COVL for all of the values ofCOVY. In general, EM fitted better than the MM. Besides, MM hasrelative errors for COVL above 3%, while the relative errors of theEM are around 1%. These relative errors in both fittings might beconsidered small for the most practical applications. The differ-ences between EM and MM for values of >COV 0.5Y are due to thefact that EM uses n10 PH phases for the fitting, while MM uses2 or 3. In contrast, the results for c 0 and >b 0. If degradation is diffusion-con-trolled, then b0.5, which gives a square root relationship; ifdegradation is by sulphate attack on concrete, b 1 (usually b2which defines a cuadratic law); corrosion of reinforcement followsa linear law (b1); and for creep in concrete, =b 1/8 (see moredetails in [4,23]). Another example is the case of fatigue inmaterials subjected to cyclic loading, which could be modeled asa cumulative deterioration shock model [41]. Finally, an interestingapplication is the case of aftershocks after a major earthquake. Inthis case, the rate of their arrival decreases over time following thewell known Omori's Law [29,30]: = + n t K t c( ) ( ) 1, where K and care constants. Then, the total number of aftershocks N(t) in thetime interval between 0 and t is given by: =N t( ) = +n s ds K t c( ) ln ( / 1)t0

. If each aftershock produces a mean

Table 3Reliability estimation of a structure subject to cumulated earthquakes, with inter-arrival times =X EXP ( 10)i X and shock sizes =Y LN ( 20,i Y COV )Y with COVYvariable. Results from Monte Carlo simulation and with the PH shock model by using the PH representations of Yi with the MM and EM algorithms.

Monte Carlosimulation =COV 0.2Y =COV 0.5Y =COV 0.8Y =COV 1.0Y =COV 2.0Y

ET: Execution time (s) 3.2 3.0 2.6 2.8 3.3MTTF 54.8 56.2 58.0 60.1 69.4COVL 0.44 0.47 0.50 0.52 0.59

PH shock model with MM algorithm =COV 0.2Y =COV 0.5Y =COV 0.8Y =COV 1.0Y =COV 2.0Y

n: Number of PH states 25 4 2 2 2K: Number of shocks 8 12 17 20 22ET: Execution time (s) 0.1 0.11 0.11 0.14 0.15MTTF (%Error) 55.2 (0.7%) 56.2 (0.1%) 58.1 (0.2%) 60.0 (0.2%) 67.4 (2.9%)COVL (%Error) 0.44 (0.1%) 0.47 (0.2%) 0.52 (3.4%) 0.55 (5.9%) 0.54 (7.9%)

PH shock model with EM algorithm =COV 0.2Y =COV 0.5Y =COV 0.8Y =COV 1.0Y =COV 2.0Y

n: Number of PH states 25 10 10 10 10K: Number of shocks 8 11 14 18 22ET: Execution time (s) 0.1 0.12 0.21 0.22 0.36MTTF (%Error) 55.2 (0.7%) 56.2 (0.1%) 58.8 (1.3%) 59.9 (0.2%) 69.6 (0.3%)COVL (%Error) 0.44 (0.1%) 0.47 (0.3%) 0.50 (0.2%) 0.53 (1.2%) 0.60 (0.9%)

Fig. 8. pdf's of the lifetime f(t) via Monte Carlo simulation and the PH shock model (with the MM and EM algorithms for the fitting).

J. Riascos-Ochoa et al. / Probabilistic Engineering Mechanics 38 (2014) 8810198damage Y , the total deterioration until time t is given by

= +D t N t K t c( ) ( ) ln ( / 1). (45)Y Y

7.2.2. Modeling degradation trends using the PH-shock modelIf we consider nonidentically distributed inter-arrival times or

shock sizes, different functional forms of D t( ) can be obtained. Thisanalysis can be carried out by using Eq. (38) to evaluate D t( ). Thenumerical solution used truncates the infinite sum to the Kthshock for which the increment in the partial sums is less than anerror parameter = 10 8. The approach followed in this analysisconsists of two steps:1. Define PH-distributions for the first inter-arrival time X1 andshock size Y1 as:

X PH Y PHT Y( , ) and ( , ). (46)1 1 1 1 1 1

Fig. 9. (a) Trends of D(t) (from Eq. (37)) for different definitions of Xk and Yk ( k 2) obtained from de distributions of X1 and Y1 (Table 4). The distributions of X1 and Y1 wereobtained by the MM algorithm by supposing = 2.5X1 days, =COVX1 =COV 0.5Y1 , and = 5Y1 . (b) pdf of the lifetime of a systemwith the previously defined degradations withinitial capacity z100 and threshold =a 0, obtained with the PH-shock algorithm Table 2.

J. Riascos-Ochoa et al. / Probabilistic Engineering Mechanics 38 (2014) 88101 992.TableDiffeTk an

each

=Xkd

X1

X1

X1

X1

kX1

k X7

akFor the next shocks ( k 2), define Xk equally distributed asg k X( ) 1 and Yk as h k Y( ) 1, i.e.:

= =X g k X Y h k Y( ) and ( ) , (47)kd

kd

1 1

where g(k) and h(k) are functions of the shock number k.Hence, the PH-representations, distributions and means of Xkand Yk are given by

= =

= =

X PH g k F t F t g k g k

Y PH h k F y F y h k h k

T

Y

( , / ( )), ( ) ( / ( )), ( ),

( , / ( )), ( ) ( / ( )), ( ).(48)

k X X X X

k Y Y Y Y

1 1

1 1

k k

k k

1 1

1 1

Note that the PH-matrices Tk and Yk change for each k, butkeep the sizes nX and nY for the first shock k1. However, theinitial probability vectors k and k remain equal to 1 and 1,respectively.As an example, suppose that =X Xkd

1 and =Y kYkd

1 for all k 1(i.e. =g k( ) 1 and =h k k( ) in Eq. (47)). Hence, X PH T( , )k 1 1 withmean =X Xk 1 and Y PH kY( , )k 1 1 with mean = kY Yk 1, k 1. Theresults by applying Eq. (38) for different PH representations of X1and Y1 show that for large ratios t( / )T the asymptotic behavior ofD t( ) is cuadratic. More precisely, the empirical result from the4rent definitions of the distributions of inter-arrival times Xk and shock sizes Yk ( k 2d Yk, and means Xk and Yk (Eq. (48)). Associated deterioration trends D t( ) (asymp

definition.

=Ykd PH-matrix Tk PH-matrix Yk Mean of Xk Xk Mean of

Y1 T1 Y1 X1 Y1

kY1 T1kY

11

X1 k Y1

k Y2 1 T1k

Y12

1 X1 k Y2 1

b Yk 1 1 T1b

Y1k 1

1 X1 bk Y1 1

Y1 Tk1

1Y1 k X1 Y1

1Y1 T

k

17 1

Y1 k X7 1 Y1

X1 1 Y1

Tak11

1Y1 ak X1 1

Y1simulations shows that: D t t( ) ( / )Y T12

21 when t( / )T . Note

that this particular case describes the deterioration trend ofconcrete due to sulphate attack, presented in Eq. (44).

In Table 4 we present some other definitions for Xk and Yk(by varying g(k) and h(k)), their corresponding PH representations(matrices Tk and Yk), the (asymptotic) deterioration trends andthe specific degradation mechanisms that can be modeled.Fig. 9(a) shows the plots of D t( ) for particular examples fromTable 4 by applying Eq. (37). For all the cases the mean of X1 was = 2.5X1 days with coefficient of variation =COV 0.5X1 , and shocksize Y1 with mean = 5Y1 and =COV 0.5Y1 . The PH-representationof these variables was obtained by the MM algorithm, which requires4 states for the fitting (Section 4.1). Fig. 9(b) shows the pdf's of thelifetime for an initial performance z100 (in appropriate unitsdepending on each application case) and threshold * =a 0.

These results show that PH shock-based deterioration can beused to model and estimate the reliability of a wide range ofdegradation mechanisms with different deterioration trends (e.g.,corrosion, cracks, creep, etc.,) and rate of shocks (e.g., aftershocksin Omori's law). This is done by relaxing the identical distributionassumption and by assuming that the random variables Xk and Yk) from the distributions of X1 and Y1 for the first shock, their associated PH-matricestotic, i.e., when t( / )X1 ) and degradation mechanism that can be modeled for

Yk Yk

D t( )tX

(1)

Trend Degradation mechanism

t( / )Y X1 1 Linear Corrosion of reinforcement

t( / )Y X12 1 1

2 Cuadratic Sulphate attack on concrete

t( / )Y X13 1 1

3 Cubic

<

bb b

1, 1

Yt T

1 ( / ) Exponential Growth of cracks in metals

t2( / )Y X1 1Square root Diffusion-controlled aging

t8( / )Y X1 18Eighth root Creep in concrete

+a ta

ln ( 1) / 1

ln,

X1Logarithmic Aftershocks arrivals

>a 1 (Omori's Law, Eq. (45))

J. Riascos-Ochoa et al. / Probabilistic Engineering Mechanics 38 (2014) 88101100are distributed proportional to X1 and Y1, respectively, withproportional factor depending on k.8. Summary and conclusions

This paper presents a model for shock-based deterioration ofsystems (e.g., structures, components, materials) based on theformalism of PH distributions. The proposed model is used to findeasy-to-evaluate expressions for the reliability quantities (relia-bility function R(t), density f(t) and moments L[ ]m of the lifetimeL); mean and moments of the deterioration D(t) at time t; andmean N t( ) and rate n t( ) of shocks arrivals. The proposed metho-dology uses the matrix-geometric properties of PH distributions toevaluate the convolutions that appear in the analytical solution,which is not an easy task. The results show excellent computa-tional efficiency and accuracy compared with Monte Carlosimulations.

A central feature of the model is to find a proper PH repre-sentation of random variables. This problem was discussed andaddressed by means of two algorithms: the EM algorithm, which isadequate when there are available datasets (artificially or empiri-cally generated); and the MM algorithm, adequate when only thefirst few moments of the variables are available. The efficiency andaccuracy of the two PH fitting algorithms were studied. The resultsgive an insight into the applicability of each algorithm and theproposed PH shock model in terms of available information as wellas their efficiency and accuracy.

The PH shock model can be applied to model differentdegradation mechanisms with non-linear deterioration trendsand through relaxing the identical distribution assumption ofinter-arrival times and/or shock sizes. Examples are the deteriora-tion by corrosion on reinforcement, cracks in metals, creep inconcrete or aftershocks sequences. The results provide valuableinformation about different degradation trends constructed fromPH distributions.

Potential challenges in the proposed PH shock model consist ofrelating the PH distribution of shock sizes and inter-arrival timeswith the interaction process between the environment and thestructure. This can imply modeling state-dependent degradationmechanisms, which could be implemented with a more elaboratePH shock model. The proposed approach can be extended toinclude interventions and maintenance, or the estimation of thereliability quantities if information of the deterioration is available(conditional probabilities).Appendix A. Approximation error for RK(t)Theorem A.1. Given >z 0, and > 0 if t 0.

To demonstrate Theorem A.1 we need the following twolemmas:

Lemma A.2. For all < z 0, there exists an integer >K 0such that t 0:

= = =

= =