4. The complex probability paradigm applied to prognostic

In this section, the novel complex probability paradigm will be presented after applying it to prognostic.

#### 4.1 The basic parameters of the new model

(Eqs. (5) and (6)) simulation is achieved for each internal pressure level (low,

Pressure mode P<sup>j</sup> (MPa) Model High (mode 1) 8 Triangular Middle (mode 2) 5 Triangular Low (mode 3) 3 Triangular

The evaluation of the remaining useful lifetime of the system is the major objective in a prognostic study. Since the RUL is the complement of the damage curve D(t), it can be deduced from it. Accordingly, at each time t, the required RUL is the length from cycle time t to the critical cycle time tC that corresponds to the

where tC is the necessary cycle time for the appearance of the first macro-cracks that means to reach failure, and t<sup>0</sup> is the initial cycle time considered in general to be

Consequently, my prognostic model computes the RULs for the three internal pressure modes that can be now simply inferred from these three curves at any

Two parameters which are C and m embody the effects of the environment. These two parameters are associated to the material environment. C and m depend on the initial crack length, on the geometry and size of the specimen, and on the testing conditions (such as the loading ratio σ). These two parameters affect the performance of the material during the process of fatigue through the crack propagation. The influencing parameters on this fatigue process, like humidity, temperature, material nature, geometry dimensions, applied load location, corrosion, water action, soil pressure and friction, atmospheric pressure, etc., can be stochastic and can be also embodied by C and m. Furthermore, it is crucial to note here that these two parameters can be as well random variables and hence can be represented by probability distributions materializing the environment stochastic and chaotic influences on the system. It is also important to mention that these two parameters are computed by the mean of experiments in real conditions. We give here some

RUL ¼ tC � t<sup>0</sup> (8)

threshold D = 1. The entire RUL is inferred using the following relation:

3.7 The effects of environment in the suggested prognostic model

as well as a 64-Bit MATLAB version 2019 software.

instant t or at any active cycle N in this manner:

For mode 3, RUL3(t) = tC<sup>3</sup> – t. For mode 2, RUL2(t) = tC<sup>2</sup> – t. For mode 1, RUL1(t) = tC<sup>1</sup> – t.

A huge amount of pressure simulations of the order of hundreds of millions are required to estimate the real system lifetime; hence, we have used an approximated model of lifetime simulation of the order of 10,000,000 iterations. Accordingly, we have considered for this purpose a high-capacity computer system: a workstation computer with parallel microprocessors, a 64-Bit operating system, a 64 GB RAM,

middle, and high).

Characteristics of each internal pressure mode.

Fault Detection, Diagnosis and Prognosis

Table 1.

3.6 RUL computation

equal to 0.

74

It is very well-known that in systems engineering, the remaining useful lifetime and the degradation prediction is profoundly linked to many aspects (like humidity, temperature, material nature, geometry dimensions, applied load location, water action, corrosion, soil pressure and friction, atmospheric pressure, etc.) that usually have a stochastic and chaotic behavior which reduces the degree of our certain system knowledge [32–35]. Consequently, the lifetime of the system becomes a random variable and is computed by the arbitrary time tC which is evaluated when sudden failure occurs due to these stochastic causes and chaotic factors. We can deduce from the CPP that we can foretell the exact probabilities of RUL and D with certitude in the whole set C ¼ R þM if we add to the probability measure of a random variable in the real set R the corresponding imaginary counterpart M since Pc = 1 perpetually and constantly. In fact, prognostic is based on the forecast of a system remaining useful lifetime at any cycle N or instant t and during the system operation. Therefore, we can make use of this novel idea and procedure to do the prognostic analysis of the system RUL and degradation prediction and evolution.

Let us consider a system degradation trajectory D(t) where we study a specific instant (or cycle) tk. The system age is measured by the number of years and by the variable tk (Figure 5). From the illustrated figures (Figures 5a and 5b), we can infer that at the system age tk of the prognostic study must give the prediction of the failure instant tC. Therefore, the RUL predicted here at the instant tk has the following value:

$$RUL(t\_k) = t\_C \underline{\cdot} t\_k \tag{9}$$

As a matter of fact, at tk = 0 (at the beginning) (point J), the system is intact, then the failure probability of the system is Pr = 0, the chaotic factor in our prognostic is null (MChf = 0) because no chaos exists yet, and our knowledge of the unharmed and undamaged system is complete and certain (DOK = 1); consequently,

$$RUL(\mathbf{0}) = \mathbf{t}\_{\mathbf{C}} - \mathbf{t}\_{\mathbf{k}} = \mathbf{t}\_{\mathbf{C}} - \mathbf{0} = \mathbf{t}\_{\mathbf{C}}.$$

If tk = tC (point L), the system is completely damaged, then RUL(tC) = tC � tC = 0, and therefore the failure probability of the system is one (Pr = 1). Failure occurs at this point. Thus, our knowledge of the totally worn-out system is perfect (DOK = 1) and the harmful task of chaos has finished; hence it is no more applicable (MChf = 0).

If 0 < tk < tC (point K, where J < K < L), the probability of occurrence of this instant and the probabilities of prediction of RUL and D are both less than 1 and are imperfect in R (0 < Pr < 1). This is the result of non-zero chaotic factors influencing the system (MChf > 0). The system degree of our knowledge which is subject to chaos is thus uncertain and is consequently less than one in R (0.5 < DOK < 1).

4.2 The new prognostic model

DOI: http://dx.doi.org/10.5772/intechopen.90157

The novel model of prognostic basic assumption will be presented now [36–53]. We assume first the cumulative probability distribution function F(t) of the random variable time t as being equal to the function of degradation itself, which means

Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines…

We mention here that we are working with discrete random functions that

1.Both D and F are cumulative functions starting from zero and ending with one.

3.Both functions are without measure units: D is an indicator quantifying system damage and degradation, as well as F which is an indicator quantifying

Afterward, we suppose that, at the instant t ¼ tk, the term Prð Þt =ψ<sup>j</sup> is the real

Probð Þt

ProbðÞ¼ t ψ<sup>j</sup> � Probð Þ tk�<sup>1</sup> ≤t≤tk

Prð Þ¼ tk ψ<sup>j</sup> � Probð Þ� t≤tk Probð Þ t≤tk�<sup>1</sup> ½ � ¼ ψ<sup>j</sup> � F tð Þ� <sup>k</sup> F tð Þ <sup>k</sup>�<sup>1</sup> ½ �

t X¼tk‐1 t¼t<sup>0</sup>

" #

Xt¼tk t¼t<sup>0</sup>

ProbðÞ¼ t D tð Þ<sup>k</sup> (12)

(13)

F tð Þ¼ <sup>k</sup> Probð Þ¼ t<sup>0</sup> ≤t ≤tk

depend on the discrete random time t of pressure cycles. This basic assumption is reasonable because:

probability of system failure and is computed as follows:

ProbðÞ � t

¼ ψ<sup>j</sup> � D tð Þ� <sup>k</sup> D tð Þ <sup>k</sup>�<sup>1</sup> ½ �

t¼t<sup>0</sup>

t¼tk�<sup>1</sup>

2.Both are non-decreasing functions.

randomness and chance.

<sup>¼</sup> <sup>ψ</sup><sup>j</sup> � <sup>X</sup><sup>t</sup>¼tk

<sup>¼</sup> <sup>ψ</sup><sup>j</sup> � <sup>X</sup><sup>t</sup>¼tk

Pr, degradation, and the CDF step function.

Figure 6.

77

Figure 5. CPP. (a) The prognostic of degradation and (b) The prognostic of RUL.

Furthermore, by applying here the CPP paradigm, we can therefore determine at any instant tk (0 ≤tk ≤tC) and, at any point between J and L inclusively, the RUL and D of the system with certitude in the set C ¼ R þM because in C we have Pc = 1 permanently.

Additionally, we can express two complementary phenomena or events E and E by their respective probabilities as follows:

$$P\_{rob}(E) = p \text{ and } P\_{rob}(\overline{E}) = q = 1 - p.$$

Therefore, let the probability Probð Þ E as a function of the time tk be defined by

$$P\_{rob}(E) = P\_{rob}(t \le t\_k) = F(t\_k) \tag{10}$$

where the classical and usual cumulative distribution function (CDF) of the random variable t is denoted by the term F tð Þ.

Since Probð Þþ E Prob E <sup>¼</sup> 1, therefore, we deduce at an instant t=tk:

$$P\_{nb}(\overline{E}) = \mathbf{1} - P\_{rb}(E) = \mathbf{1} - P\_{nb}(t \le t\_k) = P\_{nb}(t > t\_k) = \mathbf{1} - F(t\_k) \tag{11}$$

In addition, two particular instants can be defined:

t=t<sup>0</sup> = 0 which corresponds to the system raw state and which is assumed to be the initial time of functioning where D = D0.

t=tC which corresponds to the system wear-out state and which is the failure instant where D = DC = 1.

Consequently, we can state the boundary conditions as follows:

For t = t<sup>0</sup> = 0, we have D = D<sup>0</sup> ≈ 0 (the initial damage that may be nearly 0) and F tðÞ¼ F tð Þ¼ <sup>0</sup> Probð Þ¼ t≤ 0 0.

For t = tC, we have D = DC = 1 and F tðÞ¼ F tð Þ¼ <sup>C</sup> Probð Þ¼ t≤ tC 1.

We note also that since F(tk) is defined as a cumulative probability function, then F(tk) is a non-decreasing function that varies between 0 and 1. In addition, since RUL(tk) = tC � tk and tk is always increasing (0 ≤tk ≤tC), then RUL(tk) is a non-increasing remaining useful lifetime function (Figure 5b).

Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines… DOI: http://dx.doi.org/10.5772/intechopen.90157

#### 4.2 The new prognostic model

The novel model of prognostic basic assumption will be presented now [36–53]. We assume first the cumulative probability distribution function F(t) of the random variable time t as being equal to the function of degradation itself, which means

$$F(t\_k) = P\_{rob}(t\_0 \le t \le t\_k) = \sum\_{t=t\_0}^{t=t\_k} P\_{rob}(t) = D(t\_k) \tag{12}$$

We mention here that we are working with discrete random functions that depend on the discrete random time t of pressure cycles.

This basic assumption is reasonable because:


Afterward, we suppose that, at the instant t ¼ tk, the term Prð Þt =ψ<sup>j</sup> is the real probability of system failure and is computed as follows:

$$P\_r(t\_k) = \boldsymbol{\nu}\_j \times \left[ P\_{rb}(t \le t\_k) - P\_{rb}(t \le t\_{k-1}) \right] = \boldsymbol{\nu}\_j \times \left[ F(t\_k) - F(t\_{k-1}) \right]$$

$$= \boldsymbol{\nu}\_j \times \left[ D(t\_k) - D(t\_{k-1}) \right]$$

$$= \boldsymbol{\nu}\_j \times \left[ \sum\_{t=t\_0}^{t=t\_k} P\_{rb}(t) \right] - \sum\_{t=t\_0}^{t=t\_{k-1}} P\_{rb}(t) \right] \tag{13}$$

$$= \boldsymbol{\nu}\_j \times \sum\_{t=t\_{k-1}}^{t=t\_k} P\_{rb}(t) = \boldsymbol{\nu}\_j \times P\_{rb}(t\_{k-1} \le t \le t\_k)$$

Figure 6. Pr, degradation, and the CDF step function.

Furthermore, by applying here the CPP paradigm, we can therefore determine at any instant tk (0 ≤tk ≤tC) and, at any point between J and L inclusively, the RUL and D of the system with certitude in the set C ¼ R þM because in C we have

Additionally, we can express two complementary phenomena or events E and E

Therefore, let the probability Probð Þ E as a function of the time tk be defined by

where the classical and usual cumulative distribution function (CDF) of the

<sup>¼</sup> 1, therefore, we deduce at an instant t=tk:

<sup>¼</sup> <sup>1</sup> � Probð Þ¼ <sup>E</sup> <sup>1</sup> � Probð Þ¼ <sup>t</sup>≤tk Probð Þ¼ <sup>t</sup> <sup>&</sup>gt;tk <sup>1</sup> � F tð Þ<sup>k</sup> (11)

t=t<sup>0</sup> = 0 which corresponds to the system raw state and which is assumed to be

For t = t<sup>0</sup> = 0, we have D = D<sup>0</sup> ≈ 0 (the initial damage that may be nearly 0) and

We note also that since F(tk) is defined as a cumulative probability function, then F(tk) is a non-decreasing function that varies between 0 and 1. In addition, since RUL(tk) = tC � tk and tk is always increasing (0 ≤tk ≤tC), then RUL(tk) is a

t=tC which corresponds to the system wear-out state and which is the failure

Consequently, we can state the boundary conditions as follows:

For t = tC, we have D = DC = 1 and F tðÞ¼ F tð Þ¼ <sup>C</sup> Probð Þ¼ t≤ tC 1.

non-increasing remaining useful lifetime function (Figure 5b).

<sup>¼</sup> <sup>q</sup> <sup>¼</sup> <sup>1</sup> � <sup>p</sup>:

Probð Þ¼ E Probð Þ¼ t ≤tk F tð Þ<sup>k</sup> (10)

Probð Þ¼ E p and Prob E

Pc = 1 permanently.

Figure 5.

by their respective probabilities as follows:

Fault Detection, Diagnosis and Prognosis

CPP. (a) The prognostic of degradation and (b) The prognostic of RUL.

random variable t is denoted by the term F tð Þ.

the initial time of functioning where D = D0.

In addition, two particular instants can be defined:

Since Probð Þþ E Prob E

instant where D = DC = 1.

76

F tðÞ¼ F tð Þ¼ <sup>0</sup> Probð Þ¼ t≤ 0 0.

Prob E

= ψ<sup>j</sup> times the jump in F(t) or D(t) from t ¼ tk�<sup>1</sup> to t ¼ tk (Figures 6 and 7). where t ¼ 0, 1, 2, … , tk�1, tk, tkþ<sup>1</sup> ½ � , … , tC is the time of pressure cycles and t<sup>0</sup> = 0 is the initial time of pressure cycles at the simulation beginning. It corresponds to a degradation D = D(t0) = D<sup>0</sup> which is generally considered to be nearly equal to 0.

Hence, since F(tk) = D(tk) then F(t0) = D(t0) = 0.020408 ≈ 0, but F(t0) is taken all over this research work as being equal to 0;

t<sup>1</sup> = 1 = the first pressure cycle time ... tk = the kth pressure cycle time … tC = the pressure cycles time that leads to system failure = the critical pressure time. It corresponds to D = DC = 1. It follows directly that F(tC) = D(tC) = DC = 1.

ψ<sup>j</sup> is the simulation magnifying factor that depends on the pressure profile. It is ψ<sup>1</sup> ¼ 5082 for the high-pressure mode (j = 1, mode 1), ψ<sup>2</sup> ¼ 6737 for the middlepressure mode (j = 2, mode 2), and ψ<sup>3</sup> ¼ 9151 for the low-pressure mode (j = 3, mode 3).

Thus, initially we have

$$P\_r(t\_k = t\_0 = \mathbf{0}) = \boldsymbol{\nu}\_j \times F\left(t\_0\right) = \boldsymbol{\nu}\_j \times \mathbf{0} = \mathbf{0}$$

Moreover,

$$P\_r(\mathfrak{t}\_k) = \boldsymbol{\psi}\_j \times \boldsymbol{f}\_j(\mathfrak{t}\_k) \Rightarrow P\_r(\mathfrak{t}\_k)/\boldsymbol{\psi}\_j = \boldsymbol{f}\_j(\mathfrak{t}\_k),\tag{14}$$

t Xk¼tC tk¼t<sup>0</sup> fj ð Þ¼ tk

(Figure 6).

t Xk¼tC tk¼t<sup>0</sup> t Xk¼tC tk¼t<sup>0</sup>

DOI: http://dx.doi.org/10.5772/intechopen.90157

t¼t<sup>0</sup>

¼ ψ<sup>j</sup> � F tð Þ <sup>C</sup> ≈ ψ<sup>j</sup> � D tð Þ <sup>C</sup> ,

Therefore, we can deduce that

Prð Þ¼ tk <sup>ψ</sup><sup>j</sup> �Xt¼tC

¼ ψ<sup>j</sup> �

) t Xk¼tC tk¼t<sup>0</sup>

measure to probability theory. We can notice the following:

and

79

(Figures 7 and 8).

Furthermore, we have:

This implies that (Figure 9)

for every tk : 0≤tk ≤tC:

t Xk¼tC tk¼t<sup>0</sup> fj

Prð Þ tk =ψ<sup>j</sup> ¼ 1 for any pressure profile j ¼ 1, 2, 3:

(15)

This result is reasonable since Prð Þ tk =ψ<sup>j</sup> is here a probability density function

Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines…

ProbðÞ¼ t ψ<sup>j</sup> � Probð Þ t<sup>0</sup> ≤t≤ tC

since D tð Þ¼ <sup>C</sup> 1 and D tð Þ¼ <sup>0</sup> 0:020408 ≈0 and F tð Þ<sup>0</sup> is taken as ¼ 0

ð Þ¼ tk ψ<sup>j</sup> � 1 ¼ ψ<sup>j</sup>

¼ ψ<sup>j</sup> � ½F tð Þ� ¼ tC F tð Þ ¼ t<sup>0</sup> � ¼ ψ<sup>j</sup> � ½ � D tð Þ� ¼ tC D tð Þ ¼ t<sup>0</sup>

Prð Þ tk =ψ<sup>j</sup> ¼ 1, for any pressure profile j ¼ 1, 2, 3

We can understand that F(t) = D(t) is a discrete CDF where the amount of the jump is Prð Þt =ψj; then, Prð Þt =ψ<sup>j</sup> is a damage evolution and degradation function (Figures 6 and 7). And we can infer from the preceding computations that Prð Þt =ψ<sup>j</sup> is a probability density function. Accordingly, we can realize now that Prð Þt =ψ<sup>j</sup> quantifies and measures the system degradation or failure probability. Consequently, what we have achieved at this point is that we have linked degradation

0≤ Prð Þ tk =ψ<sup>j</sup> ≤ 1, 0≤ F tð Þ<sup>k</sup> ≤ 1, and ð Þ D<sup>0</sup> ≈0 ≤ D tð Þ<sup>k</sup> ≤ð Þ DC ¼ 1 ,

This, since the degradation is very flat near 0 and starts increasing with t, becoming very acute at t=tC, hence, near tC, Pr is the greatest and is equal to 1

<sup>¼</sup> <sup>ψ</sup><sup>j</sup> � D t<sup>½</sup> <sup>C</sup> � RUL tð Þ<sup>k</sup> � � D tC � RUL tð Þ <sup>k</sup>�<sup>1</sup> f g ½ � (16)

If tk ! 0 ) D ! D<sup>0</sup> ¼ 0:020408 ≈0 ) F ! 0 ) Prð Þ! tk 0

RUL tð Þ¼ <sup>k</sup> tC � tk and it corresponds to a degradation of D tð Þ<sup>k</sup> . RUL tð Þ¼ <sup>k</sup>�<sup>1</sup> tC � tk�<sup>1</sup> and it corresponds to a degradation of D tð Þ <sup>k</sup>�<sup>1</sup> .

Prð Þ¼ tk ψ<sup>j</sup> � D tð Þ� <sup>k</sup> D tð Þ <sup>k</sup>�<sup>1</sup> ½ �

if tk ! tC ) D ! DC ¼ 1 ) F ! 1 ) Prð Þ! tk 1.

where 1=ψ<sup>j</sup> is a normalizing constant that is used to reduce Prð Þ tk function to a probability density function (PDF) with a total probability equal to one. 1=ψ<sup>j</sup> is a function of the pressure mode and conditions, and it depends on the parameters in the degradation (Eqs. (5) and (6)). The decreasing values of 1=ψ<sup>j</sup> are logical since pipeline failure probabilities are decreasing with the decreasing pressure modes; hence, 1=ψ<sup>1</sup> >1=ψ<sup>2</sup> >1=ψ3. Consequently, we deduce that fj ð Þ tk is the usual probability density function (PDF) for each pressure mode j. Knowing that, from classical probability theory, we have always:

Figure 7. Pr as a function of degradation D(t).

Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines… DOI: http://dx.doi.org/10.5772/intechopen.90157

$$\sum\_{t\_k=t\_0}^{t\_k=t\_C} f\_j(t\_k) = \sum\_{t\_k=t\_0}^{t\_k=t\_C} P\_r(t\_k) / \psi\_j = \mathbf{1} \text{ for any pressure profile} \, j = \mathbf{1}, 2, 3 \dots$$

This result is reasonable since Prð Þ tk =ψ<sup>j</sup> is here a probability density function (Figure 6).

Therefore, we can deduce that

= ψ<sup>j</sup> times the jump in F(t) or D(t) from t ¼ tk�<sup>1</sup> to t ¼ tk (Figures 6 and 7). where t ¼ 0, 1, 2, … , tk�1, tk, tkþ<sup>1</sup> ½ � , … , tC is the time of pressure cycles and

corresponds to a degradation D = D(t0) = D<sup>0</sup> which is generally considered to be

pressure cycles time that leads to system failure = the critical pressure time. It corresponds to D = DC = 1. It follows directly that F(tC) = D(tC) = DC = 1.

Hence, since F(tk) = D(tk) then F(t0) = D(t0) = 0.020408 ≈ 0, but F(t0) is taken

t<sup>1</sup> = 1 = the first pressure cycle time ... tk = the kth pressure cycle time … tC = the

ψ<sup>j</sup> is the simulation magnifying factor that depends on the pressure profile. It is ψ<sup>1</sup> ¼ 5082 for the high-pressure mode (j = 1, mode 1), ψ<sup>2</sup> ¼ 6737 for the middlepressure mode (j = 2, mode 2), and ψ<sup>3</sup> ¼ 9151 for the low-pressure mode (j = 3,

Prð Þ¼ tk ¼ t<sup>0</sup> ¼ 0 ψ<sup>j</sup> � F tð Þ¼ <sup>0</sup> ψ<sup>j</sup> � 0 ¼ 0

where 1=ψ<sup>j</sup> is a normalizing constant that is used to reduce Prð Þ tk function to a probability density function (PDF) with a total probability equal to one. 1=ψ<sup>j</sup> is a function of the pressure mode and conditions, and it depends on the parameters in the degradation (Eqs. (5) and (6)). The decreasing values of 1=ψ<sup>j</sup> are logical since pipeline failure probabilities are decreasing with the decreasing pressure modes;

bility density function (PDF) for each pressure mode j. Knowing that, from classical

ð Þ) tk Prð Þ tk =ψ<sup>j</sup> ¼ fj

ð Þ tk , (14)

ð Þ tk is the usual proba-

t<sup>0</sup> = 0 is the initial time of pressure cycles at the simulation beginning. It

all over this research work as being equal to 0;

Fault Detection, Diagnosis and Prognosis

Prð Þ¼ tk ψ<sup>j</sup> � fj

hence, 1=ψ<sup>1</sup> >1=ψ<sup>2</sup> >1=ψ3. Consequently, we deduce that fj

nearly equal to 0.

mode 3).

Figure 7.

78

Pr as a function of degradation D(t).

Moreover,

Thus, initially we have

probability theory, we have always:

$$\sum\_{t\_k=t\_0}^{t\_k=t\_C} P\_r(t\_k) = \boldsymbol{\nu}\_j \times \sum\_{t=t\_0}^{t=t\_C} P\_{rab}(t) = \boldsymbol{\nu}\_j \times P\_{rab}(t\_0 \le t \le t\_C)$$

$$= \boldsymbol{\nu}\_j \times [F(t = t\_C) - F(t = t\_0)] = \boldsymbol{\nu}\_j \times [D(t = t\_C) - D(t = t\_0)]$$

$$= \boldsymbol{\nu}\_j \times F(t\_C) \approx \boldsymbol{\nu}\_j \times D(t\_C),$$

since D tð Þ¼ <sup>C</sup> 1 and D tð Þ¼ <sup>0</sup> 0:020408 ≈0 and F tð Þ<sup>0</sup> is taken as ¼ 0 (15)

$$\begin{aligned} \boldsymbol{\nu} &= \boldsymbol{\nu}\_{j} \times \sum\_{t\_k=t\_0}^{t\_k=t\_C} f\_j(t\_k) = \boldsymbol{\nu}\_j \times \mathbf{1} = \boldsymbol{\nu}\_j \\ &\Rightarrow \sum\_{t\_k=t\_0}^{t\_k=t\_C} P\_r(t\_k) / \boldsymbol{\nu}\_j = \mathbf{1}, \text{ for any pressure profile } j = \mathbf{1}, 2, 3. \end{aligned}$$

We can understand that F(t) = D(t) is a discrete CDF where the amount of the jump is Prð Þt =ψj; then, Prð Þt =ψ<sup>j</sup> is a damage evolution and degradation function (Figures 6 and 7). And we can infer from the preceding computations that Prð Þt =ψ<sup>j</sup> is a probability density function. Accordingly, we can realize now that Prð Þt =ψ<sup>j</sup> quantifies and measures the system degradation or failure probability. Consequently, what we have achieved at this point is that we have linked degradation measure to probability theory.

We can notice the following:

$$\begin{aligned} 0 \le P\_r(t\_k) / \nu\_j \le 1, \; 0 \le F(t\_k) \le 1, \text{and } (D\_0 \approx 0) \le D(t\_k) \le (D\_C = 1), \\ \text{for every } t\_k : 0 \le t\_k \le t\_C. \end{aligned}$$

and

If  $t\_k \to 0 \Rightarrow D \to D\_0 = 0.020408 \approx 0 \Rightarrow F \to 0 \Rightarrow P\_r(t\_k) \to 0$   $\text{if } t\_k \to t\_C \Rightarrow D \to D\_C = 1 \Rightarrow F \to 1 \Rightarrow P\_r(t\_k) \to 1.$ 

This, since the degradation is very flat near 0 and starts increasing with t, becoming very acute at t=tC, hence, near tC, Pr is the greatest and is equal to 1 (Figures 7 and 8).

Furthermore, we have: RUL tð Þ¼ <sup>k</sup> tC � tk and it corresponds to a degradation of D tð Þ<sup>k</sup> . RUL tð Þ¼ <sup>k</sup>�<sup>1</sup> tC � tk�<sup>1</sup> and it corresponds to a degradation of D tð Þ <sup>k</sup>�<sup>1</sup> .

This implies that (Figure 9)

$$\begin{split} P\_r(\mathfrak{t}\_k) &= \mathfrak{y}\_j \times [D(\mathfrak{t}\_k) - D(\mathfrak{t}\_{k-1})] \\ &= \mathfrak{y}\_j \times \{ D[\mathfrak{t}\_C - RUL(\mathfrak{t}\_k)] - D[\mathfrak{t}\_C - RUL(\mathfrak{t}\_{k-1})] \} \end{split} \tag{16}$$
