# Reliability Modeling and Evaluation for Complex Systems Subject to New Dependent Competing Failure Process

Authors: Huibing Hao; Chunping Li
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The objective of this study is to construct some system-level reliability models subject to new dependent competing failure process of the soft failure process and hard failure process. In those complex system-level models, the soft failure caused by continuous degradation and sudden degradation increases by random shocks and hard failure due to the same shock process; in addition, the bidirectional effects between natural degradation and random shock are considered in this study; in other words, the random shock can bring additional degradation, and the natural degradation may also impact the shock process. Therefore, the soft failure process and hard failure process are dependent. By using the cumulative shock model and Gaussian stochastic degradation process with line mean value and line standard deviation, five types of reliability models were established: (1) single component model; (2) series system model; (3) parallel system model; (4) series-parallel system model; and (5) parallel-series system model. Finally, an example of fatigue crack growth dataset is taken to verify the effectiveness of the established model and method, and some sensitivity analyses are given. The results show that the bidirectional effects can significantly accelerate the system failure, and the impact of natural degradation process to random shocks should not be ignored. In addition, the results also show that random shock and failure threshold have significant effects on the different complex systems.


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## Body

## 1. Introduction

Reliability analysis and maintenance policies for complex degradation systems have been widely used in different fields, such as in aerospace [1, 2], electronics industry [3], civil engineering [4–6], and other fields with high reliability and long-life requirements. In the above different research fields, the competitive failure model based on degradation and random shock has been deeply studied. Huang and Askin [7] studied a competitive failure reliability model in which sudden failure and degradation process were independent on each other. Lehman [8] proposed a reliability model under degradation and random shock. Klutke and Yang [9] and Li and Pham [10] studied reliability analysis and maintenance model subject to degradation and shock. Zhu et al. [11] studied maintenance policies based on an independent competitive failure reliability model. In addition, some other shock models, such as cumulative shock model [12], extreme shock model [13], mixed shock model [14], and δ-shock model [15] were also considered in competitive failure model. Most of the above competitive failure models assumed that the degradation process and the random shock process are independent on each other. However, in practice, each shock may have an impact on both failure processes, and independent assumptions are inconsistent with reality. When these failure processes depend on each other, the reliability assessment of system is a very worth studying problem.Recently, some research on reliability analysis of dependent competitive failure process (DCFP) has been carried out. Rafiee et al. [16] established a DCFP system with the degradation rate changing. Qi et al. [17] studied a DCFP system with self-recovery capability. Liu et al. [18] proposed a competitive failure process with damage self-healing ability. Gao et al. [19] studied a DCFP system with multiple species of shocks. Qiu and Cui [20] proposed a novel dependent two-stage failure process model with competing failures. Wang et al. [21] established a new DCFP model based on age and current state. Li et al. [22] proposed a DCFP system subject to different failures shifting threshold. In addition, some reliability assessment and maintenance strategies method for the complex degradation system with DCFP model have also established, such as Peng et al. [23], Wang et al. [24], Rassoul and Kamyar [25], and Jiang et al. [26]. Most of the previous studies only considered the reliability analysis of a single-component system with dependent competing failure process. There are few studies on reliability analysis of multiple components complex systems with DCFP [27, 28], and these papers only used simple linear model and extreme shock model. In addition, the above studies only assume that each shock loads will cause additional abrupt degradation, in other words, only considering the impact from random shocks to natural degradation process. In fact, the impact of natural degradation process to random shocks should also be considered, because the random shock may come more frequently with the system performance reduction.Therefore, the bidirectional effects between natural degradation and random shock are considered in this paper, and we extend the previous model presented in References [13, 27–29] to a multi-component complex system by developing some system-level reliability model. Firstly, the reliability model of single-component system with DCFP is established via the cumulative shock model and Gaussian stochastic degradation process with linear mean and linear standard deviation. Secondly, the reliability models of series and parallel systems with DCFP are conducted, respectively. Then, some complex series-parallel reliability models subjected to DCFP are also conducted. At last, an example of fatigue crack growth dataset is taken to verify the effectiveness of the established model and method.


## 2. Degradation Model of the Element

Suppose that the complex system is made up of multiple components in a defined structure, and each component contains a hard failure process and a soft failure process. Due to the same shock loads, these two failure processes are dependent. When any failure process exceeds its threshold, the element will fail.Figure1 shows that Xi(t) is the continuous degradation value of the ith component at time t. For the ith component, suppose that the jth shock load will cause additional abrupt degradation as Yij (i = 1, 2,…,n, j = 1,2,3,…), which represents the acceleration degradation value. Let the total degradation Xis(t) be made up of Xi(t) and Yij, and when Xis(t) > Hi, the soft failure occurs.Figure 1
Soft failure process.Figure2 shows that Wij represents the magnitude of the ith component caused by the jth shock load. When the cumulative magnitudes Wij of those loads surpass the hard threshold value Di, the failure occurs.Figure 2
Hard failure process.


### 2.1. Hard Failure Modelling

During the normal operation, assume that the component is continuously affected by a Poisson shock process with rateλ. Let N(t) represent the total number of shocks in (0, t], then we can get (1)PNt=j=λtjj!e−λt,j=0,1,2,….ɛ.Meanwhile, as shown in Figure2, when the cumulative magnitude is less than the threshold value Di, the hard failure will not occur. Then, we can get(2)Fi,WDi=P∑j=1NtWij<Di,i=1,2…,n,j=0,1,2,….IfWij follows the normal distribution Wij∼Nμw,σW2 and i.i.d, then each component survival probability of shock loads becomes(3)FiWDi=ΦDi−jμWjσW,i=1,2…,n,j=0,1,2,….


### 2.2. Soft Failure Modelling

In this section, suppose that each shock loads will cause an additional abrupt degradation. Let these abrupt degradations be measured sizes as {Yi1, Yi2,…} and Si(t) be the total damage size of the ith component until time t, then we can get(4)Sit=∑j=1NtYij,ifNt=0,0,ifNt=0.Figure1 shows that the overall value XiS(t) of the ith component is made up of Xi(t) and Si(t). For the ith component, the degradation process Xi(t) of the ith component can be described by Gaussian stochastic degradation process as(5)Xit∼Nμit.Assuming that the shock damage sizesYij follows the normal distribution and i.i.d as(6)Yijt∼NμiY,σiY2,i=1,2…,n;j=0,1,2,….Based on equations (4)–(7), considering the independence of Xi(t) and Si(t), we know that Xis(t) follows the normal distribution as follows:(7)Xist∼Nμit+NtμiY,σi2tNtσiY2.Suppose thatT is the soft failure time of the ith component, Hi is the threshold value of the ith component. Considering that in most engineering degradation processes, the mean and standard deviation generally increases in terms of time, then, we can get the soft failures function of the ith component as(8)Rist=PT>t=PXist<Hi=PXit+Sit<Hi=PXit<Hi|Nt=0PNt=0+∑k=1∞PXit+∑j=1NtYij<Hi|Nt=k⋅PNt=k=ΦHi−μitσit⋅e−λt+∑k=1∞ΦHi−kμiY+μitσi2t+kσiY2⋅e−λtk!λtk.


### 2.3. Reliability Modelling of DCFP

As shown in Figures1 and 2, we know that the ith component can work normally only when Xist<Hi and ∑j=1NtWij<Di. Then, the reliability function of the ith component under the model M1 can be obtained as(9)Rit=PXist<Hi,∑j=1NtWij<Di=PXit+∑j=1NtYij<Hi,∑j=1NtWij<Di=∑k=0∞PXit+∑j=1NtYij<Hi,∑j=1NtWij<Di|Nt=k⋅PNt=k=ΦHi−μitσit⋅e−λt+∑k=1∞ΦDi−kμWkσWΦHi−kμiY+μitσi2t+kσiY2⋅e−λtk!λtk.


### 2.4. New Reliability Modelling of DCFP

In the above DCFP model from equations (5) to (9), only suppose that each shock loads will cause additional abrupt degradation, in other words, only considering the impact from random shocks to natural degradation process.In fact, the impact of natural degradation process to random shocks should also be considered, because the random shock may come more frequently with the system performance reduction. Therefore, the bidirectional effects between natural degradation and random shock are considered in this paper, and the following model M2 is used to describe the degradation path of the ith component:(10)YiSt=Xt+fXitSit,where f(Xi(t)) is an increasing function of natural degradation process Xi(t), it can describe the impact of natural degradation process to random shocks. In this paper, we assume f(Xi(t)) = a Xi(t) + b, and we know that if a = 0, b = 1, the equation (11) turns into equation (5).Then, the reliability function of theith component for new DCFP is as(11)Rit=PYist<Hi,∑j=1NtWij<Di=PXit+aXit+b∑j=1NtYij<Hi,∑j=1NtWij<Di=∑k=0∞PXit1+a∑j=1NtYij+b∑j=1NtYij<Hi,∑j=1NtWij<Di|Nt=k⋅PNt=k.Based on equations (6), (7), and (12), the reliability function of the ith component under the model M2 can be obtained as(12)Rit=PXit<Hi|Nt=0PNt=0+∑k=1∞ΦDi−kμWkσWPXit<Hi−b×∑j=1NtYij1+a×∑j=1NtYij|Nt=kPNt=k,=ΦHi−μitσit⋅e−λt+∑k=1∞ΦDi−kμWkσW∫0HΦHi−bz−1+azμit1+azσitdFZnz⋅e−λtk!λtk,where Fi(z) is the distribution of the damage Yij for shocks in the damage zone and Fznz is the n-fold Stieltjes convolution with Fi(z) itself.


## 2.1. Hard Failure Modelling

During the normal operation, assume that the component is continuously affected by a Poisson shock process with rateλ. Let N(t) represent the total number of shocks in (0, t], then we can get (1)PNt=j=λtjj!e−λt,j=0,1,2,….ɛ.Meanwhile, as shown in Figure2, when the cumulative magnitude is less than the threshold value Di, the hard failure will not occur. Then, we can get(2)Fi,WDi=P∑j=1NtWij<Di,i=1,2…,n,j=0,1,2,….IfWij follows the normal distribution Wij∼Nμw,σW2 and i.i.d, then each component survival probability of shock loads becomes(3)FiWDi=ΦDi−jμWjσW,i=1,2…,n,j=0,1,2,….


## 2.2. Soft Failure Modelling

In this section, suppose that each shock loads will cause an additional abrupt degradation. Let these abrupt degradations be measured sizes as {Yi1, Yi2,…} and Si(t) be the total damage size of the ith component until time t, then we can get(4)Sit=∑j=1NtYij,ifNt=0,0,ifNt=0.Figure1 shows that the overall value XiS(t) of the ith component is made up of Xi(t) and Si(t). For the ith component, the degradation process Xi(t) of the ith component can be described by Gaussian stochastic degradation process as(5)Xit∼Nμit.Assuming that the shock damage sizesYij follows the normal distribution and i.i.d as(6)Yijt∼NμiY,σiY2,i=1,2…,n;j=0,1,2,….Based on equations (4)–(7), considering the independence of Xi(t) and Si(t), we know that Xis(t) follows the normal distribution as follows:(7)Xist∼Nμit+NtμiY,σi2tNtσiY2.Suppose thatT is the soft failure time of the ith component, Hi is the threshold value of the ith component. Considering that in most engineering degradation processes, the mean and standard deviation generally increases in terms of time, then, we can get the soft failures function of the ith component as(8)Rist=PT>t=PXist<Hi=PXit+Sit<Hi=PXit<Hi|Nt=0PNt=0+∑k=1∞PXit+∑j=1NtYij<Hi|Nt=k⋅PNt=k=ΦHi−μitσit⋅e−λt+∑k=1∞ΦHi−kμiY+μitσi2t+kσiY2⋅e−λtk!λtk.


## 2.3. Reliability Modelling of DCFP

As shown in Figures1 and 2, we know that the ith component can work normally only when Xist<Hi and ∑j=1NtWij<Di. Then, the reliability function of the ith component under the model M1 can be obtained as(9)Rit=PXist<Hi,∑j=1NtWij<Di=PXit+∑j=1NtYij<Hi,∑j=1NtWij<Di=∑k=0∞PXit+∑j=1NtYij<Hi,∑j=1NtWij<Di|Nt=k⋅PNt=k=ΦHi−μitσit⋅e−λt+∑k=1∞ΦDi−kμWkσWΦHi−kμiY+μitσi2t+kσiY2⋅e−λtk!λtk.


## 2.4. New Reliability Modelling of DCFP

In the above DCFP model from equations (5) to (9), only suppose that each shock loads will cause additional abrupt degradation, in other words, only considering the impact from random shocks to natural degradation process.In fact, the impact of natural degradation process to random shocks should also be considered, because the random shock may come more frequently with the system performance reduction. Therefore, the bidirectional effects between natural degradation and random shock are considered in this paper, and the following model M2 is used to describe the degradation path of the ith component:(10)YiSt=Xt+fXitSit,where f(Xi(t)) is an increasing function of natural degradation process Xi(t), it can describe the impact of natural degradation process to random shocks. In this paper, we assume f(Xi(t)) = a Xi(t) + b, and we know that if a = 0, b = 1, the equation (11) turns into equation (5).Then, the reliability function of theith component for new DCFP is as(11)Rit=PYist<Hi,∑j=1NtWij<Di=PXit+aXit+b∑j=1NtYij<Hi,∑j=1NtWij<Di=∑k=0∞PXit1+a∑j=1NtYij+b∑j=1NtYij<Hi,∑j=1NtWij<Di|Nt=k⋅PNt=k.Based on equations (6), (7), and (12), the reliability function of the ith component under the model M2 can be obtained as(12)Rit=PXit<Hi|Nt=0PNt=0+∑k=1∞ΦDi−kμWkσWPXit<Hi−b×∑j=1NtYij1+a×∑j=1NtYij|Nt=kPNt=k,=ΦHi−μitσit⋅e−λt+∑k=1∞ΦDi−kμWkσW∫0HΦHi−bz−1+azμit1+azσitdFZnz⋅e−λtk!λtk,where Fi(z) is the distribution of the damage Yij for shocks in the damage zone and Fznz is the n-fold Stieltjes convolution with Fi(z) itself.


## 3. DCFP Model of the Series System

### 3.1. Series System Modeling via the Element ModelM1

Figure3 shows that a series system is made up of n components. Only when each component for DCFP works properly, the system can run normally. That is, each component satisfied the total degradation Xist<Hi and the cumulative magnitude ∑j=1NtWij<Di, the series system can run normally.(13)Rt=P∑j=1NtW1j<D1,X1st<H1∩∑j=1NtW2j<D2,X2st<H2∩…∑j=1NtWnj<Dn,Xnst<Hn.Figure 3
Series system example.By using the conditional probability, (13) can also be expressed as(14)Rt=∑k=0∞P∑j=1NtW1j<D1,X1t+∑j=1NtY1j<H1∩∑j=1NtW2j<D2,X2t+∑j=1NtY2j<H2∩...∑j=1NtWnj<Dn,Xnt+∑j=1NtYnj<Hn|Nt=kPNt=k.Then, we can get the reliability function of the series system as(15)Rt=∑k=0∞∏i=1nP∑j=1NtWij<DiPXit+∑j=1kYij<Hi|Nt=kexp−λtλtkk!.According to (3) and (7), equation (15) can be shown as(16)Rt=∑k=0∞∏i=1nΦDi−kμWkσWΦHi−kμiY+μitσi2t+kσiY2⋅e−λtk!λtk.For equation (16), another expression can also be obtained according to literature [27, 30]. Firstly, we introduce a conditional system reliability given the total number of shocks N(t) as follows:(17)Rt|Nt=k=∏i=1nΦDi−kμWkσWΦHi−kμiY+μitσi2t+kσiY2.By using the law of total probability, we can get(18)Rt=∑k=0∞Rt|Nt=kPNt=k.Then, we can get(19)Rt=∑k=0∞Rt|Nt=kPNt=k=∑k=0∞Rt|Nt=k⋅e−λtk!λtk=∑k=0∞∏i=1nΦDi−kμWkσWΦHi−kμiY+μitσi2t+kσiY2⋅e−λtk!λtk.Next, we consider a special case for the series system with two components (n = 2). Based on equation (19), we can get(20)Rt=∑k=0∞ΦD1−kμWkσWΦH1−kμ1Y+μ1tσ12t+kσ1Y2ΦD2−kμWkσWΦH2−kμ2Y+μ2tσ22t+kσ2Y2e−λtk!λtk.


### 3.2. Series System Modeling via the Element ModelM2

Similarly, suppose that each component satisfied the total degradationYist<Hi and the cumulative magnitude ∑j=1NtWij<Di, the series system can run normally. Then, we can get(21)Rt=P∑j=1NtW1j<D1,Y1st<H1∩∑j=1NtW2j<D2,Y2st<H2∩...∑j=1NtWnj<Dn,Ynst<Hn.By using the conditional probability, (21) can also be expressed as(22)Rt=∑k=0∞P∑j=1NtW1j<D1,X1t+aX1t+b∑j=1NtY1j<H1∩∑j=1NtW2j<D2,X2t+aX2tb∑j=1NtY2j<H2∩...∑j=1NtWnj<Dn,Xnt+aXnt+b∑j=1NtYnj<Hn|Nt=kPNt=k.Then, we can get the reliability function of the series system as(23)Rt=∑k=0∞∏i=1nP∑j=1NtWij<DiPXit+aXit+b∑j=1kYij<Hi|Nt=kPNt=k.According to (3) and (12), the equation (23) can be shown as(24)Rt=∑k=0∞∏i=1nΦDi−kμWkσW∫0HΦHi−bz−1+azμit1+azσitdFZnz⋅e−λtk!λtk.Without loss of generality, we also consider a special case for the series system with two components (n = 2). Based on equation (24), we can get(25)Rt=∑k=0∞ΦD1−kμWkσW∫0HΦH1−bz−1+azμ1t1+azσ1tdFZ2zΦD2−kμWkσW∫0HΦH2−bz−1+azμ2t1+azσ2tdFZ2ze−λtk!λtk.


## 3.1. Series System Modeling via the Element ModelM1

Figure3 shows that a series system is made up of n components. Only when each component for DCFP works properly, the system can run normally. That is, each component satisfied the total degradation Xist<Hi and the cumulative magnitude ∑j=1NtWij<Di, the series system can run normally.(13)Rt=P∑j=1NtW1j<D1,X1st<H1∩∑j=1NtW2j<D2,X2st<H2∩…∑j=1NtWnj<Dn,Xnst<Hn.Figure 3
Series system example.By using the conditional probability, (13) can also be expressed as(14)Rt=∑k=0∞P∑j=1NtW1j<D1,X1t+∑j=1NtY1j<H1∩∑j=1NtW2j<D2,X2t+∑j=1NtY2j<H2∩...∑j=1NtWnj<Dn,Xnt+∑j=1NtYnj<Hn|Nt=kPNt=k.Then, we can get the reliability function of the series system as(15)Rt=∑k=0∞∏i=1nP∑j=1NtWij<DiPXit+∑j=1kYij<Hi|Nt=kexp−λtλtkk!.According to (3) and (7), equation (15) can be shown as(16)Rt=∑k=0∞∏i=1nΦDi−kμWkσWΦHi−kμiY+μitσi2t+kσiY2⋅e−λtk!λtk.For equation (16), another expression can also be obtained according to literature [27, 30]. Firstly, we introduce a conditional system reliability given the total number of shocks N(t) as follows:(17)Rt|Nt=k=∏i=1nΦDi−kμWkσWΦHi−kμiY+μitσi2t+kσiY2.By using the law of total probability, we can get(18)Rt=∑k=0∞Rt|Nt=kPNt=k.Then, we can get(19)Rt=∑k=0∞Rt|Nt=kPNt=k=∑k=0∞Rt|Nt=k⋅e−λtk!λtk=∑k=0∞∏i=1nΦDi−kμWkσWΦHi−kμiY+μitσi2t+kσiY2⋅e−λtk!λtk.Next, we consider a special case for the series system with two components (n = 2). Based on equation (19), we can get(20)Rt=∑k=0∞ΦD1−kμWkσWΦH1−kμ1Y+μ1tσ12t+kσ1Y2ΦD2−kμWkσWΦH2−kμ2Y+μ2tσ22t+kσ2Y2e−λtk!λtk.


## 3.2. Series System Modeling via the Element ModelM2

Similarly, suppose that each component satisfied the total degradationYist<Hi and the cumulative magnitude ∑j=1NtWij<Di, the series system can run normally. Then, we can get(21)Rt=P∑j=1NtW1j<D1,Y1st<H1∩∑j=1NtW2j<D2,Y2st<H2∩...∑j=1NtWnj<Dn,Ynst<Hn.By using the conditional probability, (21) can also be expressed as(22)Rt=∑k=0∞P∑j=1NtW1j<D1,X1t+aX1t+b∑j=1NtY1j<H1∩∑j=1NtW2j<D2,X2t+aX2tb∑j=1NtY2j<H2∩...∑j=1NtWnj<Dn,Xnt+aXnt+b∑j=1NtYnj<Hn|Nt=kPNt=k.Then, we can get the reliability function of the series system as(23)Rt=∑k=0∞∏i=1nP∑j=1NtWij<DiPXit+aXit+b∑j=1kYij<Hi|Nt=kPNt=k.According to (3) and (12), the equation (23) can be shown as(24)Rt=∑k=0∞∏i=1nΦDi−kμWkσW∫0HΦHi−bz−1+azμit1+azσitdFZnz⋅e−λtk!λtk.Without loss of generality, we also consider a special case for the series system with two components (n = 2). Based on equation (24), we can get(25)Rt=∑k=0∞ΦD1−kμWkσW∫0HΦH1−bz−1+azμ1t1+azσ1tdFZ2zΦD2−kμWkσW∫0HΦH2−bz−1+azμ2t1+azσ2tdFZ2ze−λtk!λtk.


## 4. DCFP Model of the Parallel System

### 4.1. Parallel System Modeling via the Element ModelM1

Figure4 shows that a parallel system is made up of n components. Similarly, the system can run normally when at least one component satisfied the total degradation Xist<Hi and the cumulative magnitude ∑j=1NtWij<Di.Figure 4
Parallel system example.Similarly, to a series system, we can get conditional parallel system reliability given the number shocksN(t) as(26)Rt|Nt=k=1−∏i=1n1−Rit|Nt=k.Then, the parallel system reliability is obtained as(27)Rt=∑k=0∞Rt|Nt=k⋅PNt=k=∑k=0∞1−∏i=1n1−Rit|Nt=k⋅e−λtk!λtk=∑k=0∞1−∏i=1n1−ΦDi−kμWkσWΦHi−kμiY+μitσi2t+kσiY2e−λtk!λtk=1−∑k=0∞∏i=1n1−ΦDi−kμWkσWΦHi−kμiY+μitσi2t+kσiY2e−λtk!λtk.Without loss of generality, we also consider a special case for the parallel system with two components (n = 2). Based on equation (27), we can get(28)Rt=1−∑k=0∞1−ΦD1−kμWkσWΦH1−kμ1Y+μ1tσ12t+kσ1Y21−ΦD2−kμWkσWΦH2−kμ2Y+μ2tσ22t+kσ2Y2e−λtk!λtk.


### 4.2. Parallel System Modeling via the Element ModelM2

Similarly, suppose that each component satisfied the total degradationYist<Hi and the cumulative magnitude ∑j=1NtWij<Di, the parallel system can run normally. Then, we can get(29)Rt=∑k=0∞Rt|Nt=k⋅PNt=k=∑k=0∞1−∏i=1n1−ΦDi−kμWkσW∫0HΦHi−bz−1+azμit1+azσitdFZnze−λtk!λtk=1−∑k=0∞∏i=1n1−ΦDi−kμWkσW∫0HΦHi−bz−1+azμit1+azσitdFZnze−λtk!λtk.Without loss of generality, we also consider a special case for the parallel system with two components (n = 2). Based on equation (29), we can get(30)Rt=1−∑k=0∞1−ΦD1−kμWkσW∫0HΦH1−bz−1+azμ1t1+azσ1tdFZ2z1−ΦD2−kμWkσW∫0HΦH2−bz−1+azμ2t1+azσ2tdFZ2ze−λtk!λtk.


## 4.1. Parallel System Modeling via the Element ModelM1

Figure4 shows that a parallel system is made up of n components. Similarly, the system can run normally when at least one component satisfied the total degradation Xist<Hi and the cumulative magnitude ∑j=1NtWij<Di.Figure 4
Parallel system example.Similarly, to a series system, we can get conditional parallel system reliability given the number shocksN(t) as(26)Rt|Nt=k=1−∏i=1n1−Rit|Nt=k.Then, the parallel system reliability is obtained as(27)Rt=∑k=0∞Rt|Nt=k⋅PNt=k=∑k=0∞1−∏i=1n1−Rit|Nt=k⋅e−λtk!λtk=∑k=0∞1−∏i=1n1−ΦDi−kμWkσWΦHi−kμiY+μitσi2t+kσiY2e−λtk!λtk=1−∑k=0∞∏i=1n1−ΦDi−kμWkσWΦHi−kμiY+μitσi2t+kσiY2e−λtk!λtk.Without loss of generality, we also consider a special case for the parallel system with two components (n = 2). Based on equation (27), we can get(28)Rt=1−∑k=0∞1−ΦD1−kμWkσWΦH1−kμ1Y+μ1tσ12t+kσ1Y21−ΦD2−kμWkσWΦH2−kμ2Y+μ2tσ22t+kσ2Y2e−λtk!λtk.


## 4.2. Parallel System Modeling via the Element ModelM2

Similarly, suppose that each component satisfied the total degradationYist<Hi and the cumulative magnitude ∑j=1NtWij<Di, the parallel system can run normally. Then, we can get(29)Rt=∑k=0∞Rt|Nt=k⋅PNt=k=∑k=0∞1−∏i=1n1−ΦDi−kμWkσW∫0HΦHi−bz−1+azμit1+azσitdFZnze−λtk!λtk=1−∑k=0∞∏i=1n1−ΦDi−kμWkσW∫0HΦHi−bz−1+azμit1+azσitdFZnze−λtk!λtk.Without loss of generality, we also consider a special case for the parallel system with two components (n = 2). Based on equation (29), we can get(30)Rt=1−∑k=0∞1−ΦD1−kμWkσW∫0HΦH1−bz−1+azμ1t1+azσ1tdFZ2z1−ΦD2−kμWkσW∫0HΦH2−bz−1+azμ2t1+azσ2tdFZ2ze−λtk!λtk.


## 5. DCFP Model of the Series-Parallel System

### 5.1. Series-Parallel System Modeling via the Element ModelM1

Figure5 shows a series-parallel system made up of n subsystem, and the ith subsystem has mi components. Similarly, suppose that the kth load of the jth component of the ith subsystem Wijk, assumed to be i.i.d. random variables distributed as a normal distribution, Wijk∼NμWij,σWij2, and assume that the corresponding shock damage sizes Yijk are independently distributed random variables taking the form as Yijk∼NμYij,σYij2. Suppose the degradation model of the jth component of the ith subsystem Xij(t) describes by the Gaussian process, Xij(t)∼N(μij(t), σij2t). Then, similarly, the reliability of this system at time t is the probability that the system survives each of the N(t) shock loads (∑k=1NtWijk<Dij, for i = 1, 2,…, n; j = 1, 2,…, mi; k = 1, 2,…, N(t)) and the total degradation is less than the threshold level (XSij(t) < Hij) for each component.Figure 5
Series-parallel system example.By using the condition on the number of shocks, we can get conditional series-parallel system reliability as(31)Rt|Nt=k=∏i=1n1−∏j=1mi1−Rijt|Nt=k.Then, we can get the system reliability as(32)Rt=∑k=0∞Rt|Nt=k⋅PNt=k=∑k=0∞∏i=1n1−∏j=1mi1−Rijt|Nt=k⋅e−λtk!λtk=∑k=0∞∏i=1n1−∏j=1mi1−ΦDij−kμWijkσWijΦHi−kμij+μijtσij2t+kσij2⋅e−λtk!λtk.Without loss of generality, we also consider a special case for the series-parallel system with four components (n = 2, mi = 2). Based on equation (32), we can get(33)Rt=∑k=0∞∏i=121−∏j=121−ΦDij−kμWijkσWijΦHi−kμij+μijtσij2t+kσij2⋅e−λtk!λtk.


### 5.2. Series-Parallel System Modeling via the Element ModelM2

Similarly, suppose that each component satisfied the total degradationYsijt<Hij and the cumulative magnitude ∑k=1NtWijk<Dij, the series-parallel system can run normally. Then, we can get(34)Rt=∑k=0∞Rt|Nt=k⋅PNt=k=∑k=0∞∏i=1n1−∏j=1mi1−Rijt|Nt=k⋅e−λtk!λtk=∑k=0∞∏i=1n1−∏j=1mi1−ΦDij−kμWijkσWij∫0HΦHi−bz−1+azμit1+azσijtdFZnz⋅e−λtk!λtk.Without loss of generality, we also consider a special case for the series-parallel system with four components (n = 2, mi = 2). Based on equation (21), we can get(35)Rt=∑k=0∞∏i=121−∏j=121−ΦDij−kμWijkσWij∫0HΦHi−bz−1+azμit1+azσijtdFZ2z⋅e−λtk!λtk.


## 5.1. Series-Parallel System Modeling via the Element ModelM1

Figure5 shows a series-parallel system made up of n subsystem, and the ith subsystem has mi components. Similarly, suppose that the kth load of the jth component of the ith subsystem Wijk, assumed to be i.i.d. random variables distributed as a normal distribution, Wijk∼NμWij,σWij2, and assume that the corresponding shock damage sizes Yijk are independently distributed random variables taking the form as Yijk∼NμYij,σYij2. Suppose the degradation model of the jth component of the ith subsystem Xij(t) describes by the Gaussian process, Xij(t)∼N(μij(t), σij2t). Then, similarly, the reliability of this system at time t is the probability that the system survives each of the N(t) shock loads (∑k=1NtWijk<Dij, for i = 1, 2,…, n; j = 1, 2,…, mi; k = 1, 2,…, N(t)) and the total degradation is less than the threshold level (XSij(t) < Hij) for each component.Figure 5
Series-parallel system example.By using the condition on the number of shocks, we can get conditional series-parallel system reliability as(31)Rt|Nt=k=∏i=1n1−∏j=1mi1−Rijt|Nt=k.Then, we can get the system reliability as(32)Rt=∑k=0∞Rt|Nt=k⋅PNt=k=∑k=0∞∏i=1n1−∏j=1mi1−Rijt|Nt=k⋅e−λtk!λtk=∑k=0∞∏i=1n1−∏j=1mi1−ΦDij−kμWijkσWijΦHi−kμij+μijtσij2t+kσij2⋅e−λtk!λtk.Without loss of generality, we also consider a special case for the series-parallel system with four components (n = 2, mi = 2). Based on equation (32), we can get(33)Rt=∑k=0∞∏i=121−∏j=121−ΦDij−kμWijkσWijΦHi−kμij+μijtσij2t+kσij2⋅e−λtk!λtk.


## 5.2. Series-Parallel System Modeling via the Element ModelM2

Similarly, suppose that each component satisfied the total degradationYsijt<Hij and the cumulative magnitude ∑k=1NtWijk<Dij, the series-parallel system can run normally. Then, we can get(34)Rt=∑k=0∞Rt|Nt=k⋅PNt=k=∑k=0∞∏i=1n1−∏j=1mi1−Rijt|Nt=k⋅e−λtk!λtk=∑k=0∞∏i=1n1−∏j=1mi1−ΦDij−kμWijkσWij∫0HΦHi−bz−1+azμit1+azσijtdFZnz⋅e−λtk!λtk.Without loss of generality, we also consider a special case for the series-parallel system with four components (n = 2, mi = 2). Based on equation (21), we can get(35)Rt=∑k=0∞∏i=121−∏j=121−ΦDij−kμWijkσWij∫0HΦHi−bz−1+azμit1+azσijtdFZ2z⋅e−λtk!λtk.


## 6. DCFP Model of the Parallel-Series System

### 6.1. Parallel-Series System Modeling via the Element ModelM1

Figure6 shows a parallel-series system made up of n subsystem, and the ith subsystem has mi components. The reliability of this system at time t is the probability that the system survives each of the N(t) shock loads (∑k=1NtWijk<Dij, for i = 1, 2,…, n; j = 1, 2,…, mi; k = 1, 2,…, N(t)) and the total degradation is less than the threshold level (XSij(t) < Hij) for each component.Figure 6
Parallel-series system example.By conditioning on the number of shocks, we can get conditional parallel-series system reliability as(36)Rt|Nt=k=1−∏i=1n1−∏j=1miRijt|Nt=k.Then, we can get the system reliability as(37)Rt=∑k=0∞Rt|Nt=k⋅PNt=k=∑k=0∞1−∏i=1n1−∏j=1miRijt|Nt=k⋅e−λtk!λtk=∑k=0∞1−∏i=1n1−∏j=1miΦDij−kμWijkσWijΦHi−kμij+μijtσij2t+kσij2⋅e−λtk!λtk=1−∑k=0∞∏i=1n1−∏j=1miΦDij−kμWijkσWijΦHi−kμij+μijtσij2t+kσij2e−λtk!λtk.Without loss of generality, we also consider a special case for the parallel-series system with four components (n = 2, mi = 2). Based on equation (37), we can get(38)Rt=1−∑k=0∞∏i=121−∏j=12ΦDij−kμWijkσWijΦHi−kμij+μijtσij2t+kσij2e−λtk!λtk.


### 6.2. Parallel-Series System Modeling via the Element ModelM2

Similarly, suppose that each component satisfied the total degradationYsijt<Hij and the cumulative magnitude ∑k=1NtWijk<Dij, the parallel-series system can run normally. Then, we can get(39)Rt=∑k=0∞Rt|Nt=k⋅PNt=k=∑k=0∞1−∏i=1n1−∏j=1miΦDij−kμWijkσWij∫0HΦHi−bz−1+azμit1+azσijtdFZnz⋅e−λtk!λtk=1−∑k=0∞∏i=1n1−∏j=1miΦDij−kμWijkσWij∫0HΦHi−bz−1+azμit1+azσijtdFZnze−λtk!λtk.Without loss of generality, we also consider a special case for the parallel-series system with four components (n = 2, mi = 2). Based on equation (39), we can get(40)Rt=1−∑k=0∞∏i=121−∏j=12ΦDij−kμWijkσWij∫0HΦHi−bz−1+azμit1+azσijtdFZ2ze−λtk!λtk.


## 6.1. Parallel-Series System Modeling via the Element ModelM1

Figure6 shows a parallel-series system made up of n subsystem, and the ith subsystem has mi components. The reliability of this system at time t is the probability that the system survives each of the N(t) shock loads (∑k=1NtWijk<Dij, for i = 1, 2,…, n; j = 1, 2,…, mi; k = 1, 2,…, N(t)) and the total degradation is less than the threshold level (XSij(t) < Hij) for each component.Figure 6
Parallel-series system example.By conditioning on the number of shocks, we can get conditional parallel-series system reliability as(36)Rt|Nt=k=1−∏i=1n1−∏j=1miRijt|Nt=k.Then, we can get the system reliability as(37)Rt=∑k=0∞Rt|Nt=k⋅PNt=k=∑k=0∞1−∏i=1n1−∏j=1miRijt|Nt=k⋅e−λtk!λtk=∑k=0∞1−∏i=1n1−∏j=1miΦDij−kμWijkσWijΦHi−kμij+μijtσij2t+kσij2⋅e−λtk!λtk=1−∑k=0∞∏i=1n1−∏j=1miΦDij−kμWijkσWijΦHi−kμij+μijtσij2t+kσij2e−λtk!λtk.Without loss of generality, we also consider a special case for the parallel-series system with four components (n = 2, mi = 2). Based on equation (37), we can get(38)Rt=1−∑k=0∞∏i=121−∏j=12ΦDij−kμWijkσWijΦHi−kμij+μijtσij2t+kσij2e−λtk!λtk.


## 6.2. Parallel-Series System Modeling via the Element ModelM2

Similarly, suppose that each component satisfied the total degradationYsijt<Hij and the cumulative magnitude ∑k=1NtWijk<Dij, the parallel-series system can run normally. Then, we can get(39)Rt=∑k=0∞Rt|Nt=k⋅PNt=k=∑k=0∞1−∏i=1n1−∏j=1miΦDij−kμWijkσWij∫0HΦHi−bz−1+azμit1+azσijtdFZnz⋅e−λtk!λtk=1−∑k=0∞∏i=1n1−∏j=1miΦDij−kμWijkσWij∫0HΦHi−bz−1+azμit1+azσijtdFZnze−λtk!λtk.Without loss of generality, we also consider a special case for the parallel-series system with four components (n = 2, mi = 2). Based on equation (39), we can get(40)Rt=1−∑k=0∞∏i=121−∏j=12ΦDij−kμWijkσWij∫0HΦHi−bz−1+azμit1+azσijtdFZ2ze−λtk!λtk.


## 7. Numerical Example

In this section, a case as the fatigue crack growth data of an alloy [31] is taken to verify the effectiveness of the established model and method. The dataset consists of 20 experimental samples, and Table 1 lists all the data set information. In order to verify the proposed model, relevant information about the shock process needs to be given. In this paper, reasonable assumptions are made for the information of the shock process based on real data characteristics and literature.Table 1
Fatigue crack growth data from test samples.
Test sampleCycle/10400.010.020.030.040.050.060.070.080.09Component-110.90.951.001.051.121.191.271.351.481.6420.90.940.981.031.081.141.211.281.371.4730.90.940.981.031.081.131.191.261.351.4640.90.940.981.031.071.121.191.251.341.4350.90.940.981.031.071.121.191.241.341.4360.90.940.981.031.071.121.181.231.331.4170.90.940.981.021.071.111.171.231.321.4180.90.930.971.001.061.111.171.231.301.3990.90.920.971.011.051.091.151.211.281.36100.90.920.961.001.041.081.131.191.261.34Component-210.90.930.961.001.041.081.131.181.241.3120.90.930.971.001.031.071.101.161.221.2930.90.920.970.991.031.061.101.141.201.2640.90.930.961.001.031.071.121.161.201.2650.90.920.960.991.031.061.101.161.211.2760.90.920.950.971.001.031.071.111.161.2270.90.930.960.971.001.051.081.111.161.2080.90.920.940.971.011.041.071.091.141.1990.90.920.940.970.991.021.051.081.121.16100.90.920.940.970.991.021.051.081.121.16Similar to the References [13, 31], for the all component, we suppose that Wij∼N(2, 0.5), and Yij∼N(0.02, 0.01), the soft failure threshold Hi = H = 2.0 inches, the hard failure threshold Di = D = 35 units, and the shock arrival rate λ = 0.5 × 10−4. In addition, we suppose that a = 0.8, b = 1.Figures7 and 8show fatigue crack growth paths of component 1 and component 2 in the data-set in Table 1, respectively. Figures 9 and 10 plot the mean and standard deviation of component1 and component 2, respectively. Figures 9 and 10 imply that evolution of the two parameters (mean and standard deviation) can be appropriately fitted by a linear model. Similar to Reference [30], for the degradation process, we use the formulation μi(t) = ai + bit as the mean function of the ith component, and use the formulation σi(t) = ci + dit as the standard deviation function of the ith component. Based on the dataset in Table 1, by using the least square method, the unknown parameters ai, bi, ci, and di can be obtained, the results are as follows:(41)μ1t=0.8643+5.7539×10−6t,σ1t=−0.0055+8.1343×10−7t,μ2t=0.8823+3.6158×10−6t,σ2t=−0.0017+5.5458×10−7t,where t is the number of cycles.Figure 7
Fatigue crack growth paths of component 1.Figure 8
Fatigue crack growth paths of component 2.Figure 9
Mean value and standard deviation of component 1.Figure 10
Mean value and standard deviation of component 2.


### 7.1. System Reliability about the Two Components Series System

According to (19) and (24), the reliability curves of series system with random shock case and without shock case are plotted in Figure 11. From Figure 11, we can find that, when the random shocks are not considered, the reliability of series system almost keeps 1 when t < 1.5 × 105 cycles, and when t > 1.5 × 105 cycles, the reliability declines rapidly. If those random shocks are considered under the DCFP model M1, we can find that the system reliability almost keeps 1 when t < 0.75 × 105 cycles, and when t > 0.75 × 105 cycles, the reliability declines rapidly. But, if the bidirectional effects between natural degradation and random shock are considered under the new DCFP model M2, we can find that the system reliability almost keeps 1 when t < 0.25 × 105 cycles, and when t > 0.25 × 105 cycles, the reliability declines more quickly. The results show that the random shocks can significantly accelerate the series system failure. In addition, the results also show that the bidirectional effects can significantly accelerate the series system failure. In other words, the impact of natural degradation process to random shocks should not be ignored.Figure 11
Reliability curves of different models for series system.In addition, some sensitivity analyses are also given under the new DCFP modelM2. Firstly, the system reliability curves under different soft failure thresholds are plotted in Figure 12. From Figure 12, it can be found that the soft failure threshold H has an important effect on system reliability. Secondly, the system reliability curves under different hard failure thresholds are plotted in Figure 13. From Figure 13, it can be found that the hard failure threshold D has an effect on system reliability. Compared with the hard failure threshold, the system reliability is more sensitive to the soft failure threshold. Thirdly, the sensitivity analysis of the shock arrival rate λ is also given. The system reliability curves under different λ are plotted in Figure 14. From Figure 14, we can find that the shock arrival rate λ has a great impact on the series system reliability. We can also find that the reliability curves shift to the left when the shock arrival rate λ increases from 0.5 to 1. That is, the reliability of system will be lower when there is a higher arrival rate λ.Figure 12
Sensitivity analysis ofR(t) and on H of series system.Figure 13
Sensitivity analysis ofR(t) and on D of series system.Figure 14
Sensitivity analysis ofR(t) and on λ of series system.At last, the effect analysis about the component number is given. The different component number of series system is plotted in Figure15. Without loss of generality, we assume that components 1 and 3 have the same degradation parameter with μ1(t) = μ3(t) = 0.8643 + 5.7539 × 10−6t, σ1(t) = σ3(t) = −0.0055 + 8.1343 × 10−7t, and they are different from component 2 with degradation parameter μ2(t) = 0.8823 + 3.6158 × 10−6t, σ2(t) = −0.0017 + 5.5458 × 10−7t. From Figure 15, in the series system, we can find that the reliability of component number n = 1 (1-component) is higher than component number n = 2 (2-component), and the reliability of component number n = 2 (2-component) is higher than component number n = 3 (3-component). That is to say, when the series system has the fewer components, the system has a higher reliability level.Figure 15
Different components number for series system.


### 7.2. System Reliability about the Two Components Parallel System

Similarly, according to equations (28) and (30), reliability curves of the parallel system with random shock case and without shock case are plotted in Figure 16. We can find that, when random shocks are not considered, the parallel system reliability almost keeps 1 when t < 2.2 × 105 cycles, and when t > 2.2 × 105 cycles, the reliability declines slowly. If those random shocks are considered under the DCFP model M1, we can find that the system reliability almost keeps 1 when t < 1.2 × 105 cycles, and when t > 1.2 × 105 cycles, the reliability declines rapidly. But, if the bidirectional effects between natural degradation and random shock are considered under the new DCFP model M2, we can find that the system reliability almost keeps 1 when t < 0.6 × 105 cycles, and when t > 0.6 × 105 cycles, the reliability declines more quickly. The results show that the random shocks can significantly accelerate the parallel system failure. In addition, the results also show that the bidirectional effects can significantly accelerate the parallel system failure.Figure 16
Reliability curves of different models for parallel system.In addition, some sensitivity analyses are also given under the new DCFP modelM2, such as the soft failure thresholds, the hard failure thresholds, and the shock arrival rate λ which are plotted in Figures 17–19, respectively. From Figures 17–19, we can find the soft failure threshold, the hard failure threshold and the shock arrival rate λ all have a significant effect on system reliability. But, compared with the parallel system, the reliability of the series system is not very sensitive to hard failure threshold D. Moreover, we can also find the reliability of series system will be lower when there has a higher arrival rate λ of random shocks than the parallel system.Figure 17
Sensitivity analysis ofR(t) and on H of parallel system.Figure 18
Sensitivity analysis ofR(t) and on (D) of parallel system.Figure 19
Sensitivity analysis ofR(t) and on λ of parallel system.


### 7.3. System Reliability about the Four Components Parallel-Series System

Without loss of generality, we assume that components 11 and 21 have the same degradation parameter withμ11(t) = μ21(t) = 0.8643 + 5.7539 × 10−6t, σ11(t) = σ21(t) = −0.0055 + 8.1343 × 10−7t, and they are different from component 12 and component 22 with degradation parameter μ12(t) = μ22(t) = 0.8823 + 3.6158 × 10−6t, σ12(t) = σ22(t) = −0.0017 + 5.5458 × 10−7t.Similarly, according to equations (32) and (35), reliability curves of the parallel-series system with random shock case and without shock case are plotted in Figure 20. We can find that, when random shocks are not considered, the parallel-series system reliability almost keeps 1 when t < 2.2 × 105 cycles, and when t > 2.2 × 105 cycles, the reliability declines slowly. If those random shocks are considered under the DCFP model M1, we can find that the system reliability almost keeps 1 when t < 1.2 × 105 cycles, and when t > 1.2 × 105 cycles, the reliability declines rapidly. But, if the bidirectional effects between natural degradation and random shock are considered under the new DCFP model M2, we can find that the system reliability almost keeps 1 when t < 0.6 × 105 cycles, and when t > 0.6 × 105 cycles, the reliability declines more quickly. The results show that the random shocks can significantly accelerate the parallel-series system failure. In addition, the results also show that the bidirectional effects can significantly accelerate the parallel-series system failure.Figure 20
Curves of different models for parallel-series system.In addition, the sensitivity analyses about the soft failure thresholds, the hard failure thresholds, and the shock arrival rateλ are plotted in Figures 21–23, respectively. From Figures 21–23, we can find the soft failure threshold, the hard failure threshold, and the shock arrival rate all have an important impact on system reliability.Figure 21
Sensitivity analysis onH of parallel-series system.Figure 22
Sensitivity analysis onD of parallel-series system.Figure 23
Sensitivity analysis onλ of parallel-series system.


### 7.4. System Reliability about the Four Components Series-Parallel System

Similarly, according to (34) and (36), reliability curves of the series-parallel system under the shock case and without shock case are plotted in Figure 24. We can find that, when random shocks are not considered, the parallel-series system reliability almost keeps 1 when t < 1.6 × 105 cycles, and when t > 1.6 × 105 cycles, the reliability declines rapidly. If those random shocks are considered under the DCFP model M1, we can find that the system reliability almost keeps 1 when t < 0.9 × 105 cycles, and when t > 0.9 × 105 cycles, the reliability declines rapidly. But, if the bidirectional effects between natural degradation and random shock are considered under the new DCFP model M2, we can find that the system reliability almost keeps 1 when t < 0.7 × 105 cycles, and when t > 0.7 × 105 cycles, the reliability declines more quickly. The results show that the random shocks can significantly accelerate the series-parallel system failure. In addition, the results also show that the bidirectional effects can significantly accelerate the series-parallel system failure. In other words, the impact of natural degradation process to random shocks should not be ignored.Figure 24
Curves of different models for series-parallel system.In addition, the sensitivity analyses about the soft failure thresholds, the hard failure thresholds, and the shock arrival rateλ are plotted in Figures 25–27, respectively. From Figures 25–27, we can find the soft failure threshold, the hard failure threshold, and the shock arrival rate λ all have an important impact on system reliability.Figure 25
Sensitivity analysis onH of series-parallel system.Figure 26
Sensitivity analysis onD of series-parallel system.Figure 27
Sensitivity analysis onλ of series-parallel system.


## 7.1. System Reliability about the Two Components Series System

According to (19) and (24), the reliability curves of series system with random shock case and without shock case are plotted in Figure 11. From Figure 11, we can find that, when the random shocks are not considered, the reliability of series system almost keeps 1 when t < 1.5 × 105 cycles, and when t > 1.5 × 105 cycles, the reliability declines rapidly. If those random shocks are considered under the DCFP model M1, we can find that the system reliability almost keeps 1 when t < 0.75 × 105 cycles, and when t > 0.75 × 105 cycles, the reliability declines rapidly. But, if the bidirectional effects between natural degradation and random shock are considered under the new DCFP model M2, we can find that the system reliability almost keeps 1 when t < 0.25 × 105 cycles, and when t > 0.25 × 105 cycles, the reliability declines more quickly. The results show that the random shocks can significantly accelerate the series system failure. In addition, the results also show that the bidirectional effects can significantly accelerate the series system failure. In other words, the impact of natural degradation process to random shocks should not be ignored.Figure 11
Reliability curves of different models for series system.In addition, some sensitivity analyses are also given under the new DCFP modelM2. Firstly, the system reliability curves under different soft failure thresholds are plotted in Figure 12. From Figure 12, it can be found that the soft failure threshold H has an important effect on system reliability. Secondly, the system reliability curves under different hard failure thresholds are plotted in Figure 13. From Figure 13, it can be found that the hard failure threshold D has an effect on system reliability. Compared with the hard failure threshold, the system reliability is more sensitive to the soft failure threshold. Thirdly, the sensitivity analysis of the shock arrival rate λ is also given. The system reliability curves under different λ are plotted in Figure 14. From Figure 14, we can find that the shock arrival rate λ has a great impact on the series system reliability. We can also find that the reliability curves shift to the left when the shock arrival rate λ increases from 0.5 to 1. That is, the reliability of system will be lower when there is a higher arrival rate λ.Figure 12
Sensitivity analysis ofR(t) and on H of series system.Figure 13
Sensitivity analysis ofR(t) and on D of series system.Figure 14
Sensitivity analysis ofR(t) and on λ of series system.At last, the effect analysis about the component number is given. The different component number of series system is plotted in Figure15. Without loss of generality, we assume that components 1 and 3 have the same degradation parameter with μ1(t) = μ3(t) = 0.8643 + 5.7539 × 10−6t, σ1(t) = σ3(t) = −0.0055 + 8.1343 × 10−7t, and they are different from component 2 with degradation parameter μ2(t) = 0.8823 + 3.6158 × 10−6t, σ2(t) = −0.0017 + 5.5458 × 10−7t. From Figure 15, in the series system, we can find that the reliability of component number n = 1 (1-component) is higher than component number n = 2 (2-component), and the reliability of component number n = 2 (2-component) is higher than component number n = 3 (3-component). That is to say, when the series system has the fewer components, the system has a higher reliability level.Figure 15
Different components number for series system.


## 7.2. System Reliability about the Two Components Parallel System

Similarly, according to equations (28) and (30), reliability curves of the parallel system with random shock case and without shock case are plotted in Figure 16. We can find that, when random shocks are not considered, the parallel system reliability almost keeps 1 when t < 2.2 × 105 cycles, and when t > 2.2 × 105 cycles, the reliability declines slowly. If those random shocks are considered under the DCFP model M1, we can find that the system reliability almost keeps 1 when t < 1.2 × 105 cycles, and when t > 1.2 × 105 cycles, the reliability declines rapidly. But, if the bidirectional effects between natural degradation and random shock are considered under the new DCFP model M2, we can find that the system reliability almost keeps 1 when t < 0.6 × 105 cycles, and when t > 0.6 × 105 cycles, the reliability declines more quickly. The results show that the random shocks can significantly accelerate the parallel system failure. In addition, the results also show that the bidirectional effects can significantly accelerate the parallel system failure.Figure 16
Reliability curves of different models for parallel system.In addition, some sensitivity analyses are also given under the new DCFP modelM2, such as the soft failure thresholds, the hard failure thresholds, and the shock arrival rate λ which are plotted in Figures 17–19, respectively. From Figures 17–19, we can find the soft failure threshold, the hard failure threshold and the shock arrival rate λ all have a significant effect on system reliability. But, compared with the parallel system, the reliability of the series system is not very sensitive to hard failure threshold D. Moreover, we can also find the reliability of series system will be lower when there has a higher arrival rate λ of random shocks than the parallel system.Figure 17
Sensitivity analysis ofR(t) and on H of parallel system.Figure 18
Sensitivity analysis ofR(t) and on (D) of parallel system.Figure 19
Sensitivity analysis ofR(t) and on λ of parallel system.


## 7.3. System Reliability about the Four Components Parallel-Series System

Without loss of generality, we assume that components 11 and 21 have the same degradation parameter withμ11(t) = μ21(t) = 0.8643 + 5.7539 × 10−6t, σ11(t) = σ21(t) = −0.0055 + 8.1343 × 10−7t, and they are different from component 12 and component 22 with degradation parameter μ12(t) = μ22(t) = 0.8823 + 3.6158 × 10−6t, σ12(t) = σ22(t) = −0.0017 + 5.5458 × 10−7t.Similarly, according to equations (32) and (35), reliability curves of the parallel-series system with random shock case and without shock case are plotted in Figure 20. We can find that, when random shocks are not considered, the parallel-series system reliability almost keeps 1 when t < 2.2 × 105 cycles, and when t > 2.2 × 105 cycles, the reliability declines slowly. If those random shocks are considered under the DCFP model M1, we can find that the system reliability almost keeps 1 when t < 1.2 × 105 cycles, and when t > 1.2 × 105 cycles, the reliability declines rapidly. But, if the bidirectional effects between natural degradation and random shock are considered under the new DCFP model M2, we can find that the system reliability almost keeps 1 when t < 0.6 × 105 cycles, and when t > 0.6 × 105 cycles, the reliability declines more quickly. The results show that the random shocks can significantly accelerate the parallel-series system failure. In addition, the results also show that the bidirectional effects can significantly accelerate the parallel-series system failure.Figure 20
Curves of different models for parallel-series system.In addition, the sensitivity analyses about the soft failure thresholds, the hard failure thresholds, and the shock arrival rateλ are plotted in Figures 21–23, respectively. From Figures 21–23, we can find the soft failure threshold, the hard failure threshold, and the shock arrival rate all have an important impact on system reliability.Figure 21
Sensitivity analysis onH of parallel-series system.Figure 22
Sensitivity analysis onD of parallel-series system.Figure 23
Sensitivity analysis onλ of parallel-series system.


## 7.4. System Reliability about the Four Components Series-Parallel System

Similarly, according to (34) and (36), reliability curves of the series-parallel system under the shock case and without shock case are plotted in Figure 24. We can find that, when random shocks are not considered, the parallel-series system reliability almost keeps 1 when t < 1.6 × 105 cycles, and when t > 1.6 × 105 cycles, the reliability declines rapidly. If those random shocks are considered under the DCFP model M1, we can find that the system reliability almost keeps 1 when t < 0.9 × 105 cycles, and when t > 0.9 × 105 cycles, the reliability declines rapidly. But, if the bidirectional effects between natural degradation and random shock are considered under the new DCFP model M2, we can find that the system reliability almost keeps 1 when t < 0.7 × 105 cycles, and when t > 0.7 × 105 cycles, the reliability declines more quickly. The results show that the random shocks can significantly accelerate the series-parallel system failure. In addition, the results also show that the bidirectional effects can significantly accelerate the series-parallel system failure. In other words, the impact of natural degradation process to random shocks should not be ignored.Figure 24
Curves of different models for series-parallel system.In addition, the sensitivity analyses about the soft failure thresholds, the hard failure thresholds, and the shock arrival rateλ are plotted in Figures 25–27, respectively. From Figures 25–27, we can find the soft failure threshold, the hard failure threshold, and the shock arrival rate λ all have an important impact on system reliability.Figure 25
Sensitivity analysis onH of series-parallel system.Figure 26
Sensitivity analysis onD of series-parallel system.Figure 27
Sensitivity analysis onλ of series-parallel system.


## 8. Conclusions

This paper considers the reliability of complex systems for the new DCFP model. Under the bidirectional effects between natural degradation and random shock, five types of reliability model based on Gauss stochastic degradation process and cumulative shock process were obtained, and some sensitivity analyses are given. A numerical example of fatigue crack growth dataset is taken to verify the effectiveness of the established model and method, and some sensitivity analyses are given.This paper only considers the Gauss degradation process, some other stochastic process can be used to model the degradation path in future research, such as Wiener process, Gamma process.



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