# Multi-State Reliability Analysis Based on General Wiener Degradation Process and Random Shock

Authors: Huibing Hao; Chunping Li
Journal: Shock and Vibration
 (2022)

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## Abstract

A multi-state reliability analysis model suffering from a dependent competing failure process is developed in this study, where the soft failure is described by general nonlinearity random effect. Wiener degradation process with measurement error and the hard failure is caused by random shock. Considering that the shock process not only may cause abrupt damage but also can accelerate degradation, there are some correlations between soft failure and hard failure. Based on the proposed new model, the multi-state functions are obtained under the cumulative shock model and the extreme shock model, and the system state probabilities are given under the different degradation state points. The fatigue cracks growth data example and MEMS oscillator example are given to demonstrate the proposed new model. At last, some sensitivity analyses are given to illustrate the influence of parameters on the state probability and the system reliability.


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## Body

## 1. Introduction

In many actual engineering environments, system reliability is not only related to the service time but also related to the system state, such as humidity, wear, random shock, erosion, and vibration, any of which can lead to system degradation. Considering the impact of different degrees of degradation, the systems may exist in lots of different medial states within the life cycle, and different degradation states can perform various distinctive tasks. Therefore, the multi-state reliability modeling and calculation of the complex degradation system can provide an effective technical approach to the reliability theory.The concept of multi-state reliability is defined as a complex system with a series of performance levels in its life cycle, including in perfect function state, intermediate unction state, and complete failure state [1]. Many study methods are used to deal with multi-state reliability modeling, such as the multi-state RBD method [2], Markov process method [3], semi-Markov process method [4], and some other related research about the multi-state degradation reliability modeling can be found in Refs [5–9].In real circumstances, except for performance degradation, the system components are often subject to different random shocks. We know that those random shocks may bring a faster degradation rate, resulting in the system’s performance degradation level from one state to another. There are some papers focusing on the random shock to the multi-state reliability problem. For example, Li and Pham [10] developed a generalized condition-based maintenance model subject to degradation and shock, Eryilmaz [11] studied the assessment of the multi-state shock model; Segovia and Labeau [12] studied a multi-state reliability model subject to the wear-out process and shock process; Li and Pham [13] studied a multi-state failure processes reliability model suffering from the degradation and shocks. Lin et al. [14] studied the assessment of the multi-state system under different random shocks; Pham et al. [15] presented a model for predicting the availability and mean a lifetime of multistage degraded systems with partial repairs.In practice engineering applications, because the shock process not only may cause abrupt damage but also can accelerate the degradation, there are some correlations between degradation failure and shock failure. Some multi-state reliability analyses for the system subject to the dependent competing failure process (DCFP) model have been studied [16–26], such as Wang et al. [23] constructed the DCFP model by using multi-state system reliability theory and line degradation path; Jiang et al. [24] constructed a multi-state reliability analysis for DCFP with line degradation process by using delay time theory; Li et al. [25] and Wang et al. [26] constructed reliability model for multi-state systems by using shock model and normal distribution model.Considering that the nonlinearity, uncertainty, and individual differences exist extensively in the practice degradation process, see Refs [27–30]. Therefore, some multi-state complex degradation system reliability assessment models suffering from the DCFP model are proposed in this paper, where the degradation failure process is described by a general nonlinear random effect. Wiener degradation process with measurement errors and the abrupt failure process is described by two different shock models. Based on the proposed new model, the multi-state reliability functions are given, and the system state probabilities are obtained under the different degradation state points. At last, the fatigue cracks growth data example and MEMS oscillator example are given to demonstrate the proposed new model.


## 2. Model Descriptions

As shown in Figures1 and 2, a complex system may fail due to the DCFP: soft failure caused by continuous degradation such as wear, corrosion, humidity, wear, and erosion, and hard failure caused by the stress from the shock process, such as 4random shock, vibration, and fracture. In this paper, the soft failure process can be described as nature degradation process and the additional abrupt increment by the shock loads; the soft failure occurs when the total degradation Xs(t) is above its threshold level H, where the total degradation volume contains the nature degradation and additional abrupt increment by the shock load. The hard failure can be affected by the same shock process, and hard failure happens when the shock magnitude exceeds its threshold level D. Let Yi and Wi (i = 1, 2, 3, ···) denote the abrupt damage size and the magnitude of ith shock load, respectively. These two types of failure processes are dependent because they suffer the same shock. In this paper, the cumulative shock model and the extreme shock model are used, and each shock can bring a change to the system state.Figure 1
Soft failure process.Figure 2
Hard failure process.


## 3. System Reliability Assessment Based on the Multi-State Theory

### 3.1. Shocking Process Analysis

As we know, shock is a significant reason for system failure. In practical running environments, many factors could bring all kinds of shocks to systems, and those factors may come from the external environment, such as sudden or unexpected usage loads. Suppose that the arrives of random shocks follows a Poisson distribution with parameterλ, then we have(1)PNt=i=λtii!e−λt,i=0,1,2,…,where, N(t) is the shock number in (0, t]From Figure2, suppose T is the complex system hard failure time and D is the hard threshold value, under the extreme shock model, we can get(2)Pt≤T=PW1<D,W2<D,…,WNt<D.Suppose that magnitudes of the shockWi∼Nμw,σw2, then we can get(3)Pt≤T=ΦD−μwσwNt.Similarly, under the cumulative shock model, we can get(4)Pt≤T=PW1+W2+⋯+WNt<D.And we can get(5)Pt≤T=ΦD−NtμwNtσw.


### 3.2. Degradation Process Analysis

Considering that the dynamics characterizes the random uncertainty, the nonlinearity, the uncertainty of measurement, and individual differences exist extensively in the practice degradation process, a general nonlinearity random effect. Wiener process with measurement error modelM0 is expressed as follows:(6)Xt=X0+βΛt,γ1+σBBΛt,γ2+ε,β∼Nμβ,σβ2,where, the initial value X(0) is a constant, the Λt,γ1 and Λt,γ2 are the time scale, the drift degradation rate β is a random effect parameter, the diffusion coefficient σB is the fixed effect parameter, B⋅ is the standard Brownian motion, ε is the measurement error and ε∼N0,σε2. Generally, we suppose X0=0; if not, we can use a transform X′t=Xt−X0.Ifσε=0, model M0 reduces a random effect model M1 as follows:(7)Xt=βΛt,γ1+σBBΛt,γ2,β∼Nμβ,σβ2.Ifσβ=0, the model M0 reduces a fixed effect model with measurement error M2 as follows:(8)Xt=βΛt,γ1+σBBΛt,γ2+ε.Ifσβ=0 and σε=0, the model M0 reduces a fixed effect model M3 as follows:(9)Xt=βΛt,γ1+σBBΛt,γ2.Ifσβ=0, σε=0, and γ1=γ2=γ, the model M0 reduces a fixed effect model M4 as follows:(10)Xt=βΛt,γ+σBBΛt,γ.By using a time-scale transformation, the (10) can be can be transformed into a linear Wiener process model.From (6), we can obtain(11)Xt∼NμβΛt,γ1,σβ2Λt,γ12+σB2Λt,γ2+σε2.In this section, suppose that each shock load will cause additional abrupt degradation. Let the abrupt degradation be measured sizes asY1,Y2,… and St be the total damage during (0, t], then, we can get(12)St=∑i=1NtYi,ifNt>0,0,ifNt=0.Then, we can obtain(13)XSt=Xt+St.


### 3.3. System Reliability Analysis

We know that the shocks not only can reduce the system performance directly but also can bring a faster degradation rate, resulting in the system’s performance level from one state to another. Therefore, we can describe the state transition chart as in Figure3.Figure 3
State transition charts of the system.


#### 3.3.1. Case I: Extreme Shock Modeling

From the system assumptions, we know that each shock can bring a change in the system state. That is to say, if theith (i = 1, 2, …, n) random shock occurs, the system state will divert from state i − 1 to state i as the solid line shown in Figure 3. In addition, in the actual arithmetic process, the dotted line expresses the transition path from state 0 to state j in Figure 3.Therefore, under the extreme shock, the probability at statej can be denoted as follows:(14)P10=PrNt=0,Xst<H=PrNt=0,Xt<H=PrNt=0PrXt<H,(15)P11=PrNt=1,Xst<H,W1<D=PrNt=1,Xt+St<H,W1<D=PrNt=1PrXt+Y1<HPrW1<D,(16)P12=PrNt=2,Xst<H,W1<D,W2<D=PrNt=2,Xt+St<H,W1<D,W2<D=PrNt=nPrXt+Y1+Y2<HPrW1<DPrW2<D,(17)P1n=PrNt=n,Xst<H,W1<D,W2<D,⋯,Wn<D,=PrNt=nPrXt+Y1+Y2+⋯+Yn<HPrW1<DPrW2<D⋯PrWn<D=PrNt=nPrXt+∑i=1NtYi<HPrW1<DNt.Then, the system reliability can be calculated as follows:(18)R=∑i=1nP1i.Suppose thatYi denotes the abrupt damage size with the distribution as follows:(19)Yi∼NμY,σY2i=1,2,….Then, we can get the following:(20)St=∑i=1nYi∼NnμY,nσY2.By using (11) and (20), we can get the following:(21)Xt+St∼NμβΛt,γ1+nμY,σβ2Λt,γ12+σB2Λt,γ2+σε2+nσY2.Then, we can get the following:(22)P1n=PrNt=n,Xst<H,W1<D,W2<D,⋯,Wn<D,=PrNt=nPrXt+∑i=1nYi<HPrW1<Dn=exp−λtλtnn!ΦH−μβΛt,γ1+nμYσβ2Λt,γ12+σB2Λt,γ2+σε2+iσY2ΦD−μWσWn.Then, we can get the following:(23)R=∑i=1nP1i=ΦH−μβΛt,γ1σβ2Λt,γ12+σB2Λt,γ2+σε2exp−λt+∑i=1nΦD−μWσWiΦH−μβΛt,γ1+iμYσβ2Λt,γ12+σB2Λt,γ2+σε2+iσY2exp−λtλtii!.


#### 3.3.2. Case II: Cumulative Shock Modeling

Similarly, under the cumulative shock, the probability at statej can be denoted as follows:(24)P20=PrNt=0,Xst<H=PrNt=0PrXt<H,(25)P21=PrNt=1,Xst<H,W1<D=PrNt=1,Xt+Y1<HPrW1<D=PrNt=nPrXt+Y1+Y2<HPrW1+W2<D=exp−λtλtΦH−μβΛt,γ1+2μYσβ2Λt,γ12+σB2Λt,γ2+σε2+2σY2×ΦD−2μW2σW,(26)P22=PrNt=2,Xst<H,W1+W2<D,=PrNt=2,Xt+St<H,W1+W2<D,=PrNt=nPrXt+Y1+Y2<HPrW1+W2<D,=exp−λtλt22ΦH−μβΛt,γ1+2μYσβ2Λt,γ12+σB2Λt,γ2+σε2+σY2×ΦD−2μWσW,(27)P2n=PrNt=n,Xst<H,W1+W2+⋯+Wn<D=PrNt=nPrXt+Y1+Y2+⋯+Yn<H⋅PrW1+W2+Wn<D=PrNt=nPrXt+∑i=1NtYi<H×Pr∑i=1NtWi<D=exp−λtλtnn!ΦH−μβΛt,γ1+nμYσβ2Λt,γ12+σB2Λt,γ2+σε2+nσY2×ΦD−iμWnσW.Then, we can get the following:(28)R=∑i=1nP2i=∑i=1nexp−λtλtii!ΦH−μβΛt,γ1+iμYσβ2Λt,γ12+σB2Λt,γ2+σε2+iσY2ΦD−iμWiσW.


## 3.1. Shocking Process Analysis

As we know, shock is a significant reason for system failure. In practical running environments, many factors could bring all kinds of shocks to systems, and those factors may come from the external environment, such as sudden or unexpected usage loads. Suppose that the arrives of random shocks follows a Poisson distribution with parameterλ, then we have(1)PNt=i=λtii!e−λt,i=0,1,2,…,where, N(t) is the shock number in (0, t]From Figure2, suppose T is the complex system hard failure time and D is the hard threshold value, under the extreme shock model, we can get(2)Pt≤T=PW1<D,W2<D,…,WNt<D.Suppose that magnitudes of the shockWi∼Nμw,σw2, then we can get(3)Pt≤T=ΦD−μwσwNt.Similarly, under the cumulative shock model, we can get(4)Pt≤T=PW1+W2+⋯+WNt<D.And we can get(5)Pt≤T=ΦD−NtμwNtσw.


## 3.2. Degradation Process Analysis

Considering that the dynamics characterizes the random uncertainty, the nonlinearity, the uncertainty of measurement, and individual differences exist extensively in the practice degradation process, a general nonlinearity random effect. Wiener process with measurement error modelM0 is expressed as follows:(6)Xt=X0+βΛt,γ1+σBBΛt,γ2+ε,β∼Nμβ,σβ2,where, the initial value X(0) is a constant, the Λt,γ1 and Λt,γ2 are the time scale, the drift degradation rate β is a random effect parameter, the diffusion coefficient σB is the fixed effect parameter, B⋅ is the standard Brownian motion, ε is the measurement error and ε∼N0,σε2. Generally, we suppose X0=0; if not, we can use a transform X′t=Xt−X0.Ifσε=0, model M0 reduces a random effect model M1 as follows:(7)Xt=βΛt,γ1+σBBΛt,γ2,β∼Nμβ,σβ2.Ifσβ=0, the model M0 reduces a fixed effect model with measurement error M2 as follows:(8)Xt=βΛt,γ1+σBBΛt,γ2+ε.Ifσβ=0 and σε=0, the model M0 reduces a fixed effect model M3 as follows:(9)Xt=βΛt,γ1+σBBΛt,γ2.Ifσβ=0, σε=0, and γ1=γ2=γ, the model M0 reduces a fixed effect model M4 as follows:(10)Xt=βΛt,γ+σBBΛt,γ.By using a time-scale transformation, the (10) can be can be transformed into a linear Wiener process model.From (6), we can obtain(11)Xt∼NμβΛt,γ1,σβ2Λt,γ12+σB2Λt,γ2+σε2.In this section, suppose that each shock load will cause additional abrupt degradation. Let the abrupt degradation be measured sizes asY1,Y2,… and St be the total damage during (0, t], then, we can get(12)St=∑i=1NtYi,ifNt>0,0,ifNt=0.Then, we can obtain(13)XSt=Xt+St.


## 3.3. System Reliability Analysis

We know that the shocks not only can reduce the system performance directly but also can bring a faster degradation rate, resulting in the system’s performance level from one state to another. Therefore, we can describe the state transition chart as in Figure3.Figure 3
State transition charts of the system.


### 3.3.1. Case I: Extreme Shock Modeling

From the system assumptions, we know that each shock can bring a change in the system state. That is to say, if theith (i = 1, 2, …, n) random shock occurs, the system state will divert from state i − 1 to state i as the solid line shown in Figure 3. In addition, in the actual arithmetic process, the dotted line expresses the transition path from state 0 to state j in Figure 3.Therefore, under the extreme shock, the probability at statej can be denoted as follows:(14)P10=PrNt=0,Xst<H=PrNt=0,Xt<H=PrNt=0PrXt<H,(15)P11=PrNt=1,Xst<H,W1<D=PrNt=1,Xt+St<H,W1<D=PrNt=1PrXt+Y1<HPrW1<D,(16)P12=PrNt=2,Xst<H,W1<D,W2<D=PrNt=2,Xt+St<H,W1<D,W2<D=PrNt=nPrXt+Y1+Y2<HPrW1<DPrW2<D,(17)P1n=PrNt=n,Xst<H,W1<D,W2<D,⋯,Wn<D,=PrNt=nPrXt+Y1+Y2+⋯+Yn<HPrW1<DPrW2<D⋯PrWn<D=PrNt=nPrXt+∑i=1NtYi<HPrW1<DNt.Then, the system reliability can be calculated as follows:(18)R=∑i=1nP1i.Suppose thatYi denotes the abrupt damage size with the distribution as follows:(19)Yi∼NμY,σY2i=1,2,….Then, we can get the following:(20)St=∑i=1nYi∼NnμY,nσY2.By using (11) and (20), we can get the following:(21)Xt+St∼NμβΛt,γ1+nμY,σβ2Λt,γ12+σB2Λt,γ2+σε2+nσY2.Then, we can get the following:(22)P1n=PrNt=n,Xst<H,W1<D,W2<D,⋯,Wn<D,=PrNt=nPrXt+∑i=1nYi<HPrW1<Dn=exp−λtλtnn!ΦH−μβΛt,γ1+nμYσβ2Λt,γ12+σB2Λt,γ2+σε2+iσY2ΦD−μWσWn.Then, we can get the following:(23)R=∑i=1nP1i=ΦH−μβΛt,γ1σβ2Λt,γ12+σB2Λt,γ2+σε2exp−λt+∑i=1nΦD−μWσWiΦH−μβΛt,γ1+iμYσβ2Λt,γ12+σB2Λt,γ2+σε2+iσY2exp−λtλtii!.


### 3.3.2. Case II: Cumulative Shock Modeling

Similarly, under the cumulative shock, the probability at statej can be denoted as follows:(24)P20=PrNt=0,Xst<H=PrNt=0PrXt<H,(25)P21=PrNt=1,Xst<H,W1<D=PrNt=1,Xt+Y1<HPrW1<D=PrNt=nPrXt+Y1+Y2<HPrW1+W2<D=exp−λtλtΦH−μβΛt,γ1+2μYσβ2Λt,γ12+σB2Λt,γ2+σε2+2σY2×ΦD−2μW2σW,(26)P22=PrNt=2,Xst<H,W1+W2<D,=PrNt=2,Xt+St<H,W1+W2<D,=PrNt=nPrXt+Y1+Y2<HPrW1+W2<D,=exp−λtλt22ΦH−μβΛt,γ1+2μYσβ2Λt,γ12+σB2Λt,γ2+σε2+σY2×ΦD−2μWσW,(27)P2n=PrNt=n,Xst<H,W1+W2+⋯+Wn<D=PrNt=nPrXt+Y1+Y2+⋯+Yn<H⋅PrW1+W2+Wn<D=PrNt=nPrXt+∑i=1NtYi<H×Pr∑i=1NtWi<D=exp−λtλtnn!ΦH−μβΛt,γ1+nμYσβ2Λt,γ12+σB2Λt,γ2+σε2+nσY2×ΦD−iμWnσW.Then, we can get the following:(28)R=∑i=1nP2i=∑i=1nexp−λtλtii!ΦH−μβΛt,γ1+iμYσβ2Λt,γ12+σB2Λt,γ2+σε2+iσY2ΦD−iμWiσW.


## 3.3.1. Case I: Extreme Shock Modeling

From the system assumptions, we know that each shock can bring a change in the system state. That is to say, if theith (i = 1, 2, …, n) random shock occurs, the system state will divert from state i − 1 to state i as the solid line shown in Figure 3. In addition, in the actual arithmetic process, the dotted line expresses the transition path from state 0 to state j in Figure 3.Therefore, under the extreme shock, the probability at statej can be denoted as follows:(14)P10=PrNt=0,Xst<H=PrNt=0,Xt<H=PrNt=0PrXt<H,(15)P11=PrNt=1,Xst<H,W1<D=PrNt=1,Xt+St<H,W1<D=PrNt=1PrXt+Y1<HPrW1<D,(16)P12=PrNt=2,Xst<H,W1<D,W2<D=PrNt=2,Xt+St<H,W1<D,W2<D=PrNt=nPrXt+Y1+Y2<HPrW1<DPrW2<D,(17)P1n=PrNt=n,Xst<H,W1<D,W2<D,⋯,Wn<D,=PrNt=nPrXt+Y1+Y2+⋯+Yn<HPrW1<DPrW2<D⋯PrWn<D=PrNt=nPrXt+∑i=1NtYi<HPrW1<DNt.Then, the system reliability can be calculated as follows:(18)R=∑i=1nP1i.Suppose thatYi denotes the abrupt damage size with the distribution as follows:(19)Yi∼NμY,σY2i=1,2,….Then, we can get the following:(20)St=∑i=1nYi∼NnμY,nσY2.By using (11) and (20), we can get the following:(21)Xt+St∼NμβΛt,γ1+nμY,σβ2Λt,γ12+σB2Λt,γ2+σε2+nσY2.Then, we can get the following:(22)P1n=PrNt=n,Xst<H,W1<D,W2<D,⋯,Wn<D,=PrNt=nPrXt+∑i=1nYi<HPrW1<Dn=exp−λtλtnn!ΦH−μβΛt,γ1+nμYσβ2Λt,γ12+σB2Λt,γ2+σε2+iσY2ΦD−μWσWn.Then, we can get the following:(23)R=∑i=1nP1i=ΦH−μβΛt,γ1σβ2Λt,γ12+σB2Λt,γ2+σε2exp−λt+∑i=1nΦD−μWσWiΦH−μβΛt,γ1+iμYσβ2Λt,γ12+σB2Λt,γ2+σε2+iσY2exp−λtλtii!.


## 3.3.2. Case II: Cumulative Shock Modeling

Similarly, under the cumulative shock, the probability at statej can be denoted as follows:(24)P20=PrNt=0,Xst<H=PrNt=0PrXt<H,(25)P21=PrNt=1,Xst<H,W1<D=PrNt=1,Xt+Y1<HPrW1<D=PrNt=nPrXt+Y1+Y2<HPrW1+W2<D=exp−λtλtΦH−μβΛt,γ1+2μYσβ2Λt,γ12+σB2Λt,γ2+σε2+2σY2×ΦD−2μW2σW,(26)P22=PrNt=2,Xst<H,W1+W2<D,=PrNt=2,Xt+St<H,W1+W2<D,=PrNt=nPrXt+Y1+Y2<HPrW1+W2<D,=exp−λtλt22ΦH−μβΛt,γ1+2μYσβ2Λt,γ12+σB2Λt,γ2+σε2+σY2×ΦD−2μWσW,(27)P2n=PrNt=n,Xst<H,W1+W2+⋯+Wn<D=PrNt=nPrXt+Y1+Y2+⋯+Yn<H⋅PrW1+W2+Wn<D=PrNt=nPrXt+∑i=1NtYi<H×Pr∑i=1NtWi<D=exp−λtλtnn!ΦH−μβΛt,γ1+nμYσβ2Λt,γ12+σB2Λt,γ2+σε2+nσY2×ΦD−iμWnσW.Then, we can get the following:(28)R=∑i=1nP2i=∑i=1nexp−λtλtii!ΦH−μβΛt,γ1+iμYσβ2Λt,γ12+σB2Λt,γ2+σε2+iσY2ΦD−iμWiσW.


## 4. Numerical Example

### 4.1. An Nonlinear Wiener Process about the Fatigue Crack Growth Data

In this section, fatigue crack growth data sets [23, 25, 26] are used to verify the proposed models and methods. This data set consists of 12 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 on the shock process based on real data characteristics and literature.Table 1
Fatigue crack growth data from test samples.
Test sampleCycle/104012345678910111210.900.920.971.011.051.091.151.211.281.361.441.551.7220.900.920.961.001.041.081.131.191.261.341.421.521.6730.900.930.961.001.041.081.131.181.241.311.391.491.6540.900.930.971.001.031.071.101.161.221.291.371.481.6450.900.920.970.991.031.061.101.141.201.261.311.401.5260.900.930.961.001.031.071.121.161.211.271.331.401.4970.900.920.960.991.031.061.101.161.211.271.331.401.4980.900.920.950.971.001.031.071.111.161.221.261.331.4090.900.930.960.971.001.051.081.111.161.201.241.321.38100.900.920.940.971.011.041.071.091.141.191.231.281.35110.900.920.940.970.991.021.051.081.121.161.201.251.31120.900.920.940.970.991.021.051.081.121.161.191.241.29From Table1 and Refs [23, 25, 26], we can find the degradation path of the fatigue crack growth, which is nonlinear. Therefore, the nonlinear degradation models can be used to describe the degradation path. In order to use the model M0, M1, M2, M3, and M4, we use a transformation X′t=Xt−0.9 to deal with the fatigue crack growth data and let Λt=Λt,γ=tγ in the model M0, M1, M2, M3, and M4.We know that the performance of the degradation model depends strongly on the appropriateness of the model describing a product’s degradation path. In order to compare the performances of some alternative models, Spiegelhalter et al. [31] proposed the deviance information criterion (DIC) to select the best-fitting model by using the Bayesian approach, and a smaller value of DIC indicates a better model. Based on the models M0, M1, M2, M3, and M4, by using the MCMC method, we can get the value of DIC under different models as shown in Table 2.Table 2
Comparison of different models.
ModelM0M1M2M3M4DIC−784.3−779.5−755.1−752.9−733Ranking12345From Table2, we can find the model M0 with the lowest values of DIC, therefore, model M0 is the best-fitting model. Then, by using the MCMC method, we can get the estimation of unknown parameters under the model M0 as shown in Table 3.Table 3
Estimation results of the unknown parameters.
ParameterMeanStandard errorMC error95% HPD intervalμβ0.0200200.0026681.084E-4(0.014280, 0.02499)σβ0.0063860.0015591.667 E-5(0.004103, 0.01021)σB0.0049680.0010825.871 E-5(0.003388, 0.00768)γ11.3530000.0464300.002557(1.278000, 1.47100)γ22.0420000.1944000.010460(1.620000, 2.38800)Ε6.554 E-94.958 E-82.988 E-9(0.0000, 4.818 E-8)Then, similarly to Refs [25, 26], we suppose Wi∼N(0.06,0.25), Yi∼N(0.04,0.15), H=2.2, D=2.0, and λ=1.0×10−5. According to Eqs.(24)–(27), under the cumulative shock, the probability for the component in the probability of being in state 0, 1, 2, 3, and 4 can be calculated as follows:(29)P0=exp−1.0×10−5tΦH−0.02002×t1.3530.006386×t1.3532+0.0049862×t2.042+6.554×10−92,P1=exp−1.0×10−5t1.0×10−5tΦD−0.060.5×ΦH−0.02002×t1.353−0.040.006386×t1.3532+0.0049862×t2.042+6.554×10−92+0.15,P2=exp−1.0×10−5t1.0×10−5t22ΦD−2×0.062×0.5×ΦH−0.02002×t1.353−2×0.040.006386×t1.3532+0.0049862×t2.042+6.554×10−92+2×0.15,P3=exp−1.0×10−5t1.0×10−5t36ΦD−3×0.063×0.5×ΦH−0.02002×t1.353−3×0.040.006386×t1.3532+0.0049862×t2.042+6.554×10−92+3×0.15,P4=exp−1.0×10−5t1.0×10−5t424ΦD−4×0.064×0.5,×ΦH−0.02002×t1.353−4×0.040.006386×t1.3532+0.0049862×t2.042+6.554×10−92+4×0.15.If we let the vertical axis denote the probability of the system performance, and the horizontal axis denotes time. Then, Figure4 shows the probability curve in every state i (i = 0, 1, 2, 3, 4) as a function of time t. From Figure 4, we know that when the system reaches state 2, the probability of the system performance is approximately 0.26 at 2 × 104 cycle. Moreover, with the increase of time and the number of shocks, we can find the probabilities of the system performance gradually decreasing.Figure 4
Probability curves in different states.In order to analyze the probability of the system performance better in the different states, the sensitivity analysis method is used in this paper. Take state 2 as an example, the sensitivity analysis about the different parameters is plotted in Figures5–7. From Figures 5–7, we can find that the failure threshold H(or D) and the arrivals rate λ of random shocks all have an important effect on the probability of the system performance.Figure 5
Sensitivity analysis of state 2 onλ.Figure 6
Sensitivity analysis of (P)2 on (D).Figure 7
Sensitivity analysis of (P)2 on (H).In addition, based on (27), the system reliability can be obtained as follows:(30)R=∑i=1nPi=ΦH−0.01835×t1.3960.00639×t1.3962+0.006412×t1.851exp−1.0×10−5t+∑i=1nΦD−0.060.5iΦH−0.01476×t1.491−i×0.040.00639×t1.3962+0.006412×t1.851+i×0.15×exp−1.0×10−5t−1.0×10−5tii!.Similarly, according to the parameter estimation results in Table3, the solid curve of the system reliability R(t) based on the nonlinear random effect Wiener process is plotted in Figure 8. From Figure 8, we can find that the reliability of the system almost keeps 1 when t < 0.8 × 104 cycles, and when t > 0.8 × 104 cycles, the reliability decline rapidly. The dotted curve of the system reliability R(t) based on the normal process in Ref [25] is also plotted in Figure 8. Compared with two different degradation models, the normal process assumption underestimates the system’s reliability.Figure 8
The curve of system reliabilityRt.The sensitivity analysis for the system reliabilityR(t) based on the nonlinear random effect Wiener process about the different parameters is plotted in Figures 9–11. From Figures 9–11, we can find that the failure threshold H (or D) and the arrivals rate λ of random shocks all have an important effect on the probability of the system performance. We can also find that the reliability curves shift to the left when the shock arrival rate λ increases from 1.2 to 1.8, but the reliability curves shift to the right when the failure threshold H increases from 2.0 to 2.2 or D increases from 1.8 to 1.2.Figure 9
Sensitivity analysis ofRt on H.Figure 10
Sensitivity analysis ofRt on D.Figure 11
Sensitivity analysis ofRt on λ.


### 4.2. A Real Example of MEMS Oscillators

A MEMS oscillator is a time device and is widely used in electronic systems, transfer systems, and measure systems. The electronic system is a typical system which is subjected to both mechanical and voltage shocks. As we know, on account of operating losses, the mass of MEMS oscillators will decrease over time, and the loss of mass can increase the frequency of vibration. In this paper, similarly to Ref [32, 33], we use a fixed effect Wiener process M4 as shown in (10) to describe the nature process and use the Poisson process in (1) to describe the shock process. The parameters of the Wiener process and Poisson shock process are shown in Table 4.Table 4
Parameter values of the Wiener process and Poisson process.
ParameterValuesSourcesH4100Ref [32, 33]D92Ref [32, 33]Λ0.013Ref [32, 33]Μ0.9Ref [32, 33]Σ20Ref [32, 33]μW72.6Ref [32, 33]σW6.3Ref [32, 33]μY400Ref [32, 33]σY15Ref [32, 33]γ1= γ21.0Ref [32, 33]Similarly, according to Eqs.(14)–(17), under the extreme shock, the probability of the component in the probability of being in state 0, 1, 2, 3, and 4 can be calculated as follows:(31)P0=exp−0.013tΦH−0.9×t202×t,P1=exp−0.013t0.013tΦH−0.9×t−400202×t+152ΦD−72.66.3,P2=exp−0.013t0.013t22ΦH−0.9×t−2×400202×t+2×152ΦD−72.66.32,P3=exp−0.013t0.013t36ΦH−0.9×t−3×400202×t+3×152ΦD−72.66.33,P4=exp−0.013t0.013t424ΦH−0.9×t−4×400202×t+4×152ΦD−72.66.34.Figure12 shows the probability curve of MEMS oscillator in every state i (i = 0, 1, 2, 3, 4) as a function of time t. From Figure 12, we know that when the system reaches state 4, the probability of the system performance is approximately 0.195 at 300 months. Take state 4 as an example, the sensitivity analysis about the different parameters is plotted in Figures 13–15. From Figures 13–15, we can find that the failure threshold H(or D) and the arrivals rate λ of random shocks all have an important effect on the probability of the system performance.Figure 12
Probability curves in different states.Figure 13
Sensitivity analysis of (P)4 on (H).Figure 14
Sensitivity analysis of (P)4 on (D).Figure 15
Sensitivity analysis of (P)4 on λ.Then, based on (23), the reliability of the system which is the total probability of different states can be obtained as follows:(32)R=∑i=1NtPi=ΦH−0.9×t202×texp−0.013t+∑i=1mΦD−72.66.3iΦH−0.9×t−3×400202×t+3×152×exp−1.0×10−5t−1.0×10−5tjj!.Similarly, the curve of the system reliabilityR(t) is plotted in Figure 16 by using the estimation results in Table 4. From Figure 16, we can find that the reliability of the system almost keeps 1 when t < 230 months, and when t > 230 months, the reliability declines rapidly. The sensitivity analyses about the different parameters are plotted in Figures 17–19. From Figures 17–19, we can find that the reliability curves shift to the right when parameter H increases from 3100 to 4100 or D increases from 84 to 92, but the reliability curves shift to the left when the shock arrival rate λ increases from 0.013 to 0.033.Figure 16
The curve of the system reliabilityRt.Figure 17
Sensitivity analysis ofRt on H.Figure 18
Sensitivity analysis ofRt on D.Figure 19
Sensitivity analysis ofRt on λ.


## 4.1. An Nonlinear Wiener Process about the Fatigue Crack Growth Data

In this section, fatigue crack growth data sets [23, 25, 26] are used to verify the proposed models and methods. This data set consists of 12 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 on the shock process based on real data characteristics and literature.Table 1
Fatigue crack growth data from test samples.
Test sampleCycle/104012345678910111210.900.920.971.011.051.091.151.211.281.361.441.551.7220.900.920.961.001.041.081.131.191.261.341.421.521.6730.900.930.961.001.041.081.131.181.241.311.391.491.6540.900.930.971.001.031.071.101.161.221.291.371.481.6450.900.920.970.991.031.061.101.141.201.261.311.401.5260.900.930.961.001.031.071.121.161.211.271.331.401.4970.900.920.960.991.031.061.101.161.211.271.331.401.4980.900.920.950.971.001.031.071.111.161.221.261.331.4090.900.930.960.971.001.051.081.111.161.201.241.321.38100.900.920.940.971.011.041.071.091.141.191.231.281.35110.900.920.940.970.991.021.051.081.121.161.201.251.31120.900.920.940.970.991.021.051.081.121.161.191.241.29From Table1 and Refs [23, 25, 26], we can find the degradation path of the fatigue crack growth, which is nonlinear. Therefore, the nonlinear degradation models can be used to describe the degradation path. In order to use the model M0, M1, M2, M3, and M4, we use a transformation X′t=Xt−0.9 to deal with the fatigue crack growth data and let Λt=Λt,γ=tγ in the model M0, M1, M2, M3, and M4.We know that the performance of the degradation model depends strongly on the appropriateness of the model describing a product’s degradation path. In order to compare the performances of some alternative models, Spiegelhalter et al. [31] proposed the deviance information criterion (DIC) to select the best-fitting model by using the Bayesian approach, and a smaller value of DIC indicates a better model. Based on the models M0, M1, M2, M3, and M4, by using the MCMC method, we can get the value of DIC under different models as shown in Table 2.Table 2
Comparison of different models.
ModelM0M1M2M3M4DIC−784.3−779.5−755.1−752.9−733Ranking12345From Table2, we can find the model M0 with the lowest values of DIC, therefore, model M0 is the best-fitting model. Then, by using the MCMC method, we can get the estimation of unknown parameters under the model M0 as shown in Table 3.Table 3
Estimation results of the unknown parameters.
ParameterMeanStandard errorMC error95% HPD intervalμβ0.0200200.0026681.084E-4(0.014280, 0.02499)σβ0.0063860.0015591.667 E-5(0.004103, 0.01021)σB0.0049680.0010825.871 E-5(0.003388, 0.00768)γ11.3530000.0464300.002557(1.278000, 1.47100)γ22.0420000.1944000.010460(1.620000, 2.38800)Ε6.554 E-94.958 E-82.988 E-9(0.0000, 4.818 E-8)Then, similarly to Refs [25, 26], we suppose Wi∼N(0.06,0.25), Yi∼N(0.04,0.15), H=2.2, D=2.0, and λ=1.0×10−5. According to Eqs.(24)–(27), under the cumulative shock, the probability for the component in the probability of being in state 0, 1, 2, 3, and 4 can be calculated as follows:(29)P0=exp−1.0×10−5tΦH−0.02002×t1.3530.006386×t1.3532+0.0049862×t2.042+6.554×10−92,P1=exp−1.0×10−5t1.0×10−5tΦD−0.060.5×ΦH−0.02002×t1.353−0.040.006386×t1.3532+0.0049862×t2.042+6.554×10−92+0.15,P2=exp−1.0×10−5t1.0×10−5t22ΦD−2×0.062×0.5×ΦH−0.02002×t1.353−2×0.040.006386×t1.3532+0.0049862×t2.042+6.554×10−92+2×0.15,P3=exp−1.0×10−5t1.0×10−5t36ΦD−3×0.063×0.5×ΦH−0.02002×t1.353−3×0.040.006386×t1.3532+0.0049862×t2.042+6.554×10−92+3×0.15,P4=exp−1.0×10−5t1.0×10−5t424ΦD−4×0.064×0.5,×ΦH−0.02002×t1.353−4×0.040.006386×t1.3532+0.0049862×t2.042+6.554×10−92+4×0.15.If we let the vertical axis denote the probability of the system performance, and the horizontal axis denotes time. Then, Figure4 shows the probability curve in every state i (i = 0, 1, 2, 3, 4) as a function of time t. From Figure 4, we know that when the system reaches state 2, the probability of the system performance is approximately 0.26 at 2 × 104 cycle. Moreover, with the increase of time and the number of shocks, we can find the probabilities of the system performance gradually decreasing.Figure 4
Probability curves in different states.In order to analyze the probability of the system performance better in the different states, the sensitivity analysis method is used in this paper. Take state 2 as an example, the sensitivity analysis about the different parameters is plotted in Figures5–7. From Figures 5–7, we can find that the failure threshold H(or D) and the arrivals rate λ of random shocks all have an important effect on the probability of the system performance.Figure 5
Sensitivity analysis of state 2 onλ.Figure 6
Sensitivity analysis of (P)2 on (D).Figure 7
Sensitivity analysis of (P)2 on (H).In addition, based on (27), the system reliability can be obtained as follows:(30)R=∑i=1nPi=ΦH−0.01835×t1.3960.00639×t1.3962+0.006412×t1.851exp−1.0×10−5t+∑i=1nΦD−0.060.5iΦH−0.01476×t1.491−i×0.040.00639×t1.3962+0.006412×t1.851+i×0.15×exp−1.0×10−5t−1.0×10−5tii!.Similarly, according to the parameter estimation results in Table3, the solid curve of the system reliability R(t) based on the nonlinear random effect Wiener process is plotted in Figure 8. From Figure 8, we can find that the reliability of the system almost keeps 1 when t < 0.8 × 104 cycles, and when t > 0.8 × 104 cycles, the reliability decline rapidly. The dotted curve of the system reliability R(t) based on the normal process in Ref [25] is also plotted in Figure 8. Compared with two different degradation models, the normal process assumption underestimates the system’s reliability.Figure 8
The curve of system reliabilityRt.The sensitivity analysis for the system reliabilityR(t) based on the nonlinear random effect Wiener process about the different parameters is plotted in Figures 9–11. From Figures 9–11, we can find that the failure threshold H (or D) and the arrivals rate λ of random shocks all have an important effect on the probability of the system performance. We can also find that the reliability curves shift to the left when the shock arrival rate λ increases from 1.2 to 1.8, but the reliability curves shift to the right when the failure threshold H increases from 2.0 to 2.2 or D increases from 1.8 to 1.2.Figure 9
Sensitivity analysis ofRt on H.Figure 10
Sensitivity analysis ofRt on D.Figure 11
Sensitivity analysis ofRt on λ.


## 4.2. A Real Example of MEMS Oscillators

A MEMS oscillator is a time device and is widely used in electronic systems, transfer systems, and measure systems. The electronic system is a typical system which is subjected to both mechanical and voltage shocks. As we know, on account of operating losses, the mass of MEMS oscillators will decrease over time, and the loss of mass can increase the frequency of vibration. In this paper, similarly to Ref [32, 33], we use a fixed effect Wiener process M4 as shown in (10) to describe the nature process and use the Poisson process in (1) to describe the shock process. The parameters of the Wiener process and Poisson shock process are shown in Table 4.Table 4
Parameter values of the Wiener process and Poisson process.
ParameterValuesSourcesH4100Ref [32, 33]D92Ref [32, 33]Λ0.013Ref [32, 33]Μ0.9Ref [32, 33]Σ20Ref [32, 33]μW72.6Ref [32, 33]σW6.3Ref [32, 33]μY400Ref [32, 33]σY15Ref [32, 33]γ1= γ21.0Ref [32, 33]Similarly, according to Eqs.(14)–(17), under the extreme shock, the probability of the component in the probability of being in state 0, 1, 2, 3, and 4 can be calculated as follows:(31)P0=exp−0.013tΦH−0.9×t202×t,P1=exp−0.013t0.013tΦH−0.9×t−400202×t+152ΦD−72.66.3,P2=exp−0.013t0.013t22ΦH−0.9×t−2×400202×t+2×152ΦD−72.66.32,P3=exp−0.013t0.013t36ΦH−0.9×t−3×400202×t+3×152ΦD−72.66.33,P4=exp−0.013t0.013t424ΦH−0.9×t−4×400202×t+4×152ΦD−72.66.34.Figure12 shows the probability curve of MEMS oscillator in every state i (i = 0, 1, 2, 3, 4) as a function of time t. From Figure 12, we know that when the system reaches state 4, the probability of the system performance is approximately 0.195 at 300 months. Take state 4 as an example, the sensitivity analysis about the different parameters is plotted in Figures 13–15. From Figures 13–15, we can find that the failure threshold H(or D) and the arrivals rate λ of random shocks all have an important effect on the probability of the system performance.Figure 12
Probability curves in different states.Figure 13
Sensitivity analysis of (P)4 on (H).Figure 14
Sensitivity analysis of (P)4 on (D).Figure 15
Sensitivity analysis of (P)4 on λ.Then, based on (23), the reliability of the system which is the total probability of different states can be obtained as follows:(32)R=∑i=1NtPi=ΦH−0.9×t202×texp−0.013t+∑i=1mΦD−72.66.3iΦH−0.9×t−3×400202×t+3×152×exp−1.0×10−5t−1.0×10−5tjj!.Similarly, the curve of the system reliabilityR(t) is plotted in Figure 16 by using the estimation results in Table 4. From Figure 16, we can find that the reliability of the system almost keeps 1 when t < 230 months, and when t > 230 months, the reliability declines rapidly. The sensitivity analyses about the different parameters are plotted in Figures 17–19. From Figures 17–19, we can find that the reliability curves shift to the right when parameter H increases from 3100 to 4100 or D increases from 84 to 92, but the reliability curves shift to the left when the shock arrival rate λ increases from 0.013 to 0.033.Figure 16
The curve of the system reliabilityRt.Figure 17
Sensitivity analysis ofRt on H.Figure 18
Sensitivity analysis ofRt on D.Figure 19
Sensitivity analysis ofRt on λ.


## 5. Conclusions

This paper studied a multi-state complex degradation system reliability assessment model suffering from the DCFP model with nonlinear random effect Wiener process with measurement error and random shocks. Different from the traditional multi-state system model, the transmission time from statei to i+1 is a random variable. By using the multi-state system reliability theory, the system state probabilities are obtained under the different degradation state points, and then the system reliability is obtained. Two numerical examples of fatigue crack growth and MEMS oscillators are given to illustrate the model and method. By analyzing the influence of different parameters on the state probability and the system reliability, some sensitivity analyses are obtained. In future research, some other stochastic processes and shock models are worth investigating.



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