## 2 System Description and the Soft-Failure Process Analysis of Self-Recovery Products
### Stochastic Process Model  
The reliability model based on the Wiener process is expressed as follows:  
\[X_{W}(t)=X_{W}(0)+\mu\Lambda(t;\theta_{\Lambda})+\sigma B(\tau(t;\theta_{\tau})) \tag{1}\]  
where \(X_{W}(0)\) is the initial degradation, and \(\mu\) and \(\sigma\) represent the drift and diffusion coefficients, respectively. Functions \(\Lambda(t;\theta_{\Lambda})\) and \(\tau(t;\theta_{\tau})\) describe the change in mean degradation and time transformation, respectively. The initial degradation was assumed to be \(X_{W}(0)=0\) and \(\Lambda(t;\theta_{\Lambda})=\tau(t;\theta_{\tau})=t\). On the basis of the properties of the Wiener process, we obtained \(X_{W}(t)\sim N(\mu t,\sigma^{2}t)\).  
The two Wiener processes mentioned above are expressed as follows:  
\[X_{W1}(t)=\mu_{1}t+\sigma_{1}B(t) \tag{2}\]  
\[X_{W2}(t)=\mu_{2}t+\sigma_{2}B(t) \tag{3}\]  
Then, we obtained  
\[X_{W1}(t)\sim N(\mu_{1}t,\sigma_{1}{}^{2}t) \tag{4}\]  
\[X_{W2}(t)\sim N(\mu_{2}t,\sigma_{2}^{2}t) \tag{5}\]  
When a soft failure occurs, the life of product \(T_{W}\) is equal to the first passage time for the degradation threshold:  
\[T_{W}=\inf\{t|X_{W}\geq D,t\geq 0\} \tag{6}\]  
On the basis of the Wiener process, \(X_{W1}(t)=\mu_{1}+\sigma_{1}B(t)\); \(T_{W}\) follows an inverse Gaussian distribution. Let us assume that \(T_{W1}\) represents the end time of the first Wiener  
Figure 1: Dependent competing failure processes of a product.  
process. Then, we obtain the probability density function and the cumulative distribution function of \(T_{W1}\) as follows:  
\[f_{T_{W1}}=\frac{c}{\sqrt{2\pi\sigma_{1}^{2}t^{3}}}\exp(-\frac{(c-\mu_{1}t)^{2}}{ 2\sigma_{1}^{2}t}) \tag{7}\]  
\[F_{T_{W1}}=\phi(\frac{\mu_{1}t-c}{\sigma_{1}\sqrt{t}})+\exp(\frac{2\mu_{1}c}{ \sigma_{1}^{2}})\phi(-\frac{\mu_{1}t+c}{\sigma_{1}\sqrt{t}}) \tag{8}\]  
The gamma process is expressed as \(G(t)\sim Ga(\beta(t),\eta(t))\), where \(\beta(t)\) is the scale parameter, and \(\eta(t)\) is the shape parameter. The probability density function of degradation \(X_{G}(t)\) in this phase is expressed as follows:  
\[f(X_{G}(t)|\eta(t),\beta(t))=\frac{x^{\eta(t)-1}\beta(t)^{\eta(t)}e^{-\beta(t) x}}{\Gamma(\eta(t))},x>0 \tag{9}\]  
\(T_{G}\) is denoted as \(T_{G}=\inf\{X_{G}(t)>D,t\geq 0\}\) in the gamma process. Let \(\beta(t)=\beta\) and \(\eta(t)=\alpha t\); then, the cumulative distribution function of Equation (9) can be expressed as follows:  
\[\begin{array}{ll}F_{T_{G}}(t)&=P\{T_{G}\leq t\}=P\{X_{G}(t)\geq D\}=\int_{D} ^{\infty}f(X_{G}(t)|\eta(t),\beta)dy\\ &=\int_{D}^{\infty}\frac{x^{\alpha t-1}\beta^{\alpha t}e^{-\beta x}}{\Gamma( \alpha t)}dx=\int_{D\beta}^{\infty}\frac{u^{\alpha t-1}e^{-\mu}}{\Gamma(\alpha t )}d\mu\\ &=\frac{\Gamma(\alpha t,D\beta)}{\Gamma(\alpha t)}\end{array} \tag{10}\]  
where  
\[\Gamma(a,b)=\int_{b}^{\infty}y^{a-1}e^{-y}dy \tag{11}\]  
\[\Gamma(a)=\int_{0}^{\infty}y^{a-1}e^{-y}dy \tag{12}\]  
On the basis of Equation (10), the reliability function of the gamma process can be expressed as follows:  
\[R_{G}(t)=P\{T_{G}\geq t\}=1-F_{T_{G}}(t)=1-\frac{\Gamma(\alpha t,\beta D)}{ \Gamma(\alpha t)} \tag{13}\]