## 2 System Description and the Soft-Failure Process Analysis of Self-Recovery Products
### Derivation of the Reliability Function for Self-Recovery Products  
The cumulative distribution function of degradation increment \(Y_{0i}\) caused by \(W_{i}\) is expressed as \(F_{Y_{0i}}(y)=p(Y_{0i}\leq y)\). The effective shock time by time \(t\) is denoted as \(N(t)\), and the total degradation increment is expressed as follows:  
\[S_{0}(t)=\left\{\begin{array}{ll}N(t)\\ \sum\limits_{i=1}^{N(t)}Y_{0i}&N(t)>0\\ 0&N(t)=0\end{array}\right. \tag{14}\]  
Due to the self-recovery property of the product, \(S_{0}(t)\) was larger than the actual value. Let us assume that \(S(t)\) represents the degradation increment considered in self-recovery, and the actual degradation increment of \(W_{i}\) is \(Y_{i}\). Then, to obtain the distribution function of \(Y_{i}\), the cumulative distribution function is expressed as follows:  
\[\begin{array}{ll}F_{Y_{i}}(y)&=p(Y_{i}\leq y)=1-p(Y_{i}>y)\\ &=1-\left[p(Y_{0i}>y|B_{i}\leq\tau)\times p(B_{i}\leq\tau)+p(Y_{0i}>y|B_{i}> \tau)\times p(B_{i}>\tau)\right]\end{array} \tag{15}\]Self-recovery occurs when shock interval \(B_{i}>\tau\),which means that \(Y_{0i}=0\). Thus, \(p(Y_{0i}>y)=0\). On the basis of the property of a homogeneous Poisson process, the shock interval (\(\lambda\)) followed an exponential distribution. Thus, we obtained \(p(B_{i}\leq\tau){=}1-\mathrm{e}^{-\lambda\tau}\), and Equation (15) could be expressed as follows:  
\[\begin{array}{ll}F_{Y_{i}}(y)&=1-p(Y_{0i}>y|B_{i}\leq\tau)\times p(B_{i}\leq \tau)\\ &=p(Y_{0i}\leq y\Big{|}B_{i}\leq\tau)\times p(B_{i}\leq\tau){=}(1-\mathrm{e}^{ -\lambda\tau})F_{Y_{0i}}(y)\end{array} \tag{16}\]  
Moreover, Equation (16) is given as follows:  
\[p_{Y_{i}}(y)=(1-\mathrm{e}^{-\lambda\tau})p_{Y_{0i}}(y) \tag{17}\]  
Let us assume that \(K(t)\) represents the number of shocks that caused the degradation increment. On the basis of Equation (17), the ratio of \(K(t)\) to all the shocks in \((0,t)\) can be derived as follows:  
\[p_{1}=1-e^{-\lambda\tau} \tag{18}\]  
On the basis of [PERSON]'s decomposition theorem and Equation (18), the self-recovery process \((\lambda e^{-\lambda\tau})\) and shock process \(\big{(}\big{(}1-e^{-\lambda\tau}\big{)}\big{)}\) could be determined, and both followed the Poisson process. The probability of \(k_{1}\) times self-recovery occurring by time \(t\) was calculated as follows:  
\[p\{K_{1}(t)=k_{1}\}=\frac{(e^{-\lambda\tau}\lambda t)^{k_{1}}}{k_{1}!}e^{-e^{- \lambda\tau}\lambda t} \tag{19}\]  
Considering the safety of the proposed model, the number of shocks causing the degradation increment can be expressed by rounding up the impact ration as follows:  
\[K(t)=p_{1}N(t)+1 \tag{20}\]  
On the basis of the definition of self-recovery, the product recovers from damage when \(N(t)<2\). The total degradation increment can be expressed as follows:  
\[S(t)=\begin{cases}\sum\limits_{i=1}^{K(t)}Y_{i}&N(t)\geq 2\\ 0&N(t)<2\end{cases} \tag{21}\]  
Let \(N(t)=n\); then, when \(n\geq 2\), we obtain  
\[S(t)=\sum\limits_{i=1}^{K(t)}Y_{i}\sim N((np_{1}+1)\mu_{3},(np_{1}+1)\sigma_{3 }^{2}) \tag{22}\]  
Considering the self-recovery of the product, the total degradation by time \(t\) is given as \(X_{s}(t)=X(t)+S(t)\). Reliability in competing failure is expressed as follows:  
\[\begin{array}{ll}R(t)&=\sum\limits_{n=0}^{\infty}P(X(t)+S(t)<D,\mathop{ \cap}\limits_{i=1}^{N(t)}(W_{i}<H)|P(N(t)=n))\times P(N(t)=n)\\ &=\sum\limits_{n=0}^{\infty}R_{SF}(t|N(t)=n)\,\times p^{n}(W_{i}<H)\times p(N( t)=n)\end{array} \tag{23}\]