## 3 Reliability Modeling for Self-Recovery Products in Whole Life Cycle  
(1) When \(t>T_{2}\), the product fully undergoes three phases. Degradation by time \(t\) can be derived as follows:  
\[X(t)=X_{W1}(T_{1})+X_{W2}(T_{2}-T_{1})+X_{G}(t-T_{2}) \tag{24}\]On the basis of Equations (2) and (3), Equation (24) can be expressed as follows:  
\[X(t)=\mu_{1}T_{1}+\sigma_{1}B(T_{1})+\mu_{2}(T_{2}-T_{1})+\sigma_{2}B(T_{2}-T_{1 })+X_{G}(t-T_{2}) \tag{25}\]  
Using Equations (21), (22), and (25), we can obtain the degradation by time \(t\). The probability of no soft failure occurring can be expressed as follows:  
\[ \begin{array}{l}p_{1}(X(t)+S(t)<D)\\ =p(\mu_{1}T_{1}+\sigma_{1}B(T_{1})+\mu_{2}(T_{2}-T_{1})+\sigma_{2}B(T_{2}-T_{1 })+X_{G}(t-T_{2})+S(t)<D)\\ =p(X_{G}(t-T_{2})<D-(\mu_{1}T_{1}+\sigma_{1}B(T_{1})+\mu_{2}(T_{2}-T_{1})+ \sigma_{2}B(T_{2}-T_{1})+S(t)))\end{array}  \tag{26}\]  
According to the classification shown in Equation (21), the right-hand side of Equation (26) is \(A_{11}\) when \(n\geq 2\), and \(A_{12}\) when \(n<2\). Then, we obtain  
\[A_{11}\sim N(D-\mu_{1}T_{1}-\mu_{2}(T_{2}-T_{1})-(np_{1}+1)\mu_{3},\sigma_{1}^ {2}T_{1}+\sigma_{2}^{2}(T_{2}-T_{1})+(np_{1}+1)\sigma_{3}^{2}) \tag{27}\]  
\[A_{12}\sim N(D-\mu_{1}T_{1}-\mu_{2}(T_{2}-T_{1}),\sigma_{1}^{2}T_{1}+\sigma_{2 }^{2}(T_{2}-T_{1})) \tag{28}\]  
On the basis of Equations (10) and (13), the probability of product survival in the soft-failure process is expressed as follows:  
\[p_{1}(X(t)+S(t)<D)= \begin{cases}1-\frac{\Gamma(a(t-T_{2}),\beta A_{11})}{ \Gamma(a(t-T_{2}))}&n\geq 2\\ 1-\frac{\Gamma(a(t-T_{2}),\beta A_{12})}{\Gamma(a(t-T_{2}))}&n<2\end{cases}  \tag{29}\]  
The probability of the two situations mentioned above is \(p_{11}(t)\) and \(p_{12}(t)\), respectively. \(T_{1}\) is introduced into the model by using Equation (7). Let us assume that \(T_{2}\) follows a uniform distribution \(U(T_{1},t)\); then, the probability density function of \(T_{2}\) is given as follows:  
\[f_{\mathbb{C}}(T_{2})=\frac{1}{t-T_{1}} \tag{30}\]  
On the basis of Section 2, the probability density function of cumulative degradation \(c\) in the storage stage can be derived as follows:  
\[f_{\mathbb{C}}(c)=\frac{1}{c_{2}-c_{1}}\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} (\frac{c-c_{1}}{c_{2}-c_{1}})^{a-1}(1-\frac{c-c_{1}}{c_{2}-c_{1}})^{b-1} \tag{31}\]  
In Equation (7), the probability density function of \(T_{1}\) becomes  
\[f_{T_{1}}^{W_{1}}(T_{1})=\frac{c}{\sqrt{2\pi\sigma_{1}{}^{2}t^{3}}}\exp(- \frac{(c-\mu_{1}t)^{2}}{2\sigma_{1}{}^{2}t}) \tag{32}\]  
By combining Equations (27)-(32) and (25), when \(n\geq 2\) and \(n<2\), reliability can be given as follows:  
\[R_{SF11}(t)=\int_{c_{1}}^{c_{2}}\int_{0}^{t}\int_{0}^{T_{2}}p_{11}(t)\times f_ {T_{1}}^{W_{1}}(T_{1})\times f_{\mathbb{C}}(T_{2})\times f_{\mathbb{C}}(c)dT_{ 1}dT_{2}dc \tag{33}\]  
\[R_{SF12}(t)=\int_{c_{1}}^{c_{2}}\int_{0}^{t}\int_{0}^{T_{2}}p_{12}(t)\times f_ {T_{1}}^{W_{1}}(T_{1})\times f_{\mathbb{C}}(T_{2})\times f_{\mathbb{C}}(c)dT_{ 1}dT_{2}dc \tag{34}\]The first Wiener process had no impact arrival based on the hypothesis; thus, reliability in this situation can be expressed as follows:  
\[ \begin{array}{ll}R_{1}(t)=&\sum\limits_{n=2}^{\infty}R_{SF11}(t|N(t-T_{1})=n) \,\times p^{n}(W_{i}<H)\times p(N(t-T_{1})=n)\\ &+\sum\limits_{n=0}^{1}R_{SF12}(t|N(t-T_{1})=n)\,\times p^{n}(W_{i}<H)\times p (N(t-T_{1})=n)\end{array}  \tag{35}\]  
where  
\[p(W_{i}<H)=\phi(\frac{H-\mu_{W}}{\sigma_{W}}) \tag{36}\]  
\[p(N(t-T_{1})=n)=\frac{\exp(-\lambda(t-T_{1}))(\lambda(t-T_{1}))^{n}}{n!} \tag{37}\]  
(2) When \(T_{1}<t<T_{2}\) the reliability model comprises the first and second Wiener processes. Total continuous degradation by time \(t\) is given as follows:  
\[X(t)=X_{\text{W1}}(T_{1})+X_{\text{W2}}(t-T_{1})=\mu_{1}T_{1}+\sigma_{1}B(T_{ 1})+\mu_{2}(t-T_{1})+\sigma_{2}B(t-T_{1}) \tag{38}\]  
The cumulative degradation at the end of the first Wiener process is considered a variable in interval \([c_{1},c_{2}]\). The total degradation increment occurred in the second Wiener process.