## 3 Reliability Modeling for Self-Recovery Products in Whole Life Cycle
The probability of the product surviving from the soft-failure process was calculated as follows:  
\[ \begin{array}{ll}p_{2}(X(t)+S(t)<D)\\ =p(\mu_{1}T_{1}+\sigma_{1}B(T_{1})+\mu_{2}(t-T_{1})+\sigma_{2}B(t-T_{1})+S(t)<D )\\ =p(X_{\text{W2}}(t-T_{1})<D-(c+S(t)))\end{array}  \tag{39}\]  
The right-hand side of Equation (39) is marked as \(A_{2}\) and given as follows:  
\[ \begin{array}{ll}A_{2}\sim N\Big{(}D-c-(np_{1}+1)\mu_{3},(np_{1}+1)\sigma_{ 3}^{2}\Big{)}\end{array}  \tag{40}\]  
On the basis of Equations (8) and (40), the cumulative distribution function by the end time of degradation is given as follows:  
\[ \begin{array}{ll}F_{T_{W2}}(t)&=P\{T_{\text{W2}}\leq t\}=P\{X(t)\geq D\}\\ &=\phi\Big{(}\frac{B_{2}(t-T_{1})-A_{2})}{\sigma_{2}\sqrt{t-T_{1}}}\Big{)}+ \exp\Big{(}\frac{2\mu_{3}A_{2}}{\sigma_{2}^{2}}\Big{)}\phi\bigg{(}-\frac{\mu_ {2}(t-T_{1})+A_{2}}{\sigma_{2}\sqrt{(t-T_{1})}}\Big{)}\end{array}  \tag{41}\]  
Then, Equation (41) becomes:  
\[ \begin{array}{ll}p_{2}(X(t)+S(t)<D)\\ =\begin{cases}\phi\Big{(}\frac{A_{2}-p_{2}(t-T_{1})}{\sigma_{2}\sqrt{t-T_{1}}} \Big{)}-\exp\Big{(}\frac{2\mu_{3}A_{2}}{\sigma_{2}^{2}}\Big{)}\phi\bigg{(}- \frac{\mu_{2}(t-T_{1})+A_{2}}{\sigma_{2}\sqrt{(t-T_{1})}}\bigg{)}&n\geq 2\\ \phi\Big{(}\frac{(D-c)-\mu_{2}(t-T_{1})}{\sigma_{2}\sqrt{t-T_{1}}}\Big{)}-\exp \Big{(}\frac{2\mu_{2}(D-c)}{\sigma_{2}^{2}}\Big{)}\phi\bigg{(}-\frac{\mu_{2}(t -T_{1})+(D-c)}{\sigma_{2}\sqrt{(t-T_{1})}}\Big{)}&n<2\end{cases}\end{array}  \tag{42}\]  
The probability for the two situations mentioned in Equation (42) is labeled as \(p_{21}(t)\) and \(p_{22}(t)\), respectively. By combining Equations (7), (30) and (42), we obtain:  
\[R_{SF21}(t)=\int_{c_{1}}^{c_{2}}\int_{0}^{t}p_{21}(t)\times f_{T_{1}}^{W_{1}}( T_{1})\times f_{c}(c)dT_{1}dc \tag{43}\]  
\[R_{SF22}(t)=\int_{0}^{c_{2}}\int_{0}^{t}p_{22}(t)\times f_{T_{1}}^{W_{1}}(T_{1}) \times f_{c}(c)dT_{1}dc \tag{44}\]  
Figure 3: Reliability curves with varying soft-failure threshold. Figure 2: Flowchart of the reliability solving method. Figure 4: Reliability curves with varying arrival rates of the shock process. Figure 5: Reliability curves with varying interval thresholds of self-recovery. Figure 6: Reliability curves with varying drift parameters of the second Wiener process. Figures 3 and 4 show the reliability curves with different arrival rates of the shock process and soft failure threshold, respectively. A larger arrival rate implies a lower curve. A higher arrival rate implies that more shocks arrived at the same time, and the damage caused by these shocks further accelerated the natural degradation process, which considerably decreased the product reliability. Similar changes occurred in reliabilities with a smaller soft-failure threshold. The working life of a product is longer when the initial value of the soft-failure threshold is high. Thus, an inaccurate set of soft-failure thresholds severely affects the final reliability and could produce a result that largely deviates from the actual circumstances. Therefore, the initial setting of soft-failure threshold \(D\) requires a large amount of product data and much practical experience to ensure the actual effectiveness of the reliability model. The empirical threshold given from related studies, such as national standards, provides a reference basis for this paper. Figure 5 demonstrates the influence of the self-recovery feature on the reliability of the considered products. In this model, the time interval between two shocks was set as the self-healing condition. Satisfying the condition is difficult when threshold \(\tau\) increases, and reliability is lower when there are fewer cases of self-recovery. The declining speed of the natural degradation process cuould be determined according to the drift parameters considered in the Wiener process. Larger parameter \(\mu_{2}\) corresponds to a faster degradation rate. Figure 6 shows that a smaller value of parameter \(\mu_{2}\) implied a slower decline of product reliability, resulting in a longer life.