## 3 Reliability Modeling for Self-Recovery Products in Whole Life Cycle
Figure 8: Reliability curves with varying diffusion parameters of the second Wiener process and varying scale parameters of the gamma process. Figure 7: Reliability curves with varying shape coefficients of the gamma process. Time \(T_{2}\) was random, and the reliability of the product in \((T_{1},t)\) was determined via the second Wiener and gamma processes. As shown in Figures 7 and 8, reliability increased with an increase in the shape parameter \(\alpha\) or a decrease in scale parameter \(\beta\). Equation (47) had many variables, and sampling was used in the calculation process. Thus, the computation fluctuated slightly; however, it did not affect the accuracy of the model. On the basis of the original parameters above, parameter \(\mu_{2}\) was 1.5. The dashed lines in Figure 8 comprise a set of curves obtained by changing the second Wiener process' diffusion parameter \(\sigma_{2}\). Although the differences between the curves were not significant, an increase in reliability with the increase in diffusion parameter \(\sigma_{2}\) was evident. To verify the proposed model, the curves were compared with the results of the model with only the Wiener or gamma process. Self-recovery was also considered for the verification. Using Equations (43)-(46), the reliability obtained with the Wiener process could be expressed as follows:  
\[R_{4}(t)=\left\{ \begin{array}{cc}\sum\limits_{n=2}^{\infty}R_{SF21}(t|N(t-T_ {1})=n)\,\times p^{n}(W_{i}<H)\times p(N(t-T_{1})=n)&\\ +\sum\limits_{n=0}^{1}R_{SF22}(t|N(t-T_{1})=n)\,\times p^{n}(W_{i}<H)\times p (N(t-T_{1})=n)&\\ \phi(\frac{D-\mu_{1}t}{\sigma_{1}\sqrt{t}})-\exp(\frac{2\mu_{1}D}{\sigma_{1} \sqrt{t}})\phi(-\frac{\mu_{1}t+D}{\sigma_{1}\sqrt{t}})&0<t\leq T_{1}\end{array} \right. \tag{48}\]  
Using the gamma process, Equation (26) becomes:  
\[ \begin{array}{cc}p_{2}(X(t)+S(t)<D)&=p(\mu_{1}T_{1}+\sigma_{1}B(T_{1})+X_{G} (t-T_{1})+S(t)<D)\\ &=p(X_{G}(t-T_{1})<D-(\mu_{1}T_{1}+\sigma_{1}B(T_{1})+S(t)))\end{array}  \tag{49}\]  
Similarly, when \(n\geq 2\) and \(n<2\), the right-hand side of Equation (49) was \(A_{51}\) and \(A_{52}\), respectively. Then, the distribution is given as follows:  
\[A_{51}\sim N(D-\mu_{1}T_{1}-(np_{1}+1)\mu_{3},\sigma_{1}^{2}T_{1}+(np_{1}+1) \sigma_{3}^{2}) \tag{50}\]  
\[A_{52}\sim N(D-\mu_{1}T_{1},\sigma_{1}^{2}T_{1}) \tag{51}\]  
As the model was described only with the gamma process, Equation (29) could be written as follows:  
\[p_{2}(X(t)+S(t)<D)=\left\{ \begin{array}{cc}1-\frac{\Gamma(\alpha(t-T_{1}), \beta A_{51})}{\Gamma(\alpha(t-T_{1}))}&n\geq 2\\ 1-\frac{\Gamma(\alpha(t-T_{1}),\beta A_{52})}{\Gamma(\alpha(t-T_{1}))}&n<2 \end{array} \right. \tag{52}\]  
When \(n\geq 2\) and \(n<2\), probabilities were \(p_{21}(t)\) and \(p_{22}(t)\), respectively. Then, Equations (33) and (34) are given as follows:  
\[R_{SF51}(t)=\int_{c_{1}}^{c_{2}}\int_{0}^{t}p_{21}(t)\times f_{T_{1}}^{W_{1}}( T_{1})\times f_{C}(c)dT_{1}dc \tag{53}\]  
\[R_{SF52}(t)=\int_{c_{1}}^{c_{2}}\int_{0}^{t}p_{22}(t)\times f_{T_{1}}^{W_{1}}( T_{1})\times f_{C}(c)dT_{1}dc \tag{54}\]  
Reliability when \(t>T_{1}\) is given as follows:  
\[ \begin{array}{cc}R_{53}(t)=&\sum\limits_{n=2}^{\infty}R_{SF51}(t|N(t-T_{1})= n)\,\times p^{n}(W_{i}<H)\times p(N(t-T_{1})=n)\\ &+\sum\limits_{n=0}^{1}R_{SF52}(t|N(t-T_{1})=n)\,\times p^{n}(W_{i}<H)\times p (N(t-T_{1})=n)\end{array}  \tag{55}\]Using Equation (46), the reliability modeled with the gamma process is expressed as follows:  
\[R_{5}(t)=\left\{ \begin{aligned} R_{53}(t)& t>T_{1}\\ \phi(\frac{D-\mu_{1}t}{\sigma_{1}\sqrt{t}})-\exp(\frac{2\mu_{1}D}{ \sigma_{1}{}^{2}})\phi(-\frac{\mu_{1}t+D}{\sigma_{1}\sqrt{t}})& 0<t\leq T_{1}\end{aligned} \right. \tag{56}\]  
Figure 9 contains the curves obtained with Equations (48) and (56), and the simulation and analytical results based on the proposed model. The proposed model was closer to the actual situation, indicating its better accuracy; however, it was necessary to introduce self-recovery for the analysis of specific products.