# Competing Failure Modeling for Systems under Classified Random Shocks and Degradation
## 4 Reliability Assemble
### System Reliability and Competing Failure Analysis
\begin{split} P_{\text{CF}_{1}}(H,t)&=\sum\limits_{n_{3 }=1}^{n-n_{2}}\sum\limits_{n_{2}=0}^{n}\sum\limits_{n=1}^{\infty}P\left( \begin{array}{c}\overset{n_{3}-1}{\cap}\{W_{i}<D_{0}\},W_{n_{3}}>D_{0},X_{s}(t )<H|(N(t)=n,\\ N_{2}(t)=n_{2},N_{3}(t)=n_{3})\\ &\times P(N(t)=n,N_{2}(t)=n_{2},N_{3}(t)=n_{3})\\ &=\sum\limits_{n_{3}=1}^{n-n_{2}}\sum\limits_{n_{2}=0}^{n}\sum\limits_{n=1}^ {\infty}P\left(\begin{array}{c}\overset{n_{3}-1}{\cap}\{W_{i}<D_{0}\},W_{n_{ 3}}>D_{0},X(t)+\sum\limits_{j=1}^{n_{2}}B_{j}+\sum\limits_{k=1}^{n_{3}}C_{k}<H| (N(t)=n,\\ N_{2}(t)=n_{2},N_{3}(t)=n_{3})\\ &\times P(N(t)=n)P(N_{2}(t)=n_{2})P(N_{3}(t)=n_{3})\\ &=\sum\limits_{n_{3}=1}^{n-n_{2}}\sum\limits_{n_{2}=0}^{n}\sum\limits_{n=1}^ {\infty}F_{W}(D_{0})^{n_{3}-1}(1-F_{W}(D_{0}))\Big{(}\int_{0}^{H}\int_{0}^{u}G (x-u,t)f_{B}{}^{<m>}(u-u_{1})f_{C}{}^{<n>}(u_{1})du_{1}du\Big{)}\\ &\times\frac{\exp(-\lambda t)(\lambda t)^{n}}{n!}\frac{\exp(-\lambda_{2}t)( \lambda_{2}t)^{n_{2}}}{n_{2}!}\frac{\exp(-\lambda_{3}t)(\lambda_{3}t)^{n_{3}} }{n_{3}!}\end{split}  \tag{11}\]  
For the normal variable, the expression will be  
\[ \begin{split} P_{\text{CF}_{1}}(H,t)&=\sum\limits_{n_{3}= 0}^{n-n_{2}}\sum\limits_{n_{2}=0}^{n}\sum\limits_{n=0}^{\infty}F_{\text{W}}(D_{ 0})^{n_{3}-1}(1-F_{W}(D_{0}))\Phi\bigg{(}\frac{H-(\mu_{g}t+\varphi+n_{2}\mu_{g }+n_{3}t_{C})}{\sqrt{\
u_{2}^{2}t^{2}+n_{2}\sigma_{2}n_{3}+n_{3}\sigma_{C}^{2}} }\bigg{)}\\ &\times\frac{\exp(-\lambda t)(\lambda t)^{n}}{n!}\frac{\exp(- \lambda_{2}t)(\lambda_{2}t)^{n_{2}}}{n_{2}!}\frac{\exp(-\lambda_{3}t)(\lambda_{3}t)^{ n_{3}}}{n_{3}!}\end{split}  \tag{12}\]Competing failure 2 means soft failure occurs while no hard failure occurs. The probability could be expressed by  
\[ \begin{array}{ll}P_{\text{CF}_{2}}(H,t)&=\sum\limits_{n_{3}=0}^{n-n_{2}}\sum \limits_{n_{2}=0}^{n}\sum\limits_{n=0}^{\infty}P\left(\mathop{\cap}\limits_{i=1 }^{n_{3}}\{W_{i}<D_{0}\},X_{s}(t)>H|(N(t)=n,N_{2}(t)=n_{2},N_{3}(t)=n_{3}) \right)\\ &\times P(N(t)=n,N_{2}(t)=n_{2},N_{3}(t)=n_{3})\\ &=\sum\limits_{n_{3}=0}^{n-n_{2}}\sum\limits_{n_{2}=0}^{n}\sum\limits_{n=0}^{ \infty}P\left(\mathop{\cap}\limits_{i=1}^{n_{3}}\{W_{i}<D_{0}\},X(t)+\sum \limits_{j=1}^{n_{2}}B_{j}+\sum\limits_{k=1}^{n_{3}}C_{k}>H|(N(t)=n,\\ N_{2}(t)=n_{2},N_{3}(t)=n_{3})\\ &\times P(N(t)=n)P(N_{2}(t)=n_{2})P(N_{3}(t)=n_{3})\\ &=\sum\limits_{n_{3}=0}^{n-n_{2}}\sum\limits_{n_{2}=0}^{\infty}\mathop{\cap} \limits_{n=0}^{\infty}F_{W}(D_{0})^{n}\Big{[}1-\left(\int_{0}^{H}\int_{0}^{H}G (x-u,t)f_{B}{}^{<m>}(u-u_{1})f_{C}{}^{<n>}(u_{1})du_{1}du\right)\Big{]}\\ &\times\frac{\exp(-\lambda t)(\lambda t)^{n}}{n!}\frac{\exp(-\lambda_{2}t)( \lambda_{2}t)^{n_{2}}}{n_{2}!}\frac{\exp(-\lambda_{3}t)(\lambda_{3}t)^{n_{3}}} {n_{3}!}\\ \end{array}  \tag{13}\]  
\[ \begin{array}{ll}P_{\text{CF}_{2}}(H,t)=&\sum\limits_{n_{3}=0}^{n-n_{2}} \sum\limits_{n_{2}=0}^{n}\sum\limits_{n=0}^{\infty}F_{W}(D_{0})^{n_{3}}\Big{[}1 -\Phi\left(\frac{H-(\mu_{B}t+\varphi+n_{2}\mu_{B}+n_{3}tC)}{\sqrt{\sigma_{B}t^ {2}t^{2}+2n_{2}\sigma_{B}^{2}+1\sigma_{B}tC^{2}}}\right)\Big{]}\\ &\times\frac{\exp(-\lambda t)(\lambda t)^{n}}{n!}\frac{\exp(-\lambda_{2}t)( \lambda_{3}t)^{n_{2}}}{n_{2}!}\frac{\exp(-\lambda_{3}t)(\lambda_{3}t)^{n_{3}}} {n_{3}!}\end{array}  \tag{14}\]  
Competing failure probabilities are another effective way to demonstrate the system status besides the system reliability. Even for the same system reliability value, the competing failure probabilities for the same risk mode may be different. Figure 2 introduces the problem. There are four possible combinations for soft risk mode and hard risk mode. The proportion of area 1 to the total area denotes system reliability. The system reliabilities in Figure 2A,B are the same, while the proportions of area 2 (competing failure 2) are different. It is the same as area 3. In addition, the soft failure needs to pay more attention in Figure 2A, while it becomes hard failure in Figure 2B even though the system reliabilities are the same. Thus, the competing failure probabilities are also necessary.