# Competing Failure Modeling for Systems under Classified Random Shocks and Degradation
## 4 Reliability Assemble  
### System Reliability and Competing Failure Analysis  
Systems stay reliable when neither the soft failure nor the hard failure occurs. In practice, it is a focus on how the system reliability probability changes with time. Furthermore, soft failure and hard failure compete with each other, so it is also attractive to study the competing failure probability. The competing failure probability means the failure probability for a certain failure mode (soft failure or hard failure) when the other failure mode is normal at time \(t\). If the competing failure probability of one failure mode is much larger than the other, it means that this kind of failure is more likely to occur. Engineers should pay more attention to the failure mode with a larger competing failure probability. The system survives when no failure occurs. The reliability is the probability that total degradation is smaller than threshold \(H\) and each shock size is under \(D_{0}\). \[ \begin{split} R_{System}(H,t)&=\sum\limits_{n_{3}=0}^{n -n_{2}}\sum\limits_{n_{2}=0}^{n}\sum\limits_{n=0}^{\infty}P\bigg{(}\overset{n_ {3}}{\underset{i=1}{\cap}}\{W_{i}<D_{0}\},X_{s}(t)<H|(N(t)=n,N_{2}(t)=n_{2},N_{ 3}(t)=n_{3})\bigg{)}\\ &\times P(N(t)=n,N_{2}(t)=n_{2},N_{3}(t)=n_{3})\\ &=\sum\limits_{n_{3}=0}^{n-n_{2}}\sum\limits_{n_{2}=0}^{n}\sum \limits_{n=0}^{\infty}P\bigg{(}\overset{n_{3}}{\underset{i=1}{\cap}}\{W_{i}<D_ {0}\},X(t)+\sum\limits_{j=1}^{n_{2}}B_{j}+\sum\limits_{k=1}^{n_{3}}C_{k}<H|(N( t)=n,\\ N_{2}(t)=n_{2},N_{3}(t)=n_{3})\\ &\times P(N(t)=n)P(N_{2}(t)=n_{2})P(N_{3}(t)=n_{3})\\ &=\sum\limits_{n_{3}=0}^{n-n_{2}}\sum\limits_{n_{2}=0}^{n}\sum \limits_{n=0}^{\infty}F_{W}(D_{0})^{n}\bigg{(}\int_{0}^{H}\int_{0}^{u}G(x-u,t )f_{B}{}^{<n>}(u-u_{1})f_{C}{}^{<n>}(u_{1})du_{1}du\bigg{)}\\ &\times\frac{\exp(-\lambda t)(\lambda t)^{n}}{n!}\frac{\exp(- \lambda_{2}t)(\lambda_{2}t)^{n_{2}}}{n_{2}!}\frac{\exp(-\lambda_{3}t)(\lambda_ {3}t)^{n_{3}}}{n_{3}!}\end{split}  \tag{9}\]  
Because shocks are classified into three types, while just the third type may overtake the threshold \(D_{0}\), the hard failure in Equation (9) should use \(N_{3}(t)\) and \(n_{3}\) instead of \(N(t)\) in Equation (8). If \(W_{i}\), \(B_{j}\), and \(C_{k}\) follow a normal distribution, the reliability will be expressed as Equation (10). \[ \begin{split} R_{System}(H,t)&=\sum\limits_{n_{3}=0}^{n -n_{2}}\sum\limits_{n_{2}=0}^{n}\sum\limits_{n=0}^{\infty}F_{W}(D_{0})^{n_{3} }\Phi\bigg{(}\frac{H-(\mu_{g}t+\varphi+n_{2}\mu_{g}+n_{3}t_{C})}{\sqrt{\
u_{ 2}t^{2}+n_{2}\sigma_{2}n_{3}\sigma_{2}c^{2}}}\bigg{)}\\ &\times\frac{\exp(-\lambda t)(\lambda t)^{n}}{n!}\frac{\exp(- \lambda_{2}t)(\lambda_{2}t)^{n_{2}}}{n_{2}!}\frac{\exp(-\lambda_{3}t)(\lambda_ {3}t)^{n_{3}}}{n_{3}!}\end{split}  \tag{10}\]  
There are two scenarios for competing failure probability. The first is the probability that hard failure occurs when no soft failure occurs, while the other is the probability that soft failure occurs when no hard failure occurs. Specifically, we call them competing failure 1 and competing failure 2. For competing failure 1, it requires that hard failure occurs and no soft failure occurs. As we all know, the system stops when failure occurs. Supposing there are n\({}_{3}\) times type-3 shocks, the shock failure must occur at the n3 rd shock. The reliability expression is  
\[