# Competing Failure Modeling for Systems under Classified Random Shocks and Degradation
## 3 Modeling for Risk Modes  
### Modeling for Soft Failure  
Generally, the overall degradation \(X_{s}(t)\) consists of two parts, the pure degradation \(X(t)\) caused by wastage such as wear, corrosion, and aging and the instantaneous degradation increments \(S(t)\) caused by random shocks. As soon as the total degradation exceeds its threshold, soft failure occurs. A linear degradation path \(X(t)=\varphi+\beta t\) is used to model pure degradation, where the initial degradation value \(\varphi\) and rate \(\beta\) can be constants or random variables. Random shocks arrive with a Poisson process \(\{N(t),\,t\geq 0\}\), whose rate is \(\lambda\). The size of each shock is a random variable \(W_{i}\). Each shock may cause an instantaneous increment following the same distribution, no matter how large the shock size is in previous studies. Actually, the increment caused by a shock cannot be the same or the same distribution if the shock sizes are different. It is easy to understand that a larger shock size is more likely to cause a larger instantaneous increment. For example, wear on a hinge leads to degeneration over time, and shock will also cause degradation incrementally. Shock 1 has a very small size, just like being lightly tickled by a finger. Shock 2 has a larger size, like being struck by a hammer. Shock 3 has the maximal size among these three shocks, like a heavy strike by a motor vehicle. Supposing no hard failure occurs, the damages caused by these three shocks cannot be the same, obviously. Shocks like shock 1 may cause nothing to the systems, so there will be no degradation increment. Shocks like shock 2 and shock 3 will lead to degradation increments, while the increments caused by shock 3 are even bigger than those by shock 2. Therefore, the shocks could be classified into various types according to their shock sizes. At the same time, the degradation increments caused by shocks from different shock types also occur under different distributions. The new model in the article classifies the shocks into three categories according to their shock sizes. The degradation increments vary with different categories. The shock size of \(W_{i}\)s follows a specific distribution. If one shock's size is under a determined value \(D_{2}\), it will not cause a degradation increment like \(W_{1}\), \(W_{2}\), \(W_{4}\) in Figure 1. This kind of shock is defined as shock type 1. Shock sizes between \(D_{2}\) and \(D_{1}\), like \(W_{3}\), \(W_{5}\) in Figure 1, are defined as shock type 2 with instantaneous increment \(B_{j}\). \(B_{j}\) is i.i.d. random variable with its specific distribution parameters. Similarly, shocks with size between \(D_{1}\) and hard failure thresholds will lead to another degradation increment \(C_{k}\) with different distribution parameters like \(W_{6}\) in Figure 1. Additionally, shocks like \(W_{6}\) are defined as shock type 3. Shocks arrive with a Poisson process \(\{N(t),\,t\geq 0\}\) whose rate is \(\lambda\). For shock type \(i\), it also follows the Poisson process \(\{N_{i}(t),\,t\geq 0\}(i=1,\,2,\,3)\) with rate \(\lambda_{i}\). According to stochastic processes theory, the expressions are as follows:  
\[ \begin{array}{l}N(t)=N_{1}(t)+N_{2}(t)+N_{3}(t)\\ \lambda_{1}=r_{1}\lambda,\ \lambda_{2}=r_{2}\lambda,\ \lambda_{3}=r_{3}\lambda \end{array}  \tag{1}\]  
where \(r_{i}\) is the ratio of type \(i\) shock. Additionally, \(r_{i}\) can be obtained by following expressions. \[ \begin{array}{l}r_{1}=P(W_{i}<D_{2})\\ r_{2}=P(D_{2}<W_{i}<D_{1})\\ r_{3}=P(D_{1}<W_{i}<D_{0})\end{array}  \tag{2}\]  
The cumulative instantaneous increments for degradation change into:  
\[S(t)= \begin{cases}\sum\limits_{i=1}^{N_{1}(t)}A_{i}+\sum\limits_{j=1}^{N_{2}(t )}B_{j}+\sum\limits_{k=1}^{N_{2}(t)}C_{k},&N_{i}(t)>0,\\ 0,&N_{i}(t)=0,\end{cases}  \tag{3}\]  
where \(A_{i}\) is equal to zero when no degradation occurs. \(B_{j}\) and \(C_{k}\) are i.i.d. (Independent Identically Distribution) random variables with different distribution parameters.