## 3 Reliability Modelling and Evaluation  
### Reliability Modelling and Evaluation under Triggering Mechanism 1  
In this section, we use the finite Markov chain imbedding method to conduct probabilistic analysis under triggering mechanism 1. First, we define three random variables in the sequence of \(h_{1}\) random shocks. Let \(U_{h_{1}}\) denote the accumulative number of effective shocks. Let \(V_{h_{1}}\) represent the number of \(\delta\)--ineffective shocks in the trailing run of ineffective shocks. Let \(W_{h_{1}}\) denote the number of \(\delta\)--ineffective shocks that can trigger the next self-exciting mechanism, where \(W_{h_{1}}=V_{h_{1}}-\left[\frac{V_{h_{1}}}{K_{1}}\right]\cdot k_{1}\left[ \frac{V_{h_{1}}}{K_{1}}\right]\) is the integer part of \(\frac{V_{h_{1}}}{K_{1}}\). A Markov chain \(\left\{Y_{h_{1}},h_{1}\geq 0\right\}\) that is associated with \(U_{h_{1}}\) and \(W_{h_{1}}\) is defined as \(Y_{h_{1}}=\left(U_{h_{1}},W_{h_{1}}\right)\), for \(h_{1}\geq 0\). The state space is \(\Omega_{1}=\left\{(u_{1},w_{1}):0\leq u_{1}\leq n_{1}-1\right\}\cup\left\{E_{a}\right\}\). Let \(Y_{h_{1}}=\left(u_{1},w_{1}\right)\) denote that the system has been impacted \(h_{1}\) times. Let \(u_{1}\) denote the accumulative number of effective shocks. Let \(w_{1}\) denote the number of \(\delta\)--ineffective shocks that can trigger the next self-exciting mechanism. Let \(Y_{0}=(0,0)\) denote the initial state. Let \(E_{a}\) denote the absorbing state where the accumulative number of effective shocks reaches \(n\) and the system fails. Let \(q_{1}\) denote the probability that the \(h_{1}\) - th shock is a \(\delta\)--ineffective shock whose time lag with the preceding shock is less than \(\delta\). Let \(q_{2}\) denote the probability that the \(h_{1}\) - th shock is an ineffective shock whose time lag with the preceding shock exceeds \(\delta\), where \(q_{1}=q\cdot P\left\{X_{m}<\delta\right\}\), \(q_{2}=q\cdot P\left\{X_{m}\geq\delta\right\}\) and \(q_{1}+q_{2}=q\). Table 1 shows the transition rules of the transition probability matrix \(\mathbf{\Lambda}_{1}\) of Markov chain \(\left\{Y_{h_{1}},h_{1}\geq 0\right\}\). For example, when \(n=4\), \(k_{1}=2\), \(m_{1}=1\), the state space \(\Omega_{1}\) for the Markov chain \(\left\{Y_{h_{1}},h_{1}\geq 0\right\}\) is constructed as  
\[\Omega_{1}=\left\{(0,0),(0,1),(1,0),(1,1),(2,0),(2,1),(3,0),(3,1)\right\} \cup\left\{E_{a}\right\}\]  
Based on the transition rules, we can obtain the one-step transition probability matrix:  
\[\mathbf{\Lambda}_{1}=\left( \begin{matrix}0,0\\ 0\end{matrix} \right)\left[ \begin{matrix}q_{2}&q_{1}&p&0&0&0&0&0&0\\ 0&q_{2}&p+q_{1}&0&0&0&0&0\\ 1,0\\ 1,1\end{matrix} \right)\left[ \begin{matrix}0&0&q_{2}&p+q_{1}&0&0&0&0&0&0\\ 0&0&q_{2}&q_{1}&p&0&0&0\\ 0&0&0&0&q_{2}&p+q_{1}&0&0&0\\ 0&0&0&0&0&q_{2}&q_{1}&p&0\\ 0&0&0&0&0&q_{2}&p+q_{1}&0&0\\ (3,0)&0&0&0&0&0&q_{2}&q_{1}&p\\ (3,1)&0&0&0&0&0&0&q_{2}&p+q_{1}&p\\ \end{matrix} \right].\]  
\begin{table}
\begin{tabular}{c c c} \hline
**No. ** & **Applicability** & **Condition** & **Prob. ** \\ \hline
1 & \(0\leq u_{1}<n-1,0\leq w_{1}<k_{1}-1\) & \(P\left\{Y_{h_{1}}=(u_{1},w_{1})|Y_{h_{1}-1}=(u_{1},w_{1})\right\}\) & \(q_{2}\) \\
2 & \(0\leq u_{1}<n-1,0\leq w_{1}<k_{1}-1\) & \(P\left\{Y_{h_{1}}=(u_{1}+1,0)|Y_{h_{1}-1}=(u_{1},w_{1})\right\}\  
The transition probability matrix \(\mathbf{\Lambda}_{1}\) with order \(\gamma_{1}\) can be partitioned into four submatrices:  
\[\mathbf{\Lambda}_{1}= \begin{bmatrix}\mathbf{P}_{1_{(\gamma-1)\times(\gamma-1)}}& \mathbf{Q}_{1_{(\gamma-1)\times 1}}\\ \mathbf{0}_{1_{1\times(\gamma-1)}}&\mathbf{E}_{1_{1\times 1}}\end{bmatrix}