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