## 4 Preventive Maintenance Policies
### Reliability Analysis for the Shock Model  
The simulation result of the system reliability under trigger mechanism 1 is presented in Figure 4, where \(m_{1}=1\), \(\lambda=0.55\), and \(\delta=2\). From Figure 4, we can know that as \(n\) increases, the system will fail after more shocks, and the system reliability increases. When \(k\) increases, the self-exciting mechanism will be triggered after more shocks, and the reliability of the system increases. When \(p\) increases, the probability of the effective shock arrival increases, and the reliability of the system decreases. It can be observed that the system is most reliable when \(n=5\), \(k_{1}=2\), and \(p=0.45\). The simulated system reliability under trigger mechanism 2 is presented in Figure 5, where \(m_{2}=1\), \(\lambda=0.55\), and \(\delta=2\). The effect of \(n\), \(k_{2}\) and \(p\) on the system reliability is similar to trigger mechanism 1. The reliability of the system is highest when \(n=5\), \(k_{2}=2\), and \(p=0.4\). The comparison of the simulated system reliability under trigger mechanism 1 and trigger mechanism 2 is presented in Figure 6, where \(n=5\), \(k=2\), \(m=1\), \(\lambda=0.55\), \(\delta=2\), and \(p=0.4\). From Figure 6, we can see that the system reliability under trigger mechanism 2 is higher than the system reliability under trigger mechanism 1. It can be observed that under self-exciting mechanisms, the reliability is much lower than that under the self-healing mechanism in ref. [23]. Such phenomenon is due to the fact that the damage under a self-exciting mechanism is much larger than that under a self-healing mechanism; thus, the system reliability is lower under self-exciting shock models. Furthermore, the reliability under a self-exciting mechanism is much lower than that under a protective mechanism in a shock environment [37], since the damage is reduced via the protective mechanism, which increases system reliability. Under maintenance strategy 1, the optimal solution with multiple parameters under trigger mechanism 1 is obtained, as shown in Table 3, where we can see that the increase of \(p\) leads to the decrease of the optimal value \(N^{*}\). In other words, the greater the probability of effective shocks arrival, the lower the reliability; thus, the maintenance should be carried out after fewer shocks. \(N^{*}\) increases with the rise of \(n\) or \(k\). That is to say, the system can be repaired after more shocks when the system can withstand more shocks before it fails. The system should undergo preventive maintenance after experiencing more shocks if more shocks can trigger the self-exciting mechanism. Figure 7 shows the cost function with  
Figure 4: Simulation result of the system reliability under trigger mechanism 1.  
several parameters. From Figure 7, we can see that the cost function is convex, and there exists an optimal solution. Figure 5: Simulation result of the system reliability under trigger mechanism 2. Figure 6: The comparison of different trigger mechanisms. The optimal results of different cost parameter values are shown in Table 4, where \(n=5\), \(k_{1}=2\), and \(p=0.45\). The results show that the larger \(c_{p}\) is, the larger the optimal solution \(N^{*}\) is, and the later the replacement time is. When \(c_{f}\) increases, it needs to be replaced in advance. Under maintenance strategy 2, the optimal \(t^{*}\) and the corresponding \(C_{2}(t^{*})\) are obtained, as shown in Table 5. As can be observed from Table 5, the increase of \(p\) results in the decrease of optimal \(t^{*}\), implying that the system suffers from more effective shocks, and preventive maintenance should be performed earlier. However, it leads to the increase of the average cost per unit time. The optimal \(t^{*}\) increases when \(n\) increases, which means that