## 2 Model Assumptions and Model Description  
_Problem Description_  
The system considered operates in a stochastic environment modeled by a homogeneous Poisson impact process. The time lag between adjacent shocks is independent and identically distributed following exponential distribution with parameter \(\lambda\). Arrived shocks can be classified as effective shocks, with probability \(p\) that can lead to system damage, and ineffective shocks, with probability \(1-p\), which are harmless to the system. When the accumulative number of effective shocks exceeds the threshold \(n\), the system will fail. When the time interval between an ineffective shock and the previous shock is less than the given threshold \(\delta\), it is defined as an \(\delta\)--ineffective shock. When the arrival of shocks is more frequent, the self-exciting mechanism is triggered, and the accumulative number of effective shocks will increase by a certain number. Two types of self-exciting mechanisms are considered.  
Trigger mechanism 1: Under this trigger mechanism, among the run of ineffective shocks, when the accumulative number of \(\delta\)--ineffective shocks reaches \(k_{1}\), the self-exciting mechanism will be triggered to increase the accumulative number of effective shocks by \(m_{1}\).  
Let \(X_{i},i=1,2,\ldots\) indicate the time lag between the (\(i-1\))th and \(i\)th shock. Figure 1 shows the possible evolution of the system failure process under trigger mechanism 1, where \(n=5\), \(k_{1}=2\), \(\delta=1.25\), \(m_{1}=1\). Let '1' denote effective shocks, '0' denote ineffective shocks, and '01' denote the initial state of the shock sequence. In Figure 1a, the first four shocks are \(\delta\)--ineffective shocks. Figure 1b shows the corresponding accumulative number of effective shocks. Upon the 3 rd shock, the accumulative number of \(\delta\)--ineffective shocks in the trailing run of ineffective shock reaches 2, then the self-exciting mechanism will be triggered and the accumulative number of effective shocks increases by 1. In the same way, when the 6 th shock occurs, the accumulative number of ineffective shocks in the trailing run of \(\delta\)--ineffective shocks reaches 2, the self-exciting mechanism is triggered again, and the accumulative number of effective shocks increases by 1. When the 9 th shock occurs, the accumulative number of effective shocks reaches 5, and the system fails.  
Trigger mechanism 2: Under this trigger mechanism, among the run of ineffective shocks, when the consecutive number of \(\delta\)--ineffective shocks reaches \(k_{2}\), the self-exciting mechanism is triggered to increase the accumulative number of effective shocks by \(m_{2}\).  
Figure 2 shows the failure process of the system, where \(n=5\), \(k_{2}=3\), \(\delta=1.25\), and \(m_{2}=1\). When the 4 th shock occurs, the accumulative number of \(\delta\)--ineffective shocks in the trailing run of ineffective shock reaches 2; then, the self-exciting mechanism will be triggered, and the accumulative number of effective shocks should be increased by 1. When the 4 th shock occurs, the consecutive number of ineffective shocks in the trailing run of ineffective shocks is three, so the self-exciting mechanism is triggered. When the 8 th shock occurs, the consecutive number of \(\delta\)--ineffective shocks in the trailing run of ineffective shocks reaches 2, the self-exciting mechanism is triggered, and the accumulative number of effective shocks should be increased by 1. When the 10 th shock occurs, the accumulative effective shock times of the system reaches 5, and the system fails.  
Figure 1: Failure process of the system under self-exciting trigger mechanism 1. (a denotes the type of shock and b denotes the shock occurrence time).  
Figure 2: Failure process of the system under self-exciting trigger mechanism 2. (a denotes the type of shock and b denotes the shock occurrence time).