## 1 Introduction
In addition to performing preventive maintenance, self-healing ability has recently emerged as another effective way to enhance the system reliability, due to the rapid development of technology. To evaluate the system reliability with consideration of self-healing mechanisms existing in polymer matrix composite materials, [PERSON] et al. [31] characterize healing time and healing level in the self-healing process of shocks. [PERSON] et al. [32] suggested a new type of shock models, where each delayed failure can be cured with certain probabilities. [PERSON] et al. [33] developed accumulative damage shock models and introduced the concept of self-healing effects. As another special mechanism in terms of shock damage, the damage self-exciting phenomenon, is also commonly observed in various engineering fields. For example, an aircraft with self-exciting materials may encounter turbulence that causes some damage to the fuselage. When the accumulative external shock damage exceeds the critical threshold, the aircraft will fail and cause extreme losses. If there is no effective shock in a time period, and the airframe suffers some appropriate stimulation, the self-exciting mechanism can be triggered. These stimuli can be considered as ineffective shocks and can be heat, light or electric fields. As another example, a software module includes a string of codes. The errors in software can result in software failure. We can consider the 'bugs' as a kind of shock and the software failure means the answers generated by the software are deviant from required answer. If there is no effective impact in a period of time and some ineffective bugs occur, then the self-exciting phenomenon will appear. Due to its significant impact on system failure behavior, the self-exciting phenomenon should be given full consideration in system reliability evaluation. [PERSON] et al. [34] considered that reserves are subjected to self-exciting and cross-exciting shocks and proposed a modelling method. [PERSON] et al. [35] proposed approaches to model reliability growth and provided a common evaluation method to model self-exciting point processes. [PERSON] et al. [36] established a model considering mixed shock models and self-exciting mechanisms to generalize software reliability models. Despite the popularity of the self-exciting phenomenon, little research has been dedicated to the reliability modelling of the self-exciting process in a shock environment. Although the proposed model shares some similarities with [PERSON] et al. [23], we extend the research by considering two novel self-exciting triggering mechanisms. To be specific,Zhao et al. [23] proposed a self-healing mechanism by considering that the damage caused by a valid shock can be healed when the number of invalid shocks reaches a threshold in the trailing run of invalid shocks. In contrast, we propose two novel self-exciting mechanisms by considering that the effective shocks are increased when the number of ineffective shocks attain the trigger threshold. Two types of self-exciting trigger mechanisms are considered based on the consecutive and cumulative number of ineffective shocks. To this end, this paper develops novel shock models considering two types of self-exciting triggering mechanisms. Under trigger mechanism 1, when the accumulative number of \(\delta\)--ineffective shocks reaches the trigger threshold \(k_{1}\) among the run of ineffective shocks, the system will trigger the self-exciting mechanism, and the accumulative number of effective shocks will increase by \(m_{1}\). Under trigger mechanism 2, when the consecutive number of \(\delta\)--ineffective shocks reaches the trigger threshold \(k_{2}\) among the run of ineffective shocks, the system will trigger the self-exciting mechanism, and the accumulative number of effective shocks will increase by \(m_{2}\). Based on the established shock models, we analyze the system reliability. Two maintenance policies are proposed, considering different monitoring conditions to minimize the average cost rate.