## 1 Introduction  
Internal deterioration and external shock damage are two main sources of system failure, and the shock model is commonly applied to characterize the impact of external damage on system failure behavior such as overload and vibration. Shock models have received extensive attention both theoretically and practically [1, 2]. Numerous shock models have been established during the past few decades and can be divided into five basic categories: accumulative shock [3], extreme shock [4], run shock [5], \(\delta\)--shock [6] and mixed shock [7]. In the accumulative shock model, system failure is due to the accumulative effect. Under the extreme impact model, the system will collapse as long as the single impact exceeds the critical value. The system breaks down when \(k\) consecutive shocks arrive and the magnitude exceeds the failure threshold. In the \(\delta\)--shock model, when the time interval between two adjacent shocks is smaller than a fixed threshold, the system will malfunction. In addition, shock models can be combined to construct a mixed shock model [8]. Existing research on shock models are mainly focused on reliability evaluation and optimal preventive maintenance strategy. In reliability engineering, system reliability, survival rate and failure rate function are discussed extensively. [PERSON] et al. [9] studied the reliability modelling and optimal triggering policy of protective devices for multi-state systems subject to shocks. [PERSON] et al. [10] established mixed two-stage shock models governed by a run of external shocks with certain lengths. [PERSON] et al. [11] applied the copula function in deriving the joint distribution of dependent deterioration processes. [PERSON] et al. [12] proposed reliability models for phased mission systems with finite and infinite random shocks and compared the evaluated system reliability under different shock modes. [PERSON] et al. [13] studied condition based maintenance strategy for redundant systems subject to shock damage using improved reinforcement learning. [PERSON] et al. [14] considered the system reliability with both internal deterioration and external damage and used the Monte Carlo simulation to evaluate the system reliability. In addition, some results of reliability analyses of shock models have also been studied [15; 16; 17; 18; 19]. For systems operating in a shock environment, designing appropriate maintenance policies is crucial for mitigating the failure risk caused by external shock damage. [PERSON] et al. [20] proposed a maintenance policy where the system is removed by a minimal repair when experiencing a type-I failure and a perfect repair when experiencing a type-II failure; then, a hybrid replacement policy is possible, considering system age, failure modes, and repair cost. [PERSON] et al. [21] studied an optimal preventive maintenance policy for systems subject to shock damage modelled by the shot noise process. [PERSON] [22] introduced a replacement policy to replace an element when the total damage exceeds the critical threshold. [PERSON] et al. [23] put forward three preventive maintenance strategies for systems with varying monitoring conditions, and optimization models were formulated to determine the optimal maintenance thresholds. [PERSON] et al. [24] proposed novel shock models to characterize the case that the component damage process will be accelerated with the increase of the damage degree in a shock environment. [PERSON] et al. [25] described the deterioration and shock processes of the traction power supply system by the jump process and developed a preventive maintenance strategy to minimize the average cost rate. [PERSON] et al. [26] proposed a condition-based maintenance policy for systems subject to internal degradation and external damage and evaluated the average cost per unit time by jointly optimizing the preventive thresholds and the periodic inspection frequency. Other operations and maintenance strategies in a shock environment have been reported [27; 28; 29; 30].