Reliability for systems with self-healing effect under shock models

In the paper, the concept of self-healing effects, permanent or limited duration on system health, is introduced by building a class of cumulative damage shock models in terms of the integral for counting processes including Poisson process, Hawkes process and Affine process, and so on. Based on the established shock models, the property on lifetime and the bounds of system reliability are discussed, especially for the simplest case. Then the general simulation algorithm is given for the general situation under the given cumulative damage shock models, because it is impossible to present the general analytical results on the reliability of systems. Some numerical examples are given to illustrate the algorithm and bounds of reliability. Finally, the whole works in the paper are concluded.