Discussing some approaches to delta-shock modeling

We revisit the ‘classical’ delta-shock model and generalize it to the case of renewal processes of external shocks with arbitrary inter-arrival times and arbitrary distribution of the ‘recovery’ parameter delta. Our innovative approach is based on defining the renewal points for the model and deriving the corresponding integral equations for the survival probabilities of interest that describe the setting probabilistically. As examples, the cases of exponentially distributed and constant delta are analyzed. Furthermore, delta shock modeling for systems with protection and two shock processes is considered. The first process targets the defense system and can partially destroy it. In this case, the second process that targets the main, protected system can result in its failure. The damages of the defense system are recovered during the recovery time delta. As exact solutions of the discussed problems are rather cumbersome, we provide simple and easy approximate solutions that can be implemented in practice. These results are justified under the assumption of ‘fast repair’ when the recovery time delta is stochastically much smaller than the inter-arrival times of the shock processes. The corresponding numerical examples (with discussion) illustrate our findings.