Survival of systems with protection under shocks

Survival probabilities for systems with protection operating under renewal shock process are derived. This is done using a method of simultaneous integral equations specifically developed for this setting. The solutions are given in terms of the Laplace Transforms that can be inverted numerically. The protection system’s failure under a shock means that it was not neutralized and have reached the main system. On the other hand, the neutralized shock does not affect the main system. Various specific cases are discussed as examples and ‘fast repair’ approximations are provided. Specifically, when the probabilities of failures of the protection and main systems on a single shock are sufficiently small, a simple, speaking for itself fast repair asymptotic result describes the ‘double thinning’ of the original process of shocks.