On Shock Models with Two Shock Types

We develop an innovative approach to the generalized extreme shock modelling when probability of a system’s failure under a shock depends on the time elapsed since the last shock. Shocks are modelled by the renewal process, whereas each event from the process can result in a shock of type 1 or type 2 (e.g., of electrical, thermal, mechanical, etc. nature). Our methodology is based on deriving and solving via the Laplace Transform the corresponding simultaneous integral equations. We consider different practical scenarios. For instance, in the ‘tie’ model, a failure occurs only if a shock of one type follows a shock of the other type, whereas the sequences of shocks of the same type are ‘harmless’ to a system. In the ‘match’ model, a failure occurs only if the shocks of the same type follow each other. Finally, in the ‘cure’ model, only two consecutive shocks of one harmful type can result in a failure. Therefore, a shock of the other type between them can be considered as ‘cure’. The practical examples for these scenarios are given. The ‘fast recovery’ approximations for survival probabilities are discussed and some examples are given.