Survival Under Different Types of Shocks With Random Recovery Times: The ‘Tie’, the ‘Match’ and the ‘Cure’ Models

Most results in the literature discussing the delta shock model and its variants are obtained under the assumption of one shock process and the deterministic recovery time (delta). However, systems are often exposed to different types of shocks (e.g., electrical, thermal, mechanical, etc.). In this paper, we develop an innovative approach to modelling reliability of systems operating under a renewal shock process with shocks of two types. It is based on the corresponding renewal equations. We consider three different scenarios. In the ‘tie’ model, a failure occurs (within a random recovery time delta) 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 are sufficiently close. Finally, in the ‘cure’ model only two close 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 in the Introduction. The corresponding renewal equations for each model are derived and solved via the Laplace Transform (LT). The ‘fast repair’ approximations are discussed and examples with the homogeneous Poisson processes (HPPs) are considered.