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In this paper, consecutive-k -out-of-n repairable systems are studied. We assume that both the working time, and the repair time of each component are exponentially distributed, and every component after repair is as good as new. Each component is either a key component, or an ordinary component so we can adopt a priority repair rule for key components. First, the state transition probabilities as well as their Laplace transforms for the systems are derived. Second, several fundamental reliability indices (including availability, the mean time to the first failure, reliability, and the rate of occurrence of failure) of the systems are obtained explicitly. Finally, one example is shown to explain the model, and the methodology developed in this paper.

We investigate the delta-shock model of complex systems consisting of n i.i.d. components. We first obtain a general lifetime distribution for the delta-shock model of a general complex system by reducing the system to a linear combination of parallel systems. We then consider coherent system structures including series, parallel, and k-out-of-n, then derive some useful results including reliability bounds, bounds on the mean lifetime, limiting distributions, and Laplace-Stieltes transforms

We review the principal progress of shock models during the last three decades. Our new model, the delta-shock model, and its latest developments are also introduced. Furthermore, for adapting shock models to a wider reliability field, we put forward a generalized framework for studying shock models, based on cluster point processes with cluster marks. In addition, we provide a noteworthy case under this framework as a concrete example in reliability with an insurance background, and give the asymptotic distribution of the risk process