M out of n inspected systems subject to shocks in random environment

Consider a parallel redundant system, consisting n components, that is subject to shocks. The shocks cause the components to fail with certain probabilities. Shocks arrival rate and components' failure probabilities may depend on an external Markovian environment. We consider warm and cold stand-by systems. Systems' failures are silent. The system is maintained through inspection and repair/replacement. We propose several state-dependent maintenance policies and derive system availability and cost function.