Optimal Design of Systems Subject to Two Types of Error

We consider systems such as relay circuits or juries that are composed of components, e.g., relays or jury votes. Each component is in one of two states, with probabilities that depend on the qualities of the components and the requests made of the system. The problem is to construct the system (i.e., determine the configurations of the components for which the system is in each of the two states) in a way that minimizes the probabilities of errors. We find conditions on the probabilities for which the optimal system is monotone or k out of n. We further study these conditions when the components behave independently.