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Algorithms to determine the failure frequency of a non-coherent structure function are few, and they depend on the knowledge of the prime implicants. For coherent structures, there exists a well-known approach based on criticality and the Birnbaum importance measure, which is more flexible concerning the form in which the structure function is given. This concept is extended to cover non-coherent structure functions by considering two types of criticality functions and a corresponding extension of the Birnbaum importance measure. This theory is a generalization of presently existing concepts, as it includes systems described by prime implicants as well as coherent systems. For the components, renewal frequency densities are required in addition to failure frequency densities and unavailabilities. For common component models, these are given to show that everything that can be done for coherent systems can also be achieved for non-coherent ones. This includes also the method of modularization, which is addressed in a final section. The topic of multi-state components is not addressed in this paper.
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