Stochastic bounds for generalized systems in reliability theory

Practically, it is not feasible to obtain the precise reliability of systems in a reasonable time, when the systems are large and complex. In this paper, the authors present some stochastic bounds on generalized systems of which state spaces are mathematically partially ordered sets. In the first place they introduce a notion of generalized systems and then present some stochastic bounds on the system reliability by using maximal and minimal elements of the structures of the systems. The bounds are generalization of the well-known max-min bounds on binary-state system reliability. Furthermore, the authors present the other stochastic bounds when systems are decomposed into several modules and satisfy a condition which is called MC (Maximal Coincidence) condition. They show that these bounds are tighter than the former. For a few simple systems, the authors give numerical examples and estimations of computational complexity for obtaining these stochastic bounds.