Algebraic reliability of multi-state k-out-of-n systems

In this paper, we review different definitions that multi-state k-out-of-n systems have received along the literature and study them in a unified way using the algebra of monomial ideals. We thus obtain formulas and algorithms to compute their reliability and bounds for it. We provide formulas and computer experiments for simple and generalized multi-state k-out-of-n systems and for binary k-out-of-n systems with multi-state components.