A Relationship Between Partial Derivatives of the Reliability Function of a Coherent System and its Minimal Path (Cut) Sets

A relationship between rth partial derivatives of the reliability function h(p) of a coherent system and its minimal cut (path) sets is studied (r ≥ 2). It is shown that (−1)^r(∂^rh(p)/∂p_i1, …, ∂p_ir)((−1)(∂^rh (p)/∂p_i1, …, ∂p_ir)) is nonnegative for all i₁, …, i_r and all p implies that all minimal cut (path) sets have cardinality r − 1 or less, r = 2, …, n. When r = 2, a simple characterization is obtained for series (parallel) system. Interpretations of such characterization in terms of reliability importance of components are given and a simple proof of a known characterization of series system is obtained.