ALGEBRAIC RELIABILITY OF MULTI-STATE k-OUT-OF-n SYSTEMS

Abstract

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.

A C++ class for algebraic reliability computations

“…Both sets can be formulated in algebraic terms as the minimal generators of the j-reliability ideal (lower boundary points) and maximal standard pairs (upper boundary points). For full details and a complete proof of this correspondence, see [29].…”

“…Given a coherent system S its j-reliability ideal I j (S) is generated by the monomials corresponding to its minimal j-paths. The ideal of its dual system I j (S D ) is generated by the monomials corresponding to maximal j-cuts of S and may be seen as the ideal generated by the maximal standard pairs of I j (S) [29]. These can be computed using the Alexander dual of the artinian ideal I j (S) + x M 1 +1 i , .…”

“…A binary k-out-of-n:G system is a system with n components that is in a working state whenever at least k of its components are working. The multi-state version of this kind of systems has received several definitions in the literature, see [29] for a review. A general definition is that of generalized multi-state k-out-of-n systems, see [50]:…”

“…The performance of our algorithms depend greatly on the number of minimal paths or minimal cuts, as can be seen in Figure 6 (c). There exist specialized algorithms for this kind of systems that are recursive on M , see [51,29] or based on Decision Diagrams [32]. 4.4.…”

Algebraic Analysis of Variants of Multi-State k-out-of-n Systems

“…In case that the structure function does not have an easy to describe pattern, like in general networks, the algebraic method based on monomial ideals is an approach that also produces efficient algorithms [12][13][14][15]. This latter method has been used to analyze k-out-of-n systems and variants in the binary case [13,16] and the generalized multi-state version [17]. In summary, this method assigns for each level j of performance of the system a monomial ideal whose minimal generating set is in correspondence with the set of minimal working states of the system.…”