Overview of the algebraic reliability of binary and multistate k-out-of-n and related systems

In this chapter, we present an algebraic method, based on monomial ideals, to compute the reliability of coherent systems. The first part of the chapter introduces the basics of the algebraic method and gives some examples of use. The rest of the chapter is devoted to the application of the algebraic method to binary and multistate k-out-of-n systems. We review the different definitions that binary and multistate k-out-of-n and their variant systems have received in the literature and present them in a unified way using the algebra of monomial ideals. We provide algorithms and formulas to compute the reliability of these systems and bounds for them. We include some computer experiments to evaluate the efficiency of the algebraic method showing its strengths and weaknesses.