Which Coefficients Matter Most—Consecutive $k$-Out-of-$n$:$F$ Systems Revisited

Consecutive-$k$-out-of-$n$:F systems are one of the most well-studied types of networks when discussing reliability. They have been used from safety–critical environments, such as nuclear power plants or hospital's emergency backup power supplies, to classical transportation problems, such as public water systems and oil/gas pipelines. Exact formulae for the reliability polynomial of a consecutive system are known for quite a long time. In addition, several alternatives for computing exactly the reliability polynomial are also known. However, when dealing with large consecutive systems, exact calculations become prohibitive and approximations/bounds are the common route. We begin this article by providing an in-depth review of many known bounds. Next, we focus on the coefficients of the reliability polynomial of a consecutive system in its Bernstein form. By deriving shape properties of these coefficients, we are able to identify new bounds. Our approach is uncommon for this case, as none of the previously used bounding techniques has looked closely at each and every coefficient. This is probably the reason why we obtain tight bounds with low complexity costs. Finally, detailed simulations provide strong evidence of the fidelity of the proposed bounds.