A rearrangement inequality for the longest run, with an application to network reliability

Abstract

Let X l , · ··, Χ n be independent binary variables with parameters θ l , · ··, θ n respectively, and let R denote the length of the longest run of 1's. This note concerns a new expression for and a rearrangement inequality. The inequality is applied to solve an optimal permutation problem for consecutive- k -out-of- n : F networks, and its implications on a recent conjecture of Derman et al. (1982) are discussed.