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.