A REARRANGEMENT INEQUALITY FOR THE LONGEST RUN, WITH AN APPLICATION TO NETWORK RELIABILITY

Let X//1,. . . ,X//n be independent binary variables with parameters theta //1,. . . , theta //n respectively, and let R denote the length of the longest run of 1's. This note concerns a new expression for P// theta left bracket R greater than equivalent to k right bracket for k greater than equivalent to one-half n, 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 C. Derman et al. are discussed.