Random walk and the area below its path

A simple random walk with reflected origin is considered. The walk starts at the origin and it must return to the origin at time 2n. The paper shows that the expected area below the path of this walk is If, however, the walk is required to return to the origin for the first time at time 2n, then the expected area below the path of this Bernoulli excursion is . The paper also shows that if is the order statistics based on a sample of size n from a uniform distribution over , and that if is another independent set of order statistics from the same distribution, then . It uses this result to find an average-case performance of the Earliest Due Date (EDD) heuristic for one machine scheduling problem with earliness and tardiness penalties. The paper also applies some of the results to Larson's Queue Inference Engine.