Packing random intervals

Let n random intervals I-1,...,I-n be chosen by selecting endpoints independently from the uniform distribution on [0,1]. A packing is a pairwise disjoint subset of the intervals; its wasted space is the Lebesgue measure of the points of [0,1] not covered by the packing. In any set of intervals the packing with least wasted space is computationally easy to find; but its expected wasted space in the random case is not obvious. We show that with high probability for large n, this "best" packing has wasted space O(log2n/n). It turns out that if the endpoints 0 and 1 are identified, so that the problem is now one of packing random arcs in a unit-circumference circle, then optimal wasted space is reduced to O(1/n). Interestingly, there is a striking difference between the sizes of the best packings: about log n intervals in the unit interval case, but usually only one or two arcs in the circle case.