Packing rectangles and intervals

Consider a parameter alpha > 0, and a large integer N. Assume that for p greater than or equal to 1, we are given a family I-p of [2 N-alphap] random intervals of [0,1], and consider the union I of the families I-p. A packing of I is a disjoint sub family of I. The cost of the packing is the sum of the wasted space (the measure of the part of [0,1] not covered by the packing) and the costs of the intervals of the packing, where, if I is an element of I-p, the cost of I is 2(-P) [I]. We estimate as a function of N, the expected value of the cost of an optimal packing. We discover the very surprising fact that (up to terms of smaller order) this cost behaves as N-1/2 for all 1 less than or equal to alpha less than or equal to 2. This problem is motivated by a problem of packing random rectangles.