Bin packing with items uniformly distributed over intervals [a,b]

We consider the problem of packing n items which are drawn uniformly from intervals of the form [a,b], where 0 ≪ a ≪ b ≪ 1. For a fairly large class of a and b, we determine a lower bound on the asymptotic expected number of bins used in an optimum packing. The method of proof is related to the dual of the linear programming problem corresponding to the bin packing problem. We demonstrate that these bounds are tight by providing simple packing strategies which achieve them.