On-line maximizing the number of items packed in variable-sized bins

We study on-line bin packing problem. A fixed number of n. of bins, possibly of different sizes, are given. The items arrive on-line, and the goal is to pack as many items as possible. It is known that there exists a legal packing of the whole sequence in the n bins. We consider fair algorithms that reject an item, only if it does not fit in the empty space of any bin. We show that the competitive ratio of any fair, deterministic algorithm lies between 1/2 and 2/3, and that a class of algorithms including Best-Fit has a competitive ratio of exactly n/(2n−1).