Semi-on-line scheduling on two parallel processors with an upper bound on the items

We study a variant of the on-line scheduling problem on two parallel processors. The size of the items is unknown and, as soon as an item is released, it must be immediately assigned to a processor and the assignment cannot be changed later. Optimal algorithms (with respect to competitive ratio) are known for some variants of this problem, where some practical information is given on the instance: the sum of the items is known, or a buffer is available to store a finite number of items. In these cases the best possible competitive ratio of the algorithms is 4/3. In this paper we assume that the sum of items is known in advance (supposed to equal 2) and also that the size of items does not exceed a fixed upper bound γ ≤ 1. We provide, for all the possible values of γ, a lower bound for the competitive ratio of any algorithm and propose different algorithms, for different ranges of the upper bound, for which a worst-case analysis is provided. The proposed algorithms are optimal for 1/2 ≤ γ ≤ 3/5 and 16/17 ≤ γ ≤ 1.