New bounds and algorithms for on-line scheduling: two identical processors, known sum and upper bound on the tasks

.In this paper we study a semi on-line version of the classical multiprocessor scheduling problem on two identical processors. We assume that the sum of the tasks and an upper bound gamma on the size of each task are known. Each task has to be assigned upon arrival and the assignment cannot be changed later. The objective is the minimization of the maximum completion time on the processors. In this paper we propose new algorithms and improve known lower and upper bounds on the competitive ratio. Algorithms and bounds depend on the value of gamma. An optimal algorithm, with respect to the competitive ratio, is obtained for gamma in [1/n, 2(n+1)/(n(2n+1))] or gamma = (2n-1)/(2n(n-1)), where n is any integer value, n >= 2.