Semi-online scheduling with decreasing job sizes

We investigate the problem of semi-online scheduling jobs on m identical parallel machines where the jobs arrive in order of decreasing sizes. We present a complete solution for the preemptive variant of semi-online scheduling with decreasing job sizes. We give matching lower and upper bounds on the competitive ratio for any xed number m of machines; these bounds tend to 1 2 (1+ p 3) 1:36603, as the number of machines goes to innnity. Our results are also best possible for randomized algorithms. For the non-preemptive variant of semi-online scheduling with decreasing job sizes, a result of Graham SIAM J. Appl. Math. 17(1969), 263{269] yields a (4 3 ? 1 3m) competitive deterministic non-preemptive algorithm. For m = 2 machines, we prove that this algorithm is best possible (it is 7 6 competitive). For m = 3 machines we give a lower bound of (1 + p 37)=6 1:18046. Finally, we present a randomized non-preemptive 8 7-competitive algorithm for m = 2 machines and prove that this is optimal.