On-Line Scheduling on Parallel Machines to Minimize the Makespan

This paper considers two parallel machine scheduling problems, where the objectives of both problems are to minimize the makespan, and the jobs arrive over time, on two uniform machines with speeds 1 and s (s a parts per thousand yen 1), and on m identical machines, respectively. For the first problem, the authors show that the on-line LPT algorithm has a competitive ratio of (1 + )/2 a parts per thousand 1.6180 and the bound is tight. Furthermore, the authors prove that the on-line LPT algorithm has the best possible competitive ratio if s a parts per thousand yen 1.8020. For the second problem, the authors present a lower bound of (15 - )/8 a parts per thousand 1.3596 on the competitive ratio of any deterministic on-line algorithm. This improves a previous result of 1.3473.