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 ≥ 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 + \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt 5 $$\end{document})/2 ≈ 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 ≥ 1.8020. For the second problem, the authors present a lower bound of (15 − \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt {17} $$\end{document})/8 ≈ 1.3596 on the competitive ratio of any deterministic on-line algorithm. This improves a previous result of 1.3473.