Online LPT algorithms for parallel machines scheduling with a single server

We consider two parallel machines scheduling problems with a single server. For the general case we present an online LPT algorithm with competitive ratio 2, and give a lower bound \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{\sqrt{5} + 1}{2}$\end{document}. We also apply the online LPT algorithm to the special case where all the setup times are equal to 1. We show that the competitive ratio is 1.5, and no online algorithm can has a competitive ratio less than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sqrt{2}$\end{document}.