Semi-online early work maximization problems on two hierarchical uniform machines with partial information of processing time

Abstract

In this paper, we consider four semi-online early work maximization problems on two hierarchical uniform machines \(M_1\) and \(M_2\), where machine \(M_1\) with speed \(s>0\) is available for all jobs and machine \(M_2\) with speed 1 is only available for high-hierarchy jobs. When the total size of all jobs is known, we design an optimal online algorithm with a competitive ratio of \(\min \{1+s,\frac{2+2s}{1+2s}\}\). When the total size of low-hierarchy jobs is known, we design an optimal online algorithm with a competitive ratio of \(\min {\{1+s, \frac{\sqrt{9\,s^2+10\,s+1}-s-1}{2\,s}}\}\). When the total size of high-hierarchy jobs is known, we design an optimal online algorithm with a competitive ratio of \(\min \{\sqrt{s+1}, \sqrt{s^2+2\,s+2}-s\}\). When both the total sizes of low-hierarchy and high-hierarchy jobs are known, we give a lower bound \(\frac{2s+2}{s+2}\) for the case \(s\le \frac{2}{3}\), and an optimal online algorithm with a competitive ratio of \(\frac{3s+3}{3s+2}\) for the case \(s>\frac{2}{3}\).