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

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, 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, root 9s(2)+10s+1-s-1/2s}. When the total size of high-hierarchy jobs is known, we design an optimal online algorithm with a competitive ratio of min{ root s + 1,root s(2 )+ 2s + 2 - s}. When both the total sizes of low-hierarchy and high-hierarchy jobs are known, we give a lower bound 2s+2/s+2 for the case s <= 2/3, and an optimal online algorithm with a competitive ratio of 3s+3/3s+2 for the case s > 2/3.