Abstract
This paper investigates the online scheduling on three uniform machines problem. Denote by s j the speed of each machine, j = 1 , 2 , 3 . Assume 0 < s 1 ≤ s 2 ≤ s 3 , and let s = s 2 / s 1 and t = s 3 / s 2 be two speed ratios. We show the greedy algorithm LS is an optimal online algorithm when the speed ratios ( s , t ) ∈ G 1 ∪ G 2 , where G 1 = { ( s , t ) | 1 ≤ t < 1 + 31 6 , s ≥ 3 t 5 + 2 t − 6 t 2 } and G 2 = { ( s , t ) | s ( t − 1 ) t ≥ 1 + s , s ≥ 1 , t ≥ 1 } . The competitive ratio of LS is 1 + s + 2 s t s + s t when ( s , t ) ∈ G 1 and 1 + s s t + 1 when ( s , t ) ∈ G 2 . Moreover, for the general speed ratios, we show the competitive ratio of LS is no more than m i n { 1 + s + 2 s t s + s t , 1 + s s t + 1 , 1 + s + 3 s t 1 + s + s t } and its overall competitive ratio is 2 which matches the overall lower bound of the problem.
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