An On-Line Algorithm for Some Uniform Processor Scheduling

This paper considers the problem of on-line scheduling a set of independent jobs on m uniform machines (M1, M2,..., Mm) in which machine Mi's processing speed is si=1 (i=1,..., m-1) and sm=s >1. List scheduling [Y. Cho and S. Sahni, SIAM J. Comput., 9 (1980), pp. 91--103] guarantees a worst-case performance of $\frac{3m-1}{m+1} (m\ge 3)$ and $\frac{1+\sqrt 5}2 (m=2)$ for this problem. We prove that this worst-case bound cannot be improved for m=2 and m=3 and for every $m\ge 4,$ an algorithm with worst-case performance at most $\frac{3m-1}{m+1}-\e$ is presented when sm=2,$ where e is a fixed positive number, and then we improve the bound for general sm=s > 1.