ONLINE SCHEDULING OF PARALLEL JOBS WITH BOUNDED PROCESSING TIMES ON TWO MACHINES

We study the problem of online scheduling parallel jobs with bounded processing times on 2 machines, and the objective is to minimize makespan. A parallel job requires simultaneous processing on a pre-specified, job-dependent number of machines. The problem is online in the sense that jobs are presented one by one. Once a job is presented, we must irrevocably assign it to some time slot before the next one shows up. We investigate the case where the processing times of jobs are bounded within interval [a, αa] where a > 0 and α > 1. We first prove a lower bound of competitive ratios for online algorithms equal [Formula: see text] when α ≥ 2 and [Formula: see text] when 1 < α < 2, respectively. We further prove that the Greedy algorithm proposed in Chan et al. (2008) is [Formula: see text]-competitive in the case but it cannot be better than [Formula: see text]-competitive. The results imply that when 1 < α < 2 Greedy has a competitive ratio better than 2, which is the competitive ratio of Greedy in the case without processing time bound.