We investigate the problem of on-line scheduling jobs on m identical parallel machines where preemption is allowed. The goal is to minimize the maximum flow time among all jobs, where the flow time of a job is its completion time minus its release time. We derive an on-line algorithm with competitive ratio 2−1/m. Moreover, when the algorithm must schedule jobs one by one in the order they are released, we prove that there is no randomized on-line algorithm with a better competitive ratio. Finally, we investigate the non-preemptive case and show that the First In First Out heuristic achieves the best possible competitive ratio when m = 2
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