Parallel processor scheduling with limited number of preemptions

In this paper, we compare the makespan of preemptive and i-preemptive schedules where only a limited number i of preemptions is allowed. The problem is to schedule n independent jobs on m identical processors that operate in parallel. The objective is to minimize the makespan, i. e., the completion time of the last job that finishes. We show that the ratio of the optimal i-preemptive schedule length C-max(ip)* versus the optimal preemptive schedule length C-max(p)* is bounded from above by C-max(ip)* less than or equal to ( 2 - 2/(m/(i + 1) + 1)) C-max(p)*. Furthermore, we show that the ratio of the length C-max(LPT) of a nonpreemptive schedule following the longest processing time (LPT) rule versus the optimal preemptive schedule length C-max(p)* is bounded from above by exactly the same bound when i = 0.