Asymptotic Performance and Energy Consumption of SLACK

Abstract

Scheduling n independent tasks onto m identical processors in order to minimize the makespan has been widely studied. As an alternative to classical heuristics, the Slack algorithm groups tasks by packs of m tasks of similar execution times, and schedules first the packs with the largest differences. It turns out to be very performant in practice, but only few studies have been conducted on its theoretical properties. We derive novel analytical results for Slack, and in particular, we study the performance of this algorithm from an asymptotical point of view, under the assumption that the execution times of the tasks follow a given probability distribution. The study is building on a comparison of the most heavily loaded machine compared to the least loaded one. Furthermore, we extend the results when the objective is to minimize the energy consumption rather than the makespan, since reducing the energy consumption of the computing centers is an ever-growing concern for economical and ecological reasons. Finally, we perform extensive simulations to empirically assess the performance of the algorithms with both synthetic and realistic execution time distributions.