Scheduling for Weighted Flow Time and Energy with Rejection Penalty

This paper revisits the online problem of flow-time scheduling on a single processor when jobs can be rejected at some penalty [Bansal et al. 2003]. The user cost of a job is defined as the weighted flow time of the job plus the penalty if it is rejected before completion. For jobs with arbitrary weights and arbitrary penalties, [Bansal et al. 2003] gave an online algorithm that is O((log W + log C)^2)-competitive for minimizing the total user cost when using a slightly faster processor, where W and C are the max-min ratios of job weights and job penalties, respectively. In this paper we improve this result with a new algorithm that can achieve a constant competitive ratio independent of $W$ and C when using a slightly faster processor. Note that the above results assume a processor running at a fixed speed. This paper shows more interesting results on extending the above study to the dynamic speed scaling model, where the processor can vary the speed dynamically and the rate of energy consumption is a cubic or any increasing function of speed. A scheduling algorithm has to control job admission and determine the order and speed of job execution. This paper studies the tradeoff between the above-mentioned user cost and energy, and it shows two O(1)-competitive algorithms and a lower bound result on minimizing the user cost plus energy. These algorithms can also be regarded as a generalization of the recent work on minimizing flow time plus energy when all jobs must be completed (see the survey paper [Albers 2010]).