Online Flexible Job Scheduling for Minimum Span

In this paper, we study an online Flexible Job Scheduling (FJS) problem. The input of the problem is a set of jobs, each having an arrival time, a starting deadline and a processing length. Each job has to be started by the scheduler between its arrival and its starting deadline. Once started, the job runs for a period of the processing length without interruption. The target is to minimize the span of all the jobs --- the time duration in which at least one job is running. We study online FJS under both the non-clairvoyant and clairvoyant settings. In the non-clairvoyant setting, the processing length of each job is not known for scheduling purposes. We first establish a lower bound of μ on the competitive ratio of any deterministic online scheduler, where μ is the max/min job processing length ratio. Then, we propose two O(μ)-competitive schedulers: Batch and Batch+. The Batch+ scheduler is proved to have a tight competitive ratio of (μ+1). In the clairvoyant setting, the processing length of each job is known at its arrival and can be used for scheduling purposes. We establish a lower bound of (√5+1)/2 on the competitive ratio of any deterministic online scheduler, and propose two O(1)-competitive schedulers: Classify-by-Duration Batch+ and Profit. The Profit scheduler can achieve a competitive ratio of 4+2√2. Our work lays the foundation for extending several online job scheduling problems in cloud and energy-efficient computing to jobs that have laxity in starting.