Online Scheduling of Jobs with Fixed Start Times on Related Machines

We consider online preemptive scheduling of jobs with fixed starting times revealed at those times on $$m$$m uniformly related machines, with the goal of maximizing the total weight of completed jobs. Every job has a size and a weight associated with it. A newly released job must be either assigned to start running immediately on a machine or otherwise it is dropped. It is also possible to drop an already scheduled job, but only completed jobs contribute their weights to the profit of the algorithm. In the most general setting, no algorithm has bounded competitive ratio, and we consider a number of standard variants. We give a full classification of the variants into cases which admit constant competitive ratio (weighted and unweighted unit jobs, and C-benevolent instances, which is a wide class of instances containing proportional-weight jobs), and cases which admit only a linear competitive ratio (unweighted jobs and D-benevolent instances). In particular, we give a lower bound of $$m$$m on the competitive ratio for scheduling unit weight jobs with varying sizes, which is tight. For unit size and weight we show that a natural greedy algorithm is $$4/3$$4/3-competitive and optimal on $$m=2$$m=2 machines, while for large $$m$$m, its competitive ratio is between $$1.56$$1.56 and $$2$$2. Furthermore, no algorithm is better than $$1.5$$1.5-competitive.