Preemptive online scheduling with rejection of unit jobs on two uniformly related machines

We consider preemptive online and semi-online scheduling of unit jobs on two uniformly related machines. Jobs are presented one by one to an algorithm, and each job has a rejection penalty associated with it. A new job can either be rejected, in which case the algorithm pays its rejection penalty, or it can be scheduled preemptively on the machines, in which case it may increase the maximum completion time of any machine in the schedule, also known as the makespan of the constructed schedule. The objective is to minimize the sum of the makespan of the schedule of all accepted jobs and the total penalty of all rejected jobs. We study two versions of the problem. The first one is the online problem where the jobs arrive unsorted, and the second variant is the semi-online case, where the jobs arrive sorted by a non-increasing order of penalties. We also show that the variant where the jobs arrive sorted by a non-decreasing order of penalties is equivalent to the unsorted one. We design optimal online algorithms for both cases. These algorithms have smaller competitive ratios than the optimal competitive ratio for the more general problem with arbitrary processing times (except for the case of identical machines), but larger competitive ratios than the optimal competitive ratio for preemptive scheduling of unit jobs without rejection.