Preemptive Online Scheduling with Reordering

We consider online preemptive scheduling of jobs, arriving one by one, on m identical parallel machines. A buffer of a fixed size K > 0, which assists in partial reordering of the input, is available to be used for the storage of at most K unscheduled jobs. We study the effect of using a fixed sized buffer (of an arbitrary size) on the supremum competitive ratio over all numbers of machines (the overall competitive ratio), as well as the effect on the competitive ratio as a function of m. We find a tight bound on the competitive ratio for any m. This bound is 43 for even values of m and slightly lower for odd values of m. We show that a buffer of size Θ(m) is sufficient to achieve this bound, but using K = o(m) does not reduce the best overall competitive ratio which is known for the case without reordering, e e−1 . We further consider the semi-online variant where jobs arrive sorted by non-increasing processing time requirements. In this case it turns out to be possible to achieve a competitive ratio of 1. In addition, we find tight bounds as a function of the buffer size and the number of machines for this semi-online variant. Related results for non-preemptive scheduling were recently obtained by Englert, Özmen and Westermann.