Max-min online allocations with a reordering buffer

We consider online scheduling so as to maximize the minimum load, using a reordering buffer that can store some of the jobs before they are assigned irrevocably to machines. For m identical machines, we show an upper bound of H_m-1 + 1 for a buffer of size m - 1. A competitive ratio below H _m is not possible with any fixed buffer size, and it requires a buffer of size Ω(m/log m) to get a ratio of O(log m). For uniformly related machines, we show that a buffer of size m + 1 is sufficient to get a competitive ratio of m, which is best possible for any fixed sized buffer. We show similar results (but with different constructions) for the restricted assignment model. We give tight bounds for two machines in all the three models. These results sharply contrast to the (previously known) results, which can be achieved without the usage of a reordering buffer, where it is not possible to get a competitive ratio below m already for identical machines, and it is impossible to obtain an algorithm of finite competitive ratio in the other two models, even for m = 2. Our results strengthen the previous conclusion that a reordering buffer is a powerful tool and that it allows a significant decrease in the competitive ratio of online algorithms for scheduling problems. Another interesting aspect of our results is that our algorithm for identical machines imitates the behavior of a greedy algorithm on (a specific set of) related machines, whereas our algorithm for related machines completely ignores the speeds until all jobs have arrived, and then only uses the relative order of the speeds.