Online scheduling on unbounded parallel-batch machines to minimize the makespan

We consider online scheduling on m parallel-batch machines to minimize the makespan where the batch capacity is unbounded. The processing time of a job becomes known only upon its arrival. We give an improved lower bound 1+αm on the competitive ratio, where αm is the positive solution of the equation , and establish a best possible online algorithm matching this lower bound. We also provide a new lower bound 3/2 on the competitive ratio for dense-algorithms, and present a best possible dense-algorithm matching this lower bound.