Optimal on-line algorithms for scheduling on parallel batch processing machines

We consider the problem of scheduling jobs on-line on parallel batch processing machines with dynamic job arrivals to minimize makespan. We are given m identical batch processing machines, each of which can handle up to B (the capacity of a machine) jobs simultaneously. The jobs that are processed together form a batch, and all jobs in a batch start and complete at the same time. The processing time of a batch is a constant, which is independent of the number and identity of the jobs. Each job becomes available at its arrival time, which is unknown in advance. First we deal with the unbounded case where the capacity of the batch processing machines is infinite, and derive an optimal on-line algorithm with competitive ratio 1+ g m , where g m >0 is determined by (1+ g m ) m +1 = g m +2. We then consider the bounded case where the capacity B of the batch processing machines is bounded. We provide an on-line algorithm with competitive ratio (\sqrt{5}+1)/2 (the golden ratio) and prove that the result is the best possible for all m .