Online-List Scheduling on a Single Bounded Parallel-Batch Machine to Minimize Makespan

In this paper, we consider the online-list scheduling on a single bounded parallel-batch machine to minimize makespan. In the problem, the jobs arrive online over list. The first unassigned job in the list should be assigned to a batch before the next job is released. Each batch can accommodate up to b jobs. For b = 2, we establish a lower bound 1 + gamma of competitive ratio and provide an online algorithm with a competitive ratio of root 5+1/2, where gamma is the positive root of gamma(gamma + 1)(2) = 1. For b = 3, we establish a lower bound 1 + alpha of competitive ratio and provide an online algorithm with a competitive ratio of 2, where a is the positive root of the equation (1 + alpha)(1 + alpha(2)) = 2.