Improved online algorithms for the batch scheduling of equal-length jobs with incompatible families to maximize the weighted number of early jobs

We consider the online scheduling of equal-length jobs with incompatible families on m identical batch machines. Each job has a release time, a deadline and a weight. Each batch machine can process up to b jobs (which come from the same family) simultaneously as a batch, where b is called the capacity of the machines. Our goal is to determine a preemption-restart schedule which maximizes the weighted number of early jobs. For this problem, Li et al. (Inf Process Lett 112:503-508, 2012) provided an online algorithm of competitive ratio 3+2 root 2 for both b = infinity and b < infinity. In this paper, we study two special cases of this problem. For the case that m = 2, we first present a lower bound 2, and then provide an online algorithm with a competitive ratio of 3 for both b = infinity and b < infinity. For the case in which m = 3, b = infinity and all jobs come from a common family, we present an online algorithm with a competitive ratio of (8 + 2 root 15)/3 approximate to 5.249.