Research on the parallel–batch scheduling with linearly lookahead model

In this paper, we consider the new online scheduling model with linear lookahead intervals, which has the character that at any time $ t $, one can foresee the jobs that will coming in the time interval $ (t, \lambda t+\beta ] $ with $ \lambda\geq1, \beta\geq 0 $. We consider online scheduling of unit length jobs on $ m $ identical parallel-batch machines under this new lookahead model to minimize the maximum flowtime and the makespan, respectively. We give some lower bounds for these problems with the batch capacity $ b = \infty $ and $ b<\infty $, respectively. And for the bounded model to minimize makespan, we give an online algorithm with competitive ratio $ 1+\alpha $ for $ 1\leq \lambda <4/3, 0\leq \beta\leq \frac{4-3\lambda}{6} $ and $ \frac{3}{2} $ for $ \lambda\geq1, 0\leq\beta<1 $, where $ \alpha $ is the positive root of $ \lambda\alpha^2+(\lambda+\beta)\alpha+\beta-1 = 0 $. The online algorithm is best possible when $ 1\leq \lambda <4/3, 0\leq \beta\leq \frac{4-3\lambda}{6} $.