Online algorithms for scheduling unit length jobs on parallel-batch machines with lookahead

Abstract

We consider online scheduling of unit length jobs on m parallel-batch machines with lookahead to minimize makespan. A parallel-batch machine can handle up to b jobs simultaneously as a batch with processing time equal to the maximum processing time of the jobs included in the batch. In the lookahead model, at a time instant t, an online algorithm can foresee all the jobs that will arrive in the time segment (t,t+β]. In this paper, we deal with two variants: the unbounded model with b=∞ and the bounded model with b<∞. For the unbounded model, we present an optimal online algorithm for β⩾1/m, and provide a best possible online algorithm of competitive ratio 1+αm for 0⩽β<1/m, where αm is the positive root of (1+α)(m+1)=α+2−β∑i=1m(1+α)i. For the bounded model, we establish a lower bound with a form of piecewise function, and provide an online algorithm with competitive ratios 1+α⁎ for 0⩽β⩽16 and 32 for β>16, respectively, where α⁎ is the positive root of α2+(β+1)α+β−1=0. The online algorithm is the best possible when 0⩽β⩽16.