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Online algorithms for scheduling unit length jobs on parallel-batch machines with lookahead. (English) Zbl 1237.68036

Summary: 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 \((t,t+\beta ]\). In this paper, we deal with two variants: the unbounded model with \(b=\infty \) and the bounded model with \(b<\infty \). For the unbounded model, we present an optimal online algorithm for \(\beta \geq 1/m\), and provide a best possible online algorithm of competitive ratio \(1+\alpha_m\) for \(0\leq \beta <1/m\), where \(\alpha_m\) is the positive root of \((1+\alpha)^{(m+1)} = \alpha + 2 - \beta\sum_{i=1} ^m (1+\alpha)^i\). For the bounded model, we establish a lower bound with a form of a piecewise function, and provide an online algorithm with competitive ratios \(1+\alpha ^{\ast}\) for \(0\leq\beta\leq\frac{1}{6}\) and \(\frac{3}{2}\) for \(\beta>\frac{1}{6}\), respectively, where \(\alpha^{\ast}\) is the positive root of \(\alpha ^{2}+(\beta +1)\alpha +\beta - 1=0\). The online algorithm is the best possible when \(0\leq\beta\leq\frac{1}{6}\).

MSC:

68M20 Performance evaluation, queueing, and scheduling in the context of computer systems
68W27 Online algorithms; streaming algorithms
90B35 Deterministic scheduling theory in operations research
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References:

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