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A multiprocessing system composed of identical units is considered. This system is executing a set of partially ordered tasks, with known execution times, using a non-preemptive scheduling strategy. Lower bounds on the number of processors required to compute the tasks before a deadline, and on the minimum time to execute the tasks with a fixed number of processors, are of great value for the determination of the corresponding optimal schedules. In this paper, methods for the efficient computation of the lower bounds obtained by Fernández and Bussell are discussed. Computational improvements for the case of general partial orders are reported, and further reductions of the number of operations are shown to be possible for special graphs (trees, independent chains, independent tasks).
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