We study minimum execution time scheduling of computer task systems where each task may require control of several processors during each step of its execution. Such a task system consists of: m identical processors, tasks T1, …, Tn, and a partial order on those tasks which represents precedence constraints. Associated with each task T1 is a positive integral execution time τ1 and a degree of concurrency q1 ∈ 𝒞 where 𝒞 ⊆ {1, …, m}. Task T1 must execute for τ1, steps and for each of those τ1 steps it requires q1 processors. Minimum length schedules for systems in which all task execution times are equal (concurrent UET task systems) are studied. Three sets of results are given. First, we show that scheduling concurrent UET task systems is NP-complete even if there are only three processors and each task has a degree of concurrency of either 1 or 2. Secondly, given any concurrent UET task system let r be the maximum degree of concurrency of any task. We show that the ratio of the length of an arbitrary list...
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