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This paper generalizes the notion of utilization bounds for schedulability of aperiodic tasks to the case of distributed resource systems. In the basic model, aperiodically arriving tasks are processed by multiple stages of a resource pipeline within end-to-end deadlines. The authors consider a multidimensional space in which each dimension represents the instantaneous utilization of a single stage. A feasible region is derived in this space such that all tasks meet their deadlines as long as pipeline resource consumption remains within the feasible region. The feasible region is a multidimensional extension of the single-resource utilization bound giving rise to a bounding surface in the utilization space rather than a scalar bound. Extensions of the analysis are provided to nonindependent tasks and arbitrary task graphs. We evaluate the performance of admission control using simulation, as well as demonstrate the applicability of these results to task schedulability analysis in the total ship computing environment envisioned by the US navy.