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The utilization bound for real-time rate monotonic (RM) scheduling on uniprocessors is extended to multiprocessors with partitioning-based scheduling. This allows fast schedulability tests to be performed on multiprocessors and quantifies the influence of key parameters, such as the number of processors and task sizes on the schedulability of the system. The multiprocessor utilization bound is a function of the allocation algorithm, so among all the allocation algorithms there exists at least one allocation algorithm providing the minimum multiprocessor utilization bound, and one allocation algorithm providing the maximum multiprocessor utilization bound. We prove that the multiprocessor utilization bound associated with the allocation heuristic worst fit (WF) coincides with that minimum if we use Liu and Layland's bound (LLB) as the uniprocessor schedulability condition. In addition, we present a class of allocation algorithms sharing the same multiprocessor utilization bound which coincides with the aforementioned maximum using LLB. The heuristics first fit decreasing (FFD) and best fit decreasing (BFD) belong to this class. Thus, not even an optimal allocation algorithm can guarantee a higher multiprocessor utilization bound than that of FFD and BFD using LLB. Finally, the pessimism of the multiprocessor utilization bounds is estimated through extensive simulations.