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This paper is an extended version of a paper that appeared in the proceedings of the IEEE Real-Time Systems Symposium 2009. This paper has been updated with respect to advances made in schedulability analysis, and contains a number of significant additional results.The paper addresses the problem of priority assignment in multiprocessor real-time systems using global fixed task-priority pre-emptive scheduling.We prove that Audsley's Optimal Priority Assignment (OPA) algorithm, originally devised for uniprocessor scheduling, is applicable to the multiprocessor case, provided that three conditions hold with respect to the schedulability tests used. Our empirical investigations show that the combination of optimal priority assignment policy and a simple compatible schedulability test is highly effective in terms of the number of tasksets deemed to be schedulable.We also examine the performance of heuristic priority assignment policies such as Deadline Monotonic, and an extension of the TkC priority assignment policy called DkC that can be used with any schedulability test. Here we find that Deadline Monotonic priority assignment has relatively poor performance in the multiprocessor case, while DkC priority assignment is highly effective.