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In fixed-priority scheduling, the priority of a job,.once assigned, may not change. A new fixed-priority algorthm for scheduling systems of periodic tasks upon identical multiprocessors is proposed. This algorithm has an achievable utilization of (m+1)/2 upon m unit-capacity processors. It is proven that this algorithm is optimal from the perspective of achievable utilization in the sense that no fixed-priority algorithm for scheduling periodic task systems upon identical multiprocessors may have an achievable utilization greater than (m+1)/2.