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We present a response time analysis for global fixed-priority preemptive scheduling of constrained-deadline tasks upon a uniform multiprocessor where each processor can be characterized by a different speed. A fixed-priority scheduler assigns the jobs with the highest priorities to the fastest processors. Since determining whether all tasks can meet their deadlines is generally intractable even with identical processors, we propose two sufficient schedulability tests that calculate upper bounds on the task’s worst-case response time within polynomial and pseudo-polynomial time. The proposed tests leverage the linear programming model to upper bound the interference of the higher-priority tasks. Furthermore, we identify specific conditions and platforms upon which the problem can be solved more efficiently within linear time. These formulations are used to iteratively evaluate and refine possible solutions until a safe upper bound on the task’s worst-case response time is found. Additionally, we demonstrate that, with specific minor modifications, the proposed tests are compatible with Audsley’s optimal priority assignment. Experimental evaluations performed on synthetic task sets show that the proposed approach outperforms the state-of-the-art methods.