Guaranteeing satisfaction of timing constraints of a real time system is an important aspect in hard real time systems design. The utilization test is an effective means to determine schedulability of a real time system prior to implementation and to assign the time budget to individual tasks in the design phase. In this thesis, a novel approach is proposed to compute the exact utilization bound for a task set with known task periods and deadlines using a fixed priority scheduling algorithm. Compared to Park's period-specific utilization bound algorithm, this approach can obtain a more accurate utilization bound without knowledge of the exact task computation times. In addition, this thesis presents two case studies that apply utili...