Utilization bound is a well-known concept first introduced in the seminal paper of Liu & Layland which provides a simple and practical way to test the schedulability of a real-time task set. The original utilization bound for the fixed-priority scheduler was given as a function of the number of tasks in the periodic task set. In this paper, we define the utilization bound as a function of the information about the task set. By making use of more than just the number of tasks, we obtain various improvements over the Liu & Layland bound. In particular, we shall give a more intuitive derivation of the bound as a function of the number of harmonic chains in the task periods which allows us to derive a simpler algorithm to calculate such bounds. We derive algorithms that yield better bounds as a function of the period parameters of the task set. We also give a generalization of the bound for tasks whose deadlines are smaller than their periods.