The utilization bound for multiprocessor systems using the rate-monotonic scheduling algorithm and first fit allocation policy proposed by Lopez, et al. offers the best performance among all O(m) complexity schedulability tests. In this paper, a utilization bound is derived for the same target problem. The main difference between these two bounds lies in the technique to verify the schedulability of task sets on uniprocessors; the schedulability test here is performed based on the hyperbolic bound proposed by Bini, et al. The new bound surpasses the existing one under quite a lot parameter settings, and the combination of these two schedulability test methods, which are compatible with each other, can significantly improve the number of schedulable task sets with little extra overhead.