Scheduling algorithms play a significant role in the design of real-time Computing systems. Rate-monotonic, deadline-monotonic, relative urgency and least laxity algorithms are the most commonly used algorithms. It is already proved that rate-monotonic (RM) is an optimal fixed priority algorithm and many practical safety verification algorithms are developed [1, 2, 3, 4]. Here, an improvement to lower bound processor load factor for some special cases is presented. We then established a modified version of the rate-monotonic algorithm named delayed rate-monotonic (DRM) algorithm that uses two ready states, instead of one. For two special cases we showed full processor utilization is possible. Two algorithms, namely DRM and RM, are then compared and it is shown that there are many systems that are safe with DRM algorithm but are unsafe with RM algorithm. Also, for a set of randomly generated systems number of task preemptions for the two algorithms is compared. It is concluded that DRM is very promising and it has the potential to be considered for future research.