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In this paper we study hard real-time systems composed of independent periodic preemptive tasks where we assume that tasks are scheduled by using Liu & Layland's pioneering model following the rate monotonic analysis (RMA). For such systems, the designer must guarantee that all the deadlines of all the tasks are met, otherwise dramatic consequences occur. Certainly, guaranteeing deadlines is not always achievable because the preemption is approximated when using this analysis, and this approximation may lead to a wrong real-time execution whereas the schedulability analysis concluded that the system was schedulable. To cope with this problem the designer usually allows margins which are difficult to assess, and thus in any case lead to a waste of resources. This paper makes multiple contributions. First, we show that, when considering the cost of the preemption during the analysis, the critical instant does not occur upon simultaneous release of all tasks. Second, we provide a technique which counts the exact number of preemptions of each instance for all the tasks of a given system. Finally, we present an RMA extension which takes into account the exact cost due to preemption in the schedulability analysis rather than an approximation, thus yielding a new and stronger schedulability condition which eliminates the waste of resources since margins are not necessary.