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Real-time systems usually consist of a set of periodic and sporadic tasks. Periodic tasks can be divided into two classes: synchronous and asynchronous. The first type does not define the task first release, contrary to the second. Hence, synchronous periodic tasks are assumed to be released at the worst instant: the critical instant. The schedulability test is reduced to check a single execution of the task under analysis. The integration of sporadic tasks is also straightforward: they are treated as a periodic task with maximum arrival frequency. On the other hand, asynchronous periodic tasks require a test for each release in the hyper period and the integration of sporadic tasks is not trivial: the worst release instant is unknown a priori. However, they do not assume that the tasks are released at the worst instant.This paper presents a new schedulability analysis method based on the Response Time Analysis (RTA) to determine the worst response time of both asynchronous periodic and sporadic tasks, scheduled by a fixed-priority preemptive algorithm with general deadlines. It also presents another method that enables the introduction of a user configurable degree of pessimism, reducing the hyper period dependency.