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This paper presents linear time, on-line algorithms which guarantee and jointly schedule firm aperiodic, hard sporadic and periodic tasks in fixed-priority real-time systems. We develop and capitalize on a methodology which computes the spare capacity Z(a,b) exactly in time Theta(n), for arbitrary schedule intervals [a, b), which, to the best of our knowledge, is the first linear time algorithm reported in the literature. Previous state of the art methods incur pseudopolynomial time to guarantee on-line a single aperiodic and incur coninuous overhead for slack maintenance. Our method guarantees and schedules firm tasks to receive FIFO or EDF service, incurring a one-time linear cost, of Theta(n) and Theta(n+k), respectively, where k is the number of pending firm tasks.