The integration of both real-rime systems and fault-tolerant systems has been emerged as one of the greatest challenges of this decade. It is called a responsive system, which has the objective to optimize both timeliness and reliability. The performance measure in responsive systems is responsiveness that tells how probable a system executes correctly on time with faults occurred. While there have been some achievements in communication protocols and specification, we believe that scheduling problems in responsive systems are not understood deeply and sufficiently, yet. In this paper, we discuss the scheduling problem in responsive systems. At first, we investigate the issues in the scheduling and propose the precise definition of the responsiveness. We also suggest a scheduling algorithm called Responsive Earliest Deadline First (REDF) for preemptive-aperiodic tasks in a uniprocessor system. We show that REDF is optimal to obtain the maximum responsiveness, and the time complexity is analyzed to be O (N x 2(N)). By illustrating a contradictory example, it is shown that REDF can be enhanced if a constraint on tasks is released.