This paper presents an optimal checkpoint strategy for fault-tolerance in real-time systems. In our environment, multiple real-time tasks with arbitrary periods are scheduled in the system by Rate Monotonic (RM) algorithm, and checkpoints are inserted at a constant interval in each task while the width of interval is different with respect to the task. We propose a method to determine the optimal checkpoint interval for each task so that the probability of completing all the tasks is maximized. Whenever a fault occurs to a checkpoint interval of a task, the execution time of the task would be prolonged by rollback and re-execution of checkpoints. Our scheme includes the schedulability test to examine whether a task can be completed with an extended execution time. A numerical experiment is conducted to demonstrate the applicability of the proposed scheme.