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The heterogeneous multiprocessor task partitioning with replication problem involves determining a mapping of recurring tasks upon a set consisting of different processing units in such a way that all tasks meet their timing constraints and no two replicas of the same task are assigned to the same processing unit. The replication requirement improves the resilience of the real-time system to a finite number of processor failures. This problem is NP-hard in the strong sense. We develop a Fully Polynomial-Time Approximation Scheme (FPTAS) for this problem.