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In this paper, we present a novel energy-aware scheduling (EAS) algorithm which statically schedules both communication transactions and computation tasks onto heterogeneous network-on-chip (NoC) architectures under real-time constraints. Our algorithm automatically assigns tasks onto different processing elements and then schedules their execution. At the same time, the algorithm also takes into consideration the exact communication delay by scheduling communication transactions in parallel. As the main contribution, we first formulate the problem of concurrent communication and task scheduling for heterogeneous NoC architectures and then propose an efficient heuristic to solve it. Experimental results show that significant energy savings can be achieved by using our energy-aware scheduler while meeting the specified performance constraints. For instance, for a complex multimedia application, 44% energy savings have been observed, on average, compared to the schedules generated by a standard earliest-deadline-first scheduler.