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Inspired by the recent developments of the Internet of Things (IoT) relay and mobile edge computing (MEC), a hospital/home-based medical monitoring framework is proposed, in which the intensive computing tasks from the implanted sensors can be efficiently executed by on-body wearable devices or a coordinator-based MEC (C-MEC). In this paper, we first propose a wireless relay-enabled task offloading mechanism that consists of a network model and a computation model. Moreover, to manage the computation resources among all relays, a task offloading decision model and the best task offloading recipient selection function is given. The performance evaluation considers different computation schemes under the predetermined link quality condition regarding the selected vital quality of service (QoS) metrics. After demonstrating the channel characterization and network topology, the performance evaluation is implemented under different scenarios regarding the network lifetime of all relays, network residual energy status, total number of locally executed packets, path loss (PL), and service delay. The results show that data transmission without the offloading scheme outperforms the offload-based technique regarding network lifetime. Moreover, the high computation capacity scenario achieves better performance regarding PL and the total number of locally executed packets.