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Remote health monitoring is a powerful tool to provide preventive care and early intervention for populations-at-risk. Such monitoring systems are becoming available nowadays due to recent advancements in Internet-of-Things (IoT) paradigms, enabling ubiquitous monitoring. These systems require a high level of quality in attributes such as availability and accuracy due to patients critical conditions in the monitoring. Deep learning methods are very promising in such health applications to obtain a satisfactory performance, where a considerable amount of data is available. These methods are perfectly positioned in the cloud servers in a centralized cloud-based IoT system. However, the response time and availability of these systems highly depend on the quality of Internet connection. On the other hand, smart gateway devices are unable to implement deep learning methods (such as training models) due to their limited computational capacities. In our previous work, we proposed a hierarchical computing architecture (HiCH), where both edge and cloud computing resources were efficiently exploited, allocating heavy tasks of a conventional machine learning method to the cloud servers and outsourcing the hypothesis function to the edge. Due to this local decision making, the availability of the system was highly improved. In this paper, we investigate the feasibility of deploying the Convolutional Neural Network (CNN) based classification model as an example of deep learning methods in this architecture. Therefore, the system benefits from the features of the HiCH and the CNN, ensuring a high-level availability and accuracy. We demonstrate a real-time health monitoring for a case study on ECG classifications and evaluate the performance of the system in terms of response time and accuracy.