This paper proposes a three-computing-layer architecture consisting of Edge, Fog, and Cloud for remote health vital signs monitoring. The novelty of this architecture is in using the Narrow-Band IoT (NB-IoT) for communicating with a large number of devices and covering large areas with minimum power consumption. Additionally, the architecture reduces the communication delay as the edge layer serves the health terminal devices with initial decisions and prioritizes data transmission for minimizing congestion on base stations. The paper also investigates different authentication protocols for improving security while maintaining low computation and transmission time. For data analysis, different machine learning algorithms, such as decision tree, support vector machines, and logistic regression, are used on the three layers. The proposed architecture is evaluated using CloudSim, iFogSim, and ns3-NB-IoT on real data consisting of medical vital signs. The results show that the proposed architecture reduces the NB-IoT delay by 59.9%, the execution time by an average of 38.5%, and authentication time by 35.1% for a large number of devices. This paper concludes that the NB-IoT combined with edge, fog, and cloud computing can support efficient remote health monitoring for large devices and large areas.