The Internet of Things has the potential of transforming health systems through the collection and analysis of patient physiological data via wearable devices and sensor networks. Such systems can offer assisted living services in real-time and offer a range of multimedia-based health services. However, service downtime, particularly in the case of emergencies, can lead to adverse outcomes and in the worst case, death. In this paper, we propose an e-health monitoring architecture based on sensors that relies on cloud and fog infrastructures to handle and store patient data. Furthermore, we propose stochastic models to analyze availability and performance of such systems including models to understand how failures across the Cloud-to-Thing continuum impact on e-health system availability and to identify potential bottlenecks. To feed our models with real data, we design and build a prototype and execute performance experiments. Our results identify that the sensors and fog devices are the components that have the most significant impact on the availability of the e-health monitoring system, as a whole, in the scenarios analyzed. Our findings suggest that in order to identify the best architecture to host the e-health monitoring system, there is a trade-off between performance and delays that must be resolved.