A large spectrum of healthcare applications, ranging from continuous blood sugar level monitoring to sleep apnea detection are nowadays facilitated by modern mobile gadgets. Wearable and ambient sensors generate enormous amounts of physiological data that demand high computation power for real-time processing and large storage area for recording the personal data. In order to conserve the energy on the battery-limited mainstream mobile devices of the end-users, the execution of the healthcare applications may be offloaded to a remote server. While cloud computing provides unlimited pool of resources for latency-tolerant services such as training a machine learning model, the personalization of healthcare services and the delay sensitivity of continuous health assessment necessitate a computation infrastructure in the vicinity of the end-users. As a remedy to address various demands of a wide range of pervasive healthcare applications, we propose a multi-tier computing and communication architecture composed of end-user devices, edge servers, and legacy cloud data-centers. The dynamic management of this architecture, policies to be applied within the network and the orchestration of the healthcare services are carried out by the concept of programmable networks, in the form of software-defined networking (SDN). As a concrete demonstration of our ideas, a fall risk assessment service is implemented and an experimental study is conducted to evaluate its accuracy and the performance of the multi-tier architecture. The results indicate that the proposed architecture is feasible to enable real-time healthcare services and has significant performance advantages over traditional cloud-based approaches.