Healthcare data streams originate from various sensors and Internet of Things (IoT) devices deployed in medical equipment and healthcare facilities as well as worn by patients. These vast volumes of data need to be leveraged to improve patient care, optimize processes, and help health sector stakeholders and applications make faster and more informed decisions. Many healthcare applications use the power of the cloud for data processing. However, time-sensitive healthcare applications cannot tolerate sending data streams to the cloud for processing due to unacceptable high latency and network bandwidth requirements. Healthcare facilities and caregivers need the ability to efficiently stream data and process data streams in real-time at the edge. This paper describes a five-tier architecture that aims to deal with the streaming and processing of data generated by the various devices and equipment of healthcare facilities and systems to enable the creation of smart healthcare applications. The architecture is based on emerging and established technologies, including IoT, edge/fog computing, data integration techniques, cloud computing, and data analytics. The proposed architecture will facilitate the creation of healthcare applications for real-time event detection, notification of alerts, and building monitoring dashboards. Fog node components include an advanced and widely recognized distributed messaging system, Apache Kafka, and the popular stream processing engine, Apache Storm, capable of processing large amounts of data.