In recent years, Internet of Things (IoT) advancements have led to the development of vastly improved remote healthcare services. Scalability, high bandwidth, low latency, and low power consumption are all essential features of the applications that make these services possible. An upcoming healthcare system and wireless sensor network that can fulfil these needs is based on fifth-generation network slicing. For better resource management, organizations can implement network slicing, which partitions the physical network into distinct logical slices according to quality of service (QoS) needs. Based on the findings of this research, an IoT–fog–cloud architecture is proposed for use in e-Health services. The framework is made up of three different but interconnected systems: a cloud radio access network, a fog computing system, and a cloud computing system. A queuing network serves as a model for the proposed system. The model’s constituent parts are then subjected to analysis. To assess the system’s performance, we run a numerical example simulation using Java modelling tools and then analyze the results to identify the key performance parameters. The analytical formulas that were derived ensure the precision of the results. Finally, the results show that the proposed model improves eHealth services’ quality of service in an efficient way by selecting the right slice compared to the traditional systems.