Arithmetic Modelling and Routing Algorithms for an Edge–Fog–Cloud Continuum Structured as a Full k-ary Tree

Remote computing is currently widely deployed in production networks, although its performance could be optimised through an appropriate architectural design. Accordingly, a mathematical model of an edge-fog-cloud hierarchy is presented, where all devices are organised as a full complete k-ary tree. When devices across layers are sequentially indexed, integer division and modular arithmetic can be used to determine the devices and ports involved along the path between a source and a destination end device. Two pseudocode algorithms are proposed, where one considers only a single cloud server, while the other extends the approach to multiple cloud servers. The key properties of both algorithms are simplicity and scalability, highlighting their technical relevance in remote computing architectures. The proposed model may have a practical impact on IoT, edge AI, and network design as it simplifies how packets are forwarded between end devices in remote computing environments.