Spatiotemporal Analysis of Parallelized Computing at the Extreme Edge

Low-latency computational-task execution can be achieved by leveraging device-to-device offloading and parallel processing over nearby extreme edge devices (EEDs), a paradigm known as extreme edge computing (EEC). However, EEC performance is challenged by device spatial randomness with intermittent wireless connectivity, limited device computing power, time-varying availability, and device failures. This paper introduces a novel spatiotemporal analytical framework for EEC by integrating stochastic geometry with an absorbing continuous-time Markov chain (ACTMC) to capture the interplay between communication and computation. Modeling a large-scale millimeter-wave network, we derive tractable expressions for the average task response delay and the task completion probability under both random and location-aware EED selection. Numerical results quantify the impact of location-awareness and unveil the existence of an optimal task segmentation that minimizes delay, which depends on network parameters and EED capabilities. We also demonstrate that device failures and EED scarcity exacerbate delay, which can be mitigated through a collaborative load-balancing approach between EEC and Multi-Access Edge Computing (MEC) schemes. Simulations and sensitivity analyses validate the proposed framework and offer design insights for optimizing system performance.