Large language models have significantly transformed multiple fields with their exceptional performance in natural language tasks, but their deployment in resource-constrained environments like edge networks presents an ongoing challenge. Decentralized techniques for inference have emerged, distributing the model blocks among multiple devices to improve flexibility and cost effectiveness. However, energy limitations remain a significant concern for edge devices. We propose a sustainable model for collaborative inference on interconnected, battery-powered edge devices with energy harvesting. A semi-Markov model is developed to describe the states of the devices, considering processing parameters and average green energy arrivals. This informs the design of scheduling algorithms that aim to minimize device downtimes and maximize network throughput. Through empirical evaluations and simulated runs, we validate the effectiveness of our approach, paving the way for energy-efficient decentralized inference over edge networks.