Generative artificial intelligence (GAI) is a promising technique towards 6G networks, and generative foundation models such as large language models (LLMs) have attracted considerable interest from academia and telecom industry. This work considers a novel edge-cloud deployment of foundation models in 6G networks. Specifically, it aims to minimize the service delay of foundation models by radio resource allocation and task offloading, i.e., offloading diverse content generation tasks to proper LLMs at the network edge or cloud. In particular, we first introduce the communication system model, i.e., allocating radio resources and calculating link capacity to support generated content transmission, and then we present the LLM inference model to calculate the delay of content generation. After that, we propose a novel in-context learning method to optimize the task offloading decisions. It utilizes LLM's inference capabilities, and avoids the difficulty of dedicated model training or fine-tuning as in conventional machine learning algorithms. Finally, the simulations demonstrate that the proposed edge-cloud deployment and in-context learning task offloading method can achieve satisfactory generation service quality without dedicated model training or fine-tuning.
