Large language model (LLM)-powered agentic AI services increasingly demand low-latency inference, motivating the deployment of LLMs across distributed edge servers. However, heterogeneous communication and computing capabilities, together with dynamically evolving inference states, make the edge server selection for each incoming request time-varying and tightly coupled across slots. In this paper, we investigate an online request scheduling framework for edge LLM inference that jointly minimizes long-term average end-to-end latency and regulates workload distribution across heterogeneous edge servers. Two main challenges arise in this context. First, conventional latency models cannot accurately capture the fine-grained dynamics of multi-stage LLM execution. Second, the latency consequence of a scheduling decision is observed only after request completion, making immediate decision evaluation difficult. To address these challenges, we develop a cross-slot inference model that captures transmission, prefill, iteration-level decoding, and key-value (KV) cache evolution for each diverse request, and characterize server workload through a KV cache memory-time consumption metric. We propose the LYREO approach that transforms the long-term load-balancing constraint via Lyapunov optimization and employs reward redistribution with sequencebased return prediction to convert delayed outcomes into timely learning signals for earlier decisions. Simulations under various configurations demonstrate that LYREO consistently achieves lower latency and more balanced load distribution than representative learning-based and heuristic baseline schemes.

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