This paper presents a novel two-phase maskable proximal policy optimization (TP-MPPO) algorithm, which maximizes the system goodput counting request throughput with strict service level objective (SLO) compliance for large language model (LLM) inference services in wireless edge networks. In the first phase of TP-MPPO, we optimize the task offloading decisions by MPPO with action masking mechanism, effectively avoiding exploring invalid actions and reducing the action space. In the second phase, closed-form solutions are derived for uplink bandwidth allocation; a greedy algorithm is designed for downlink bandwidth allocation to provide immediate rewards for the MPPO in the next round. The two stages alternate till convergence. Simulation results demonstrate that TP-MPPO can improve the system reward by 33.3%--87.5% compared to its benchmarks and achieve the highest goodput.

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