We study a long-term formulation of link scheduling in large-scale wireless networks. Existing approaches maximize the instantaneous sum rate; however, while wireless networks exhibit highly dynamic behavior, end users generally experience performance in expectation. This motivates the optimization of time-averaged performance metrics. We formulate link scheduling as a constrained optimization problem in which the objective is to maximize the sum rate averaged over a series of time slots. Furthermore, we impose a minimum transmission requirement that guarantees each link is scheduled for at least a fixed fraction of the total time. We represent the network by modeling users and links as nodes and edges and construct the corresponding conflict graph to capture interference. We operate in the Lagrangian dual domain and parameterize the scheduling policy with a Graph Neural Network (GNN). To address the challenge of long-term performance optimization, we adopt a state-augmentation technique: by incorporating the Lagrangian dual variables as dynamic inputs to the GNN, the policy can gradually adapt its scheduling decisions to satisfy the time-average constraints. We demonstrate the effectiveness of the proposed policy through numerical simulations.