This paper considers learning outcomes in games where the outcomes of previous rounds have cumulative effects, such as routers. Previous work by Gaitonde and Tardos used an unrealistic model without buffers on servers, showing that timestamps and priorities are necessary for system stability. This paper presents a modified model that increases the realism of the model by adding a small buffer on the servers and not using timestamps or priorities, allowing for higher traffic throughput. Through theoretical analysis and simulations, we show that the system stability can be maintained even with a certain percentage increase in server capacity compared to centralized coordination. In particular, we prove that this result holds even when the servers randomly select packets that arrive simultaneously.