The disclosed embodiments relate to a packet-processing system. This system includes an input which is configured to receive packets, wherein the packets include control-message (CM) packets and traffic packets. It also includes a pipeline to process the packets, wherein the pipeline includes access points for accessing an engine which services requests for packets, wherein CM packets and traffic packets access the engine through different access points. The system additionally includes an arbiter to schedule packets entering the pipeline. While scheduling the packets, the arbiter is configured to account for empty slots in the pipeline to ensure that when CM packets and traffic packets initiate accesses to the engine through different access points, the accesses do not cause an overflow at an input queue for the engine.