The load never goes away in these graphs - the load is a constant sinusoid that's not actually shown on the graph. The blue line is queue depth. You are right

The load never goes away in these graphs - the load is a constant sinusoid that's not actually shown on the graph. The blue line is queue depth. You are right that it's not unloading. In general, you'd expect the response (green line) of a PID controller to a sinusoid load to be another sinusoid. In this case, because the maximum error is bounded on one side(the setpoint is 0, and queue length can never get below 0), it can only unload at a certain rate. It's typically the job of the Ki term to bring the long-term error towards zero. In this case, I think it's tuned small enough + the bounded asymmetric maximum error to result in it never unloading.