Before revision:  Yet, the matchings that QPS-r computes have the same quality as maximal matchings in the following sense: Using such matchings as crossbar schedules results in exactly the same aforementioned provable throughput and delay guarantees as using maximal matchings, as we show using Lyapunov stability analysis .
After revision:  Using Lyapunov stability analysis, we show that the throughput and delay guarantees as using maximal matchings, as we show using Lyapunov stability analysis .