Recent results have demonstrated unprecedented wavelength-tuning speed and repetition rate performance of semiconductor ring lasers incorporating scanning filters. However, several unique operational characteristics of these lasers have not been adequately explained, and the lack of an accurate model has hindered optimization. We numerically investigated the characteristics of these sources, using a semiconductor optical amplifier (SOA) traveling-wave Langevin model, and found good agreement with experimental measurements. In particular, we explored the role of the SOA refractive-index nonlinearities in determining the intracavity frequency-shift-broadening and the emitted power dependence on scan speed and direction. Our model predicts both continuous-wave and pulse operation and shows a universal relationship between the output power of lasers that have different cavity lengths and the filter peak frequency shift per round trip, therefore revealing the advantage of short cavities for high-speed biomedical imaging.