The mouthpiece is one of the key components in determining the sound and the playability of the saxophone. Assuming plane-wave propagation inside the mouthpiece, two different models are proposed for acoustic modeling of the saxophone mouthpiece. First, a Transfer Matrix (TM) model is derived from a finite element model and is then validated by comparing the calculated input impedance of a mouthpiece-cone structure with measurements. Then, by adapting a methodology traditionally used in modeling the human vocal tract, a Geometric Model (GM) is used to approximate the mouthpiece as a series of acoustic tubes of varying cross-sectional areas, and its accuracy is examined by comparing it with the TM mouthpiece model. We couple both models to a measured input impedance of an alto saxophone and provide a comparison of both methods and their impact on different fingering configurations. Finally, we discuss applications in sound synthesis and in mouthpiece design.