Based on the distinguishing features of multi-tier millimeter wave (mmWave) networks such as different transmit powers, different directivity gains from directional beamforming alignment and path loss laws for line-of-sight (LOS) and non-line-of-sight (NLOS) links, we introduce a normalization model to simplify the analysis of multi-tier mmWave cellular networks. The highlight of the model is that we convert a multi-tier mmWave cellular network into a single-tier mmWave network, where all the base stations (BSs) have the same normalized transmit power 1 and the densities of BSs scaled by LOS or NLOS scaling factors respectively follow piecewise constant function which has multiple demarcation points. On this basis, expressions for computing the coverage probability are obtained in general case with beamforming alignment errors and the special case with perfect beamforming alignment in the communication. According to corresponding numerical exploration, we conclude that the normalization model for multi-tier mmWave cellular networks fully meets requirements of network performance analysis, and it is simpler and clearer than the untransformed model. Besides, an unexpected but sensible finding is that there is an optimal beam width that maximizes coverage probability in the case with beamforming alignment errors.