The rise of e-hailing taxis has significantly altered urban transportation and resulted in a competitive taxi market with both traditional street-hailing and e-hailing taxis. The new mobility services provide similar door-to-door rides as the traditional one and there is competition across these various services. In this study, we propose an innovative queueing network model for the competitive taxi market and capture the interactions not only within the taxi market but also between the taxi market and urban road system. An example is designed based on data from New York City. Numerical results show that the proposed modeling structure, together with the corresponding stationary limits, can capture dynamics within high demand areas using multiple data sources. Overall, this study shows how the queueing network approach can measure both the taxi and urban road system performance at an aggregate level. The model can be used to estimate not only the waiting/searching time during passenger-vehicle matching but also the delays in the urban road network. Furthermore, the model can be generalized to study the control and management of taxi markets.