The rise of e-hailing taxis has significantly altered urban transportation system and led to a competitive taxi market with both traditional street-hailing and e-hailing taxis. The various taxi services provide similar door-to-door rides, thus compete with each other. Meanwhile, the expanding fleet size of e-hailing taxis, together with considerable number of traditional taxicabs, influence the urban road network performance, which can also in turn affect taxi mode choice and operation. In this study, we propose an innovative modeling structure 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.

The model is built on a network linking a certain number of queues, derived from both the taxi and urban road system. Considering both the passenger and vehicle arrivals, we utilize an assembly-like queue $\mathit{SM}/M/1$ for passenger-vehicle matching within the taxi system, which controls how many and how frequently vehicles drive from the taxi system to the urban road system. A common multi-server $M/M/c$ queue that can account for road capacity is proposed for the urban road system and a feedback of network states is sent back to the taxi system. Then the stationary state distributions, as well as asymptotic properties, of the queueing network are discussed.

An example is designed based on data from New York City. Numerical results show that the proposed modeling structure, together with the corresponding approximation method, can capture dynamics within high demand areas and, in particular, a spatially balanced system. Overall, this study shows how the queueing network approach can measure aggregated performance of various taxi services, for instance, 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.

电子打车出租车的兴起极大地改变了城市交通系统，并导致了传统的街头打车和电子打车出租车的竞争。各种出租车服务提供类似的门到门游乐设施，因此彼此竞争。同时，电子叫车出租车的车队规模扩大，加上相当数量的传统出租车，影响了城市道路网的性能，这反过来又会影响出租车的模式选择和运营。在这项研究中，我们为竞争性出租车市场提出了一种创新的建模结构，不仅捕获了出租车市场内的相互作用，而且还捕获了出租车市场与城市道路系统之间的相互作用。

该模型建立在连接一定数量队列的网络上，该队列既来自出租车也来自城市道路系统。考虑到乘客和车辆的到来，我们利用了类似组装的队列$\mathit{SM}/\mathrm{中号}/\mathrm{1个}$用于出租车系统中的客车匹配，它控制从出租车系统到城市道路系统的行驶车辆的数量和频率。通用的多服务器$\mathrm{中号}/\mathrm{中号}/C$针对城市道路系统，提出了可以解释道路通行能力的队列，并将网络状态的反馈发送回出租车系统。然后讨论了排队网络的稳态分布以及渐近性质。

根据纽约市的数据设计了一个示例。数值结果表明，所提出的建模结构以及相应的近似方法可以捕获高需求区域内的动力学，尤其是空间平衡的系统。总的来说，这项研究表明了排队网络方法如何能够衡量各种出租车服务的综合性能，例如，不仅是客车匹配期间的等待/搜索时间，还包括城市道路网的延误。此外，该模型可以推广研究出租车市场的控制和管理。