There have been conflicting results in the literature regarding the congestion impacts of shared mobility systems with for-hire vehicles (FHVs). To the best of our knowledge, there is no physically meaningful and mathematically tractable model to explain these conflicting results or devise efficient management schemes for such mobility systems. In this paper, we attempt to fill the gap by presenting a compartmental model for passenger trip and vehicle dynamics in shared mobility systems with FHVs and discussing the impacts of different fleet-size management schemes.

To develop the compartmental model, we first divide passenger trips into four compartments: planned, waiting, traveling, and completed. We describe the dynamics of the waiting trips by the point queue model, and those of the traveling trips by an extended bathtub model. As the traditional bathtub model for vehicular trips, the extended bathtub model is derived in a relative space with respect to individual trips’ distances to their destinations. However, different from the traditional bathtub model, vehicular dynamics and trip dynamics in the extended bathtub model are not overlapping, as the dynamics of FHVs are controlled by the fleet-size management scheme; but they are related, as traveling trips travel with occupied FHVs, and empty FHVs supply seats to waiting trips. Within this modeling framework, the matching process between waiting passengers and FHVs is modeled at the aggregate level, such that the passenger trip flow from the waiting compartment to the traveling compartment equals the minimum of the waiting trips’ demand of seats and the supply of seats determined by the completion rate of traveling trips and the fleet-size management scheme. In addition to the pooling ratio, the deadhead miles, the detour miles caused by pooling services, and other extra miles associated with the matching process are captured by another exogenous parameter, namely, the extra mileage ratio. With these assumptions and simplifications, the resulting compartmental model is a deterministic, coupled queueing model, which can be written as a system of differential equations. We also present the sufficient and necessary condition on the fleet-size management scheme for the model to be well-defined.

With the parsimonious, closed-form compartmental model, we demonstrate theoretically that limiting the wait time leads to a fleet-size management scheme equivalent to that of the privately operated vehicles (POVs), i.e., the POV scheme. In such a system, the completion rate depends on the extra trip mileage ratio, as well as the pooling ratio. With 100% autonomous FHVs, the optimal fleet size that minimizes the total costs occurs at the maximum flow-rate and the free-flow speed. With mixed POVs and FHVs, we extend the compartmental model and numerically solve for the optimal fleet sizes under different market penetration rates. This study reconciles the conflicting results in the literature. We find that, with a low pooling ratio, the overall system’s performance can be deteriorated or improved, depending on the fleet-size management scheme: with the POV scheme, the system could become more congested; but with an appropriate fleet-size cap, the system’s performance can be substantially improved. A major policy implication of this study is that implementing a cap for the FHV fleet size is a viable measure to mitigate the congestion effects of extra deadhead and detour miles caused by FHVs.

关于共享出行系统与租赁车辆 (FHV) 的拥堵影响，文献中存在相互矛盾的结果。据我们所知，没有物理意义和数学上易于处理的模型来解释这些相互矛盾的结果或为此类移动系统设计有效的管理方案。在本文中，我们试图通过在具有 FHV 的共享移动系统中提出乘客出行和车辆动力学的分区模型并讨论不同车队规模管理方案的影响来填补这一空白。

为了开发分区模型，我们首先将乘客旅行分为四个分区：计划、等待、旅行和完成。我们通过点队列模型描述等待行程的动态，并通过扩展浴缸模型描述旅行行程的动态。作为传统的汽车出行的浴缸模型，扩展浴缸模型是在相对空间中推导出的关于个人出行到目的地的距离。然而，与传统浴缸模型不同，扩展浴缸模型中的车辆动力学和行程动力学并不重叠，因为FHV的动力学由车队规模管理方案控制；但它们是相关的，因为旅行旅行与被占用的 FHV 一起旅行，而空的 FHV 为等待旅行提供座位。在这个建模框架内，等候乘客与 FHV 之间的匹配过程在聚合层面建模，使得从等候舱到旅行舱的乘客旅行流量等于等候旅行的座位需求和由完成率决定的座位供应中的最小值旅行和车队规模管理计划。除了拼车率之外，空头里程、拼车服务引起的绕道里程以及与匹配过程相关的其他额外里程都被另一个外生参数捕获，即额外里程比。有了这些假设和简化，产生的隔室模型是一个确定性的、耦合的排队模型，可以写成一个微分方程组。

通过简约的封闭式隔间模型，我们从理论上证明了限制等待时间会导致车队规模管理方案与私人运营车辆 (POV) 的管理方案相同，即 POV 方案。在这样的系统中，完成率取决于额外的行程里程比率，以及共享比率。使用 100% 自主 FHV，可在最大流量和自由流动速度下实现总成本最小化的最佳车队规模。通过混合 POV 和 FHV，我们扩展了分区模型，并在不同的市场渗透率下对最佳车队规模进行了数值求解。本研究调和了文献中相互矛盾的结果。我们发现，在池化率较低的情况下，整个系统的性能可能会恶化或提高，具体取决于车队规模管理方案：使用 POV 方案，系统可能会变得更加拥挤；但通过适当的机队规模上限，可以显着提高系统的性能。本研究的一个主要政策含义是，对 FHV 车队规模实施上限是减轻 FHV 造成的额外死角和绕行里程的拥堵影响的可行措施。