Interactions among different modes or vehicle classes in urban road networks affect the network performance in different and complex ways. Thus, an answer to the question of “how many cars are too many for a city?” is not trivial. However, multi-modal macroscopic fundamental diagrams (MFD) offer a novel opportunity to answer this question. So far, no methodology exists to estimate multi-modal MFDs resulting from arbitrary multi-modal interactions. In this paper, we propose a methodology to capture additional delays in the shape of the MFD and derive an approach for estimating multi-modal MFDs thereof. The influence on the MFD shape is established using the two-fluid theory of urban traffic by defining pairwise copula functions between travel times of each mode. In contrast to many existing approaches, the presented approach retains individual mode's speed information. We show the applicability of the approach with a tri-modal case of bicycles, buses, and cars with empirical data from Amsterdam (The Netherlands) and London (United Kingdom). Although the approach is not limited to this specific tri-modal case, we use the example to discuss the initial policy question by deriving optimal modal splits for a given accumulation of travelers. Last, we compare the new approach to existing estimation methods for bi-modal MFDs describing car and bus traffic.@en

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.@en