Agent-based models have been extensively used to simulate the behavior of travelers in transportation systems because they allow for realistic and versatile modeling of interactions. However, traditional agent-based models suffer from high computational costs and rely on tracking physical locations, raising privacy concerns. This paper proposes an efficient formulation for the agent-based bathtub model (AB²M) in the relative space, where each agent's trajectory is represented by a time series of the remaining distance to its destination. The AB²M can be understood as a microscopic model that tracks individual trips’ initiation, progression, and completion and is an exact numerical solution of the bathtub model for generic (time-dependent) trip distance distributions. The model can be solved for a deterministic set of trips with a given demand pattern (defined by the start time of each trip and its distance), or it can be used to run Monte Carlo simulations to capture the average behavior and variations of stochastic demand patterns. To enhance the computational efficiency, we introduce a priority queue formulation for AB²M, eliminating the need to update trip positions at each time step and allowing us to run large-scale scenarios with millions of individual trips in seconds. We systematically explore the scaling properties of AB²M and discuss the introduction of biases and numerical errors. Finally, we analyze the upper bound of the computational complexity of the AB²M and the benefits of the priority queue formulation and downscaling on the computational cost. The systematic exploration of scaling properties of the modeling of individual agents in the relative space with the AB²M further enhances its applicability to large-scale transportation systems and opens up opportunities for studying travel time reliability, scheduling, and mode choices.

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.

The introduction of autonomous vehicles (AV) will increase the vehicle heterogeneity on our roads. It is claimed that these vehicles will be able to achieve lower spacings for the same speed than human driven ones. Therefore, a good understanding of the influence of heterogeneous driver behavior on macroscopic traffic flow characteristics is crucial. This paper presents a stochastic Lighthill-Whitham Richards model by introducing heterogeneous, i.e., vehicle dependent, jam densities. The model is solved in Lagrangian coordinates, and the nature of the model allows for investigating the impact of driver heterogeneity on macroscopic relations of traffic flow, both through simulations and analytically. The results show that both static and dynamic macroscopic characteristics of the model, such as average speed, capacity drop and flow rate evolution at a bottleneck, are consistent with the deterministic version with an equivalent jam density, which is the harmonic mean of the distribution. Establishing the theoretical way to average the parameters will allow us to develop some control strategies for connected AV in a mixed environment to control the platoon behavior and drive traffic flow to the desired state. In this line, this paper discusses the relation between the desired AV characteristics and the market penetration rate to maximize the flow rate at bottlenecks by reducing the capacity drop effects. This further motivates future research on the technology development for AVs.

Sags and tunnels are major bottlenecks, where the road capacity is reduced, and the “capacity drop” phenomenon occurs; however, there is no simple model or theory that can explain the formation and other characteristics of capacity drop. This paper presents a car-following model, which is equivalent to a continuum model in the Lagrangian coordinates. The model is built on two main assumptions: (i) inhomogeneous fundamental diagrams with location-dependent time gaps, and (ii) bounded acceleration. We first demonstrate that the stationary speed profiles, the low acceleration rates, the dropped capacity, and the approximate time duration of the capacity drop formation in the model are consistent with empirical observations. Then the impacts on the stationary states and dropped capacity of the numerical viscosity caused by the discretization method are investigated, and it is shown that the dropped capacity converges to the theoretical value. Further, a one-dimensional iterated function system is proposed to model the formation mechanism of the capacity drop, which is derived by investigating the spatial pattern of equilibrium and bounded acceleration traffic states that arises in a lead-vehicle problem. Utilizing this model, we uncover a set of properties of the capacity drop such as existence, uniqueness, global convergence, and convergence speed. Finally, the model is applied to analyze the impacts of heterogeneous drivers. The model and insights in this study will help to develop control and management schemes to alleviate capacity drop effects with connected and autonomous vehicles in the future.

Some studies consider variable speed limit (VSL) control as a viable option to prevent traffic breakdown at bottlenecks by limiting the mainline flow with reduced speed limits. However, few studies consider the location of the application area as a design variable of the problem. This paper explains why the location of a VSL control area is crucial to prevent the capacity drop phenomenon at lane drop bottlenecks. We first define two types of stationary states, congested and uncongested, inside a lane drop bottleneck assuming the Lighthill-Whitham Richards model with bounded acceleration. In particular, the characteristics of these stationary states and their admissible conditions are discussed thoroughly. If the speed limit imposed is low enough, the location of the VSL application area is irrelevant to ensure an uncongested stationary state inside the bottleneck. However, for a given range of speed limits, the location of the VSL application area should be designed carefully to allow for uncongested stationary states and prevent the occurrence of the capacity drop. We formulate an optimization problem and show that, contrary to the general belief, the larger the speed limit, the farther the VSL application area should be from the bottleneck. Finally, the results are extended to other types of bottlenecks, such as sag or tunnel bottlenecks. To the best of our knowledge, this is the first study to analytically identify, formulate, and solve the optimal location problem for variable speed limit application areas. It makes fundamental contributions to both traffic flow theory (by analyzing the stationary states for VSL-controlled bottlenecks) and traffic control (by determining the optimal location of a VSL application area). Moreover, the results presented are of practical relevance because they can help to establish some guidelines for practitioners to implement VSL control strategies.

When there is upstream congestion the discharging flow-rate of a tunnel or sag bottleneck can drop, which leads to additional traffic jams. Therefore, control strategies such as variable speed limit (VSL) have been developed aiming to prevent or mitigate upstream traffic congestion. Understanding traffic dynamics at bottlenecks, especially the mechanism of capacity drop, is critical for developing such models. Many studies are centered on the control algorithm design of VSL. However, there are few studies that systematically anayze the effect that the VSL application area has on the control effectiveness. This paper extends to sag and tunnel bottlenecks the theoretical framework to analytically solve the optimal location of the speed limit application area (first developed in Martínez and Jin (2018)). Moreover, we prove that the optimization formulation can be simplified. Consequently, it can be applied to further bounded acceleration models than the constant one. Finally, for an open-loop control with a constant speed limit for the Kobotonoke tunnel bottleneck, we validate the analytic definition of optimal location by preventing capacity drop in numerical simulations.