Recently different formulations of the first-order Lighthill-Whitham-Richards (LWR) model have been identified in different coordinates and state variables. However, relationships between higher-order continuum and car-following traffic flow models are still not well understood. In this study, we first categorize traffic flow models according to their coordinates, state variables, and orders in the three-dimensional representation of traffic flow and propose a unified approach to convert higher-order car-following models into continuum models and vice versa. The conversion method consists of two steps: equivalent transformations between the secondary Eulerian (E-S) formulations and the primary Lagrangian (L-P) formulations, and approximations of L-P derivatives with anisotropic (upwind) finite differences. We use the method to derive continuum models from general second- and third-order car-following models and derive car-following models from second-order continuum models. Furthermore, we demonstrate that corresponding higher-order continuum and car-following models have the same fundamental diagrams, and that the string stability conditions for vehicle-continuous car-following models are the same as the linear stability conditions for the corresponding continuum models. A numerical example verifies the analytical results. In a sense, we establish a weak equivalence between continuum and car-following models, subject to errors introduced by the finite difference approximation. Such an equivalence relation can help us to pick out anisotropic solutions of higher-order models with non-concave fundamental diagrams.

All existing day-to-day dynamics of departure time choice at a single bottleneck are unstable, and this has led to doubt over the existence of a stable user equilibrium in the real world. However, empirical observations and our personal driving experience suggest stable stationary congestion patterns during a peak period. In this paper, we attempt to reconcile the discrepancy by presenting a stable day-to-day dynamical system for drivers' departure time choice at a single bottleneck. In our model, the decision variable in the execution stage is still drivers' departure times on the next day, but in the planning stage before the execution stage, drivers determine their departure times in order to arrive at the destination at better times with lower scheduling costs. We first define within-day traffic dynamics with the point queue model, costs, the departure time user equilibrium (DTUE), and the arrival time user equilibrium (ATUE). We then identify three behavioral principles in the planning stage: (i) drivers choose their departure and arrival times in a backward fashion (backward choice principle); (ii) after choosing the arrival times, they update their departure times to balance the total costs (cost-balancing principle); (iii) they choose their arrival times to reduce their scheduling costs or gain their scheduling payoffs (scheduling cost-reducing or scheduling payoff-gaining principle). In this sense, drivers' departure and arrival time choices are driven by their scheduling payoff choice. With a single tube or imaginary road model, we convert the nonlocal day-to-day arrival time shifting problem to a local scheduling payoff shifting problem. After introducing a new variable for the imaginary density, we apply the Lighthill-Whitham-Richards (LWR) model to describe the day-to-day dynamics of scheduling payoff choice and present splitting and costbalancing schemes to determine arrival and departure flow rates accordingly. We define the scheduling payoff user equilibrium (SPUE) as the stationary state of the LWR model, formulate a new optimization problem for the SPUE, and prove the global stability of the SPUE and, therefore, ATUE and DTUE by using Lyapunov's second method in which the objective function in the optimization formulation is the potential function. We also develop the corresponding discrete models for numerical solutions and use one numerical example to demonstrate the effectiveness and stability of the new day-to-day dynamical model. Different from existing ones, the new adjustment mechanism leads to stable day-to-day departure time choice dynamics by guaranteeing that drivers have better choices of departure/arrival times with larger scheduling payoffs on the next day, and such better choices are not over-chosen because of the constraint imposed by the single tube's cross-section area, which is equal to the jam density in the LWR model. This study is the first step for understanding stable day-to-day dynamics for departure time choice, and many follow-up studies are possible and warranted.

Diverging junctions are important network bottlenecks, and a better understanding of diverging traffic dynamics has both theoretical and practical implications. In this paper, we present a new framework for constructing kinematic wave solutions to the Riemann problem of kinematic wave models of diverging traffic with jump initial conditions. Within this framework, the function space of weak solutions is enlarged to include interior states that occur with stationary discontinuities, and various discrete flux functions in cell transmission models of diverging traffic are used as entropy conditions to pick out unique physical solutions. Then in demand-supply space, we prove that the Riemann problem can be uniquely solved for existing discrete flux functions, in the sense that stationary states and, therefore, shock or rarefaction waves on all links, can be uniquely determined under given initial conditions. We show that the two diverge models by Lebacque and Daganzo are asymptotically equivalent. We also prove that the supply proportional and priority-based diverge models are locally optimal evacuation strategies. With numerical examples, we demonstrate the validity of the analytical solutions of interior states, stationary states, and corresponding kinematic waves. This study presents a unified framework for analyzing traffic dynamics arising at diverging junctions and could be helpful for developing emergency evacuation strategies.

Our daily driving experience and empirical observations suggest that traffic patterns in a road network are relatively stationary during peak periods. In numerous transportation network studies, there has been an implicit conjecture that stationary states exist in a network when origin demands, route choice proportions, and destination supplies are constant. In this study, we first rigorously formulate the conjecture within the framework of a network kinematic wave theory with an invariant junction model. After defining stationary states, we derive a system of algebraic equations in 3-tuples of stationary link flow-rates, demands, and supplies. We then introduce a new definition of junction critical demand levels based on effective demands and supplies. With a map in critical demand levels, we show that its fixed points and, therefore, stationary states exist with the help of Brouwer's fixed point theorem. For two simple road networks, we show that the map is well-defined and can be used to solve stationary states with a brute-force method. Finally we summarize the study and present some future extensions and applications.

The kinematic wave model of traffic flow on a road network is a system of hyperbolic conservation laws, for which the Riemann solver is of physical, analytical, and numerical importance. In this paper, we present a new Riemann solver at a general network junction in the demand-supply space. In the Riemann solutions, traffic states on a link include the initial, stationary, and interior states, and a discrete Cell Transmission Model flux function in interior states is used as an entropy condition, which is consistent with fair merging and first-in-first-out diverging rules. After deriving the feasibility conditions for both stationary and interior states, we obtain a set of algebraic equations, and prove that the Riemann solver is well-defined, in the sense that the stationary states, the out-fluxes of upstream links, the in-fluxes of downstream links, and kinematic waves on all links can be uniquely solved. In addition, we show that the resulting global flux function in initial states is the same as the local one in interior states. Hence we presents a new definition of invariant junction models, in which the global and local flux functions are the same. We also present a simplified framework for solving the Riemann problem with invariant junction flux functions.

In (Jin, 2018), a new day-to-day dynamical system was proposed for drivers' departure time choice at a single bottleneck. Based on three behavioral principles, the nonlocal departure and arrival times choice problems were converted to the local scheduling payoff choice problem, whose day-to-day dynamics are described by the Lighthill-Whitham-Richards (LWR) model on an imaginary road of increasing scheduling payoff. Thus the departure time user equilibrium (DTUE), the arrival time user equilibrium (ATUE), and the scheduling payoff user equilibrium (SPUE) are uniquely determined by the stationary state of the LWR model, which was shown to be locally, asymptotically stable with analysis of the discrete approximation of the LWR model and through a numerical example. In this study attempt to analytically prove the global stability of the SPUE, ATUE, and DTUE. We first generalize the conceptual models for arrival time and scheduling payoff choices developed in (Jin, 2018) for a single bottleneck with a generalized scheduling cost function, which includes the cost of the free-flow travel time. Then we present the LWR model for the day-to-day dynamics for the scheduling payoff choice as well as the SPUE. We further formulate a new optimization problem for the SPUE and demonstrate its equivalent to the optimization problem for the ATUE in (Iryo and Yoshii, 2007). Finally we show that the objective functions in the two optimization formulations are equal and can be used as the potential function for the LWR model and prove that the stationary state of the LWR model, and therefore, the SPUE, DTUE, and ATUE, are globally, asymptotically stable, by using Lyapunov's second method. Such a globally stable behavioral model can provide more efficient departure time and route choice guidance for human drivers and connected and autonomous vehicles in more complicated networks.

Vickrey (1991, 2020) proposed a bathtub model for the evolution of trip flows served by privately operated vehicles inside a road network based on three premises: (i) treatment of the road network as a single bathtub; (ii) the speed-density relation at the network level, also known as the network fundamental diagram of vehicular traffic, and (iii) the time-independent negative exponential distribution of trip distances. However, the distributions of trip distances are generally time-dependent in the real world, and Vickrey's model leads to unreasonable results for other types of trip distance distributions. Thus there is a need to develop a bathtub model with more general trip distance distribution patterns. In this study, we present a unified framework for modeling network trip flows with general distributions of trip distances, including negative exponential, constant, and regularly sorting trip distances studied in the literature. In addition to tracking the number of active trips as in Vickrey's model, this model also tracks the evolution of the distribution of active trips’ remaining distances. We derive four equivalent differential formulations from the network fundamental diagram and the conservation law of trips for the number of active trips with remaining distances not smaller than any value. Then we define and discuss the properties of stationary and gridlock states, derive the integral form of the bathtub model with the characteristic method, and present two numerical methods to solve the bathtub model based on the differential and integral forms respectively. We further study equivalent formulations and solutions for two special types of distributions of trip distances: time-independent negative exponential or deterministic. In particular, we present six equivalent conditions for Vickrey's bathtub model to be applicable.

Fundamental to many transportation network studies, traffic flow models can be used to describe traffic dynamics determined by drivers' car-following, lane-changing, merging, and diverging behaviors. In this study, we develop a deterministic queueing model of network traffic flow, in which traffic on each link is considered as a queue. In the link queue model (LQM), the demand and supply of a link queue are defined in the queue size (number of vehicles), and its in- and out-flows are computed from junction flux functions corresponding to macroscopic merging and diverging rules. The new model is a system of ordinary differential equations that is mathematically tractable and computationally efficient and can capture queue spillbacks and interactions among links. We further demonstrate that the LQM is fundamentally different from the cell transmission model (CTM) and link transmission model (LTM) for a road segment, a signalized ring road, and a diverge-merge network, with respect to the shock and rarefaction waves, network fundamental diagram, and stability property. In a sense, the new model is a spacecontinuous approximation of the kinematic wave model and can be a useful addition to the multiscale modeling framework of network traffic flow. The model has been applied to formulate and solve network traffic control and observation problems.

In this paper, for continuous, linearly-controllable quadratic control systems with a single input, an explicit, constructive method is proposed for studying their Brunovsky forms, initially studied in [W. Kang and A. J. Krener, Extended quadratic controller normal form and dynamic state feedback linearization of nonlinear systems, SIAM Journal on Control and Optimization, 30:1319-1337, 1992]. In this approach, the computation of Brunovsky forms and transformation matrices and the proof of their existence and uniqueness are carried out simultaneously. In addition, it is shown that quadratic transformations in the aforementioned paper can be simplified to prevent multiplicity in Brunovsky forms. This method is extended for studying discrete quadratic systems. Finally, computation algorithms for both continuous and discrete systems are summarized, and examples demonstrated.