A stochastic model of traffic flow: Theoretical foundations

In a variety of applications of traffic flow, including traffic simulation, real-time estimation and prediction, one requires a probabilistic model of traffic flow. The usual approach to constructing such models involves the addition of random noise terms to deterministic equations, which could lead to negative traffic densities and mean dynamics that are inconsistent with the original deterministic dynamics. This paper offers a new stochastic model of traffic flow that addresses these issues. The source of randomness in the proposed model is the uncertainty inherent in driver gap choice, which is represented by random state dependent vehicle time headways. A wide range of time headway distributions is allowed. From the random time headways, counting processes are defined, which represent cumulative flows across cell boundaries in a discrete space and continuous time conservation framework. We show that our construction implicitly ensures non-negativity of traffic densities and that the fluid limit of the stochastic model is consistent with cell transmission model (CTM) based deterministic dynamics.     

A flexible traffic stream model and its three representations of traffic flow

To connect microscopic driving behaviors with the macro-correspondence (i.e., the fundamental diagram), this study proposes a flexible traffic stream model, which is derived from a novel car-following model under steady-state conditions. Its four driving behavior-related parameters, i.e., reaction time, calmness parameter, speed- and spacing-related sensitivities, have an apparent effect in shaping the fundamental diagram. Its boundary conditions and homogenous case are also analyzed in detail and compared with other two models (i.e., Longitudinal Control Model and Intelligent Driver Model). Especially, these model formulations and properties under Lagrangian coordinates provide a new perspective to revisit the traffic flow and complement with those under Eulerian coordinate. One calibration methodology that incorporates the monkey algorithm with dynamic adaptation is employed to calibrate this model, based on real-field data from a wide range of locations. Results show that this model exhibits the well flexibility to fit these traffic data and performs better than other nine models. Finally, a concrete example of transportation application is designed, in which the impact of three critical parameters on vehicle trajectories and shock waves with three representations (i.e., respectively defined in x-t, n-t and x-n coordinates) is tested, and macro- and micro-solutions on shock waves well agree with each other. In summary, this traffic stream model with the advantages of flexibility and efficiency has the good potential in level of service analysis and transportation planning.     

A chance-constrained based stochastic dynamic traffic assignment model: Analysis, formulation and solution algorithms

This paper is concerned with the system optimum-dynamic traffic assignment (SO-DTA) problem when the time-dependent demands are random variables with known probability distributions. The model is a stochastic extension of a deterministic linear programming formulation for SO-DTA introduced by Ziliaskopoulos (Ziliaskopoulos, A.K., 2000. A linear programming model for the single destination system optimum dynamic traffic assignment problem, Transportation Science, 34, 1–12). The proposed formulation is chance-constrained based and we demonstrate that it provides a robust SO solution with a user specified level of reliability. The model provides numerous insights and can be a useful tool in producing robust control and management strategies that account for uncertainty in applications where SO-DTA is relevant (e.g. evacuation modeling, computing alternate routes around freeway incidents and establishing lower bounds on network performance).     

Estimation of traffic variables using a linear model of traffic flow

Estimation of traffic velocity and the number of vehicles on adjacent sections of a limited access highway is examined. The method evaluated is based upon application of Kalman Filtering Methods to a linear state variable model of traffic flow. The estimator utilizes velocity and flow measurements at selected points along the highway. The flow measurement is a nonlinear function of the state variables and necessitates linearization about the one step ahead prediction of the state (extended Kalman Filter) or about nominal values of the state variables. It is shown that performance using Lincoln Tunnel data is comparable in either case to that of methods previously reported and provides a substantial savings in storage requirements. Also demonstrated is the fact that flow at an internal measurement point may be deleted from the observation vector without a serious effect on performance. This would arise, for example, if control of traffic were to be exercised at such a point.     

Personalized real-time traffic information provision: Agent-based optimization model and solution framework

The advancement of information and communication technology allows the use of more sophisticated information provision strategies for real-time congested traffic management in a congested network. This paper proposes an agent-based optimization modeling framework to provide personalized traffic information for heterogeneous travelers. Based on a space–time network, a time-dependent link flow based integer programming model is first formulated to optimize various information strategies, including elements of where and when to provide the information, to whom the information is given, and what alternative route information should be suggested. The analytical model can be solved efficiently using off-the-shelf commercial solvers for small-scale network. A Lagrangian Relaxation-based heuristic solution approach is developed for medium to large networks via the use of a mesoscopic dynamic traffic simulator.     

Empirical analysis of heterogeneous traffic flow and calibration of porous flow model

Heterogeneous traffic flow, characterized by a free inter-lane exchange, has become an important issue in addressing congestion in urban areas. It is of particular interest in many developing countries, that experience a strong increase in motorcycle use. New approaches to the heterogeneous non-lane-based flow have been proposed. However insufficient empirical verification has been made to estimate vehicle interaction, that is necessary for an accurate representation of mixed-flow conditions. In this paper, we focus on the porous flow approach to capture the complex interactions. The parameters from this approach are estimated from empirical observations. Video data was recorded and processed to capture vehicle interactions at a number of road sections in Surabaya City, Indonesia. The specific behavior of each vehicle in the traffic flow was captured by developing the pore size–density distributions, analyzing the class-specific critical pore sizes, and producing the class specific speed–density and flow–density diagrams. The results reveal how critical pore sizes are based on pore size–density distributions, the flow diagram for each vehicle class, and how traffic flow relationships for motorcyclists and the other vehicles exhibit significant differences. It is concluded that the proposed approach can represent the specific behavior of the motorcyclist in heterogeneous traffic flow, in both the situations of with- and without an exclusive lane for motorcycles, can clarify motorcyclist’s behavior in terms of passenger car unit of motorcycle, and can therefore support policy making on the improvement of urban transport.     

Hysteresis in traffic flow revisited: An improved measurement method

This paper presents new insights on the hysteresis phenomenon in congested freeway traffic. It is found that existing theories based on different driver behavior for acceleration and deceleration are incomplete. The data suggests that one needs to consider aggressive and timid driver behavior as well. These findings are based on an improved method for measuring traffic flow variables from trajectory data consistently with kinematic wave theory.     

The variational formulation of a non-equilibrium traffic flow model: Theory and implications

The analysis and numerical solution of non-equilibrium traffic flow models in current literature are almost exclusively carried out in the hyperbolic conservation law framework, which requires a good understanding of the delicate and non-trivial Riemann problems for conservation laws. In this paper, we present a novel formulation of certain non-equilibrium traffic flow models based on their isomorphic relation with optimal control problems. This formulation extends the minimum principle observed by the LWR model. We demonstrate that with the new formulation, generic initial-boundary conditions can be conveniently handled and a simplified numerical solution scheme for non-equilibrium models can be devised. Besides deriving the variational formulation, we provide a comprehensive discussion on its mathematical properties and physical implications.     

Analytical and grid-free solutions to the Lighthill-Whitham-Richards traffic flow model

In this article, we propose a computational method for solving the Lighthill-Whitham-Richards (LWR) partial differential equation (PDE) semi-analytically for arbitrary piecewise-constant initial and boundary conditions, and for arbitrary concave fundamental diagrams. With these assumptions, we show that the solution to the LWR PDE at any location and time can be computed exactly and semi-analytically for a very low computational cost using the cumulative number of vehicles formulation of the problem. We implement the proposed computational method on a representative traffic flow scenario to illustrate the exactness of the analytical solution. We also show that the proposed scheme can handle more complex scenarios including traffic lights or moving bottlenecks. The computational cost of the method is very favorable, and is compared with existing algorithms. A toolbox implementation available for public download is briefly described, and posted at http://traffic.berkeley.edu/project/downloads/lwrsolver.     

Improving urban traffic flow by restraint of traffic: The case of Lagos,Nigeria

Traffic restraint techniques, widely used to combat traffic congestion in the developed countries, have been tried in Lagos, Nigeria. This paper examines the impact of the use of odd and even numbered vehicles on alternate days in reducing congestion on the urban roads. The result of the analysis indicates that the restraint technique worked only during the first few months of its introduction. The basic causes of the subsequent failure are outlined and possible solutions suggested.     

Multilane first-order traffic flow model with endogenous representation of lane-flow equilibrium

In this study, we develop a multilane first-order traffic flow model for freeway networks. In the model, lane changing is considered as a stochastic behavior that can decrease an individual driver’s disutility or cost, and is represented as dynamics toward the equilibrium of lane-flow distribution along with longitudinal traffic dynamics. The proposed method can be differentiated from those in previous studies because in this study, the motivation of lane changing is explicitly considered and it is treated as a utility defined by the current macroscopic traffic state. In addition, the entire process of lane changing is computed macroscopically by an extension of the kinematic wave theory employing IT principle; moreover, in the model framework, the lane-flow equilibrium curve is endogenously generated because of self-motivated lane changes. Furthermore, the parsimonious representation enables parameter calibration using the data collected from conventional loop detectors. The calibration of the data collected at four different sites, including a sag bottleneck, on the Chugoku expressway in Japan reveals that the proposed method can represent the lane-flow distribution of any observation site with high accuracy, and that the estimated parameters can reasonably explain the multilane traffic dynamics and the bottleneck phenomena uphill of sag sections.     

A hybrid deep learning based traffic flow prediction method and its understanding

Deep neural networks (DNNs) have recently demonstrated the capability to predict traffic flow with big data. While existing DNN models can provide better performance than shallow models, it is still an open issue of making full use of spatial-temporal characteristics of the traffic flow to improve their performance. In addition, our understanding of them on traffic data remains limited. This paper proposes a DNN based traffic flow prediction model (DNN-BTF) to improve the prediction accuracy. The DNN-BTF model makes full use of weekly/daily periodicity and spatial-temporal characteristics of traffic flow. Inspired by recent work in machine learning, an attention based model was introduced that automatically learns to determine the importance of past traffic flow. The convolutional neural network was also used to mine the spatial features and the recurrent neural network to mine the temporal features of traffic flow. We also showed through visualization how DNN-BTF model understands traffic flow data and presents a challenge to conventional thinking about neural networks in the transportation field that neural networks is purely a “black-box” model. Data from open-access database PeMS was used to validate the proposed DNN-BTF model on a long-term horizon prediction task. Experimental results demonstrated that our method outperforms the state-of-the-art approaches.     

Spillback congestion in dynamic traffic assignment: A macroscopic flow model with time-varying bottlenecks

In this paper, we propose a new model for the within-day Dynamic Traffic Assignment (DTA) on road networks where the simulation of queue spillovers is explicitly addressed, and a user equilibrium is expressed as a fixed-point problem in terms of arc flow temporal profiles, i.e., in the infinite dimension space of time’s functions. The model integrates spillback congestion into an existing formulation of the DTA based on continuous-time variables and implicit path enumeration, which is capable of explicitly representing the formation and dispersion of vehicle queues on road links, but allows them to exceed the arc length. The propagation of congestion among adjacent arcs will be achieved through the introduction of time-varying exit and entry capacities that limit the inflow on downstream arcs in such a way that their storage capacities are never exceeded. Determining the temporal profile of these capacity constraints requires solving a system of spatially non-separable macroscopic flow models on the supply side of the DTA based on the theory of kinematic waves, which describe the dynamic of the spillback phenomenon and yield consistent network performances for given arc flows. We also devise a numerical solution algorithm of the proposed continuous-time formulation allowing for “long time intervals” of several minutes, and give an empirical evidence of its convergence. Finally, we carry out a thorough experimentation in order to estimate the relevance of spillback modeling in the context of the DTA, compare the proposed model in terms of effectiveness with the Cell Transmission Model, and assess the efficiency of the proposed algorithm and its applicability to real instances with large networks.     

Symmetries in the kinematic wave model and a parameter-free representation of traffic flow

This paper identifies a family of linear transformations where conservation laws are invariant. In the case of a triangular fundamental diagram, it is shown that for a subset of these transformations, flow, total distance traveled and total delay are invariant. This means that for capacity or delay computations one may choose the transformation—i.e., the shape of the triangular diagram—that simplifies the problem the most, which does not require knowing the actual fundamental diagram. This is appealing also for delay-optimizing control problems since they may be solved using an isosceles fundamental diagram, which provides the most efficient numerical methods. Examples are given.     

Entropy weighted average method for the determination of a single representative path flow solution for the static user equilibrium traffic assignment problem

The formulation of the static user equilibrium traffic assignment problem (UETAP) under some simplifying assumptions has a unique solution in terms of link flows but not in terms of path flows. Large variations are possible in the path flows obtained using different UETAP solution algorithms. Many transportation planning and management applications entail the need for path flows. This raises the issue of generating a meaningful path flow solution in practice. Past studies have sought to determine a single path flow solution using the maximum entropy concept. This study proposes an alternate approach to determine a single path flow solution that represents the entropy weighted average of the UETAP path flow solution space. It has the minimum expected Euclidean distance from all other path flow solution vectors of the UETAP. The mathematical model of the proposed entropy weighted average method is derived and its solution stability is proved. The model is easy to interpret and generalizes the proportionality condition of Bar-Gera and Boyce (1999). Results of numerical experiments using networks of different sizes suggest that the path flow solutions for the UETAP using the proposed method are about identical to those obtained using the maximum entropy approach. The entropy weighted average method requires low computational effort and is easier to implement, and can therefore serve as a potential alternative to the maximum entropy approach in practice.     

Stop-and-go traffic analysis: Theoretical properties,environmental impacts and oscillation mitigation

This study aims (i) to analyze theoretical properties of a recently proposed describing-function (DF) based approach (Li and Ouyang, 2011; Li et al., 2012) for traffic oscillation quantification, (ii) to adapt it for estimating fuel consumption and emission from traffic oscillation and (iii) to explore vehicle control strategies of smoothing traffic with advanced technologies. The DF approach was developed to predict traffic oscillation propagation across a platoon of vehicles following each other by a nonlinear car-following law with only the leading vehicle’s input. We first simplify the DF approach and prove a set of properties (e.g., existence and uniqueness of its solution) that assure its prediction is always consistent with observed traffic oscillation patterns. Then we integrate the DF approach with existing estimation models of fuel consumption and emission to analytically predict environmental impacts (i.e., unit-distance fuel consumption and emission) from traffic oscillation. The prediction results by the DF approach are validated with both computer simulation and field measurements. Further, we explore how to utilize advantageous features of emerging sensing, communication and control technologies, such as fast response and information sharing, to smooth traffic oscillation and reduce its environmental impacts. We extend the studied car-following law to incorporate these features and apply the DF approach to demonstrate how these features can help dampen the growth of oscillation and environmental impact measurements. For information sharing, we convert the corresponding extended car-following law into a new fixed point problem and propose a simple bisecting based algorithm to efficiently solve it. Numerical experiments show that these new car-following control strategies can effectively suppress development of oscillation amplitude and consequently mitigate fuel consumption and emission.     

A new probability statistical model for traffic noise prediction on free flow roads and control flow roads

A new traffic noise prediction approach based on a probability distribution model of vehicle noise emissions and achieved by Monte Carlo simulation is proposed in this paper. The probability distributions of the noise emissions of three types of vehicles are obtained using an experimental method. On this basis, a new probability statistical model for traffic noise prediction on free flow roads and control flow roads is established. The accuracy of the probability statistical model is verified by means of a comparison with the measured data, which has shown that the calculated results of L_eq, L₁₀, L₅₀, L₉₀, and the probability distribution of noise level occurrence agree well with the measurements. The results demonstrate that the new method can avoid the complicated process of traffic flow simulation but still maintain high accuracy for the traffic noise prediction.     

A hybrid model predictive control for traffic flow stabilization and pollution reduction of freeways

In this work a control system is developed and analyzed for the suppression of moving jamwaves and the reduction of pollutant concentrations near motorways. The system is based on the second-order macroscopic freeway traffic model METANET, joined by an emission dispersion model, introduced in a previous work of the authors. For the control tasks dedicated controllers are designed, both using the nonlinear model predictive control method. The control objectives require a distinction in the utilized control measures, thus different controllers are designed and used in predefined control modes. The first mode of the controller is responsible for keeping pollutant concentrations below prescribed limits under stable conditions. The second mode of the controller is working in case of a shockwave threat, aiming for traffic stabilization. Between the control modes switching is based on an appropriate rule set that satisfies the stability of the controlled system. The hybrid controller structure is realized by a finite automata. A complex case study is presented for the evaluation of the suggested controller.