Flow-density Relation Research Articles

Macroscopic Traffic Modeling Using Probe Vehicle Data: A Machine Learning Approach

The macroscopic fundamental diagram (MFD) captures an orderly relationship among traffic flow, density, and speed at the network level. It is a simple yet powerful tool for modeling traffic dynamics in large urban networks with broad application in traffic control and management. However, empirically derived MFDs in urban regions require high-resolution traffic data from the network. Having the network flow and vehicular density estimated at the (granular) census tract level using vehicle probe data, we apply machine learning methods to predict the MFDs across U.S. urban areas and capture the impacts of location-specific input features on the network flow–density relationships at a large scale. The results show that, among the four tested machine learning approaches (Random Forest, XGBoost, Support Vector Machine, and Neural Network), XGBoost delivers the best performance in predicting network traffic flow based on vehicular density and location attributes. Using interaction Shapley Additive explanation (SHAP) values and partial correlation analysis, we examine the factors influencing MFD shapes across different locations. Our empirical findings reveal that across U.S. urban areas, network topology, transportation infrastructure, and land use are primary factors shaping MFD curves, while demand and trip-related factors play a lesser role. Specifically, higher ranking roads, centrality, and development levels correlate positively with network capacity and critical density, whereas negative associations are observed for network connectivity, mixed-use development, and road roughness levels.

How do age compositions affect pedestrian dynamics on stairways? Findings from controlled experiments

Understanding pedestrian dynamics for different age compositions is paramount for enhancing their safety level on stairways in normal and evacuation conditions. In this study, we illustrate the findings of a series of single-file pedestrian flow experiments on stairways. This experiment was carried out with 131 participants having different age compositions, including preschool students (4–5 years), primary school students (8–10 years), middle school students (12–14 years), middle-aged pedestrians (27–57 years), and the elderly pedestrians (63–81 years). We collect data using three experimental settings: individual experiments with age homogeneity, single-file pedestrian experiments with age homogeneity, and single-file pedestrian experiments with age-mixed. Our findings show that age plays a substantive non-linear impact on individual unconstrained ascending and descending speeds. In addition, we developed a linear regression model that predicts the pedestrians’ free speed of different ages on stairways. Then, the effects of age compositions on pedestrian dynamics with fundamental diagrams (i.e., speed-density and specific flow-density relations) are analysed, and the maximum specific flow of age homogeneity is higher than that of age-mixed scenarios. Finally, we illustrate how the headway distance impacts participants’ ascending and descending speeds in both age-homogeneity and age-mixed conditions. As such, these new experimental results can provide the foundation for future calibration and validation of pedestrian evacuation models as well as insight into safety guidelines.

Impact of detour on traffic flow in branching Koch curve network with bottleneck

There are various routes with and without detours in city traffic network with a complex connectivity. Branching Koch curve fractal has such complex connectivity with singly, doubly, and multiply connecting links. We consider branching Koch curve as a city traffic network. We study the effect of detour routes (bypasses) on macroscopic traffic flow in branching Koch curve network with a bottleneck. The bottleneck effect depends on whether or not there are detour routes. We propose a mathematical dynamic model for the traffic flow with a bottleneck in the directed Koch curve network. The macroscopic dynamic equations of vehicular densities are derived in the Koch curve network. Vehicular densities and currents are obtained at all roads in the network by solving the density equations numerically. The fundamental diagrams (flow-density relations) are derived. The traffic currents (flows) depend highly on the position of the bottleneck due to the connectivity of the fractal network. It is shown where and how traffic congestion is induced by the bottleneck on the directed fractal network.

Development and experimental validation of a humanoid pedestrian model that captures stepping behavior and body rotation

This paper presents a novel humanoid pedestrian model (HPM) incorporating stepping behavior and body rotation. The HPM is composed of two main components: (a) body modeling and (b) gait planning. A pedestrian is represented as a three-dimensional skeleton with 11 degrees of freedom in the body modeling component, which provides a universal approach for explaining the mathematical correlations between joint rotation angles and critical gait parameters such as step length, step width, and projected shoulder width. A framework for designing gaits that accounts for step-synchronization behavior and the effect of body rotation on stepping behavior is offered by the gait planning process. To validate the model, two types of experiments were conducted: nine sets of single-file experiments and 10 sets of bidirectional flow experiments, in which pedestrians walked in a 0.5-m-wide circular corridor and rotated their bodies to avoid collisions. It was suggested by the fundamental diagrams that body rotation reduces the walking speed of a pedestrian and consequently affects the overall flow rate. Furthermore, it was found that pedestrians were more resistant to moving forward in narrow bidirectional flow environments and tend to wait for a larger gap in front to take a longer or faster step. This behavior led to the formation of stop-and-go waves in the narrow-corridor scenario. The simulation results were consistent with the experimental findings in terms of flow-density relationships and the reproduction of stop-and-go waves. Additionally, synchronized steps were detected in the simulation and quantitatively compared with a publicly available dataset. The HPM offers a new perspective on modeling pedestrian dynamics and emphasizes the necessity of accounting for micro-characteristics at the step level in pedestrian models.

On cellular automata model of traffic flow with Lennard-Jones potential

ABSTRACT By introducing the Lennard-Jones potential into the interaction of vehicles, a new cellular automata model of traffic flow is proposed, in which the fixed probability of randomisation of cellular automata is replaced by the state transition probability of cell sites. The proposed model considers that the driver is affected by the interactional potential of vehicles before him/her, and the decision-making process is related to the value of the interactional potential. The model is more suitable for the driver’s random decision-making process which is based on the stochastic behaviour of traffic flow. The fundamental diagram (flow density relationship) of the proposed model is obtained, and the complex traffic phenomenon is reproduced through numerical simulation.

Positivity-Preserving Discontinuous Galerkin Methods on Triangular Meshes for Macroscopic Pedestrian Flow Models

The macroscopic models for solving the pedestrian flow problem can be generally classified into two categories as follows: first-order continuum models and high-order continuum models. In first-order continuum models, the density satisfies the mass conservation law, the speed is defined by a flow-density relationship, and the desired directional motion of pedestrians is determined by an Eikonal-type equation. In contrast, in high-order models, the velocity is governed by the momentum conservation law. In this study, we summarize existing first-order and high-order models and rewrite them in the form of unified scalar or system hyperbolic conservation laws. Next, we apply high-order discontinuous Galerkin methods with a positivity-preserving limiter on unstructured triangular meshes to solve the conservation law and a second-order fast-sweeping scheme to solve the Eikonal equations. Our method can efficiently model real-life complex computational regions and avoid nonphysical solutions and simulation blow-ups. Finally, numerical examples are presented to demonstrate the accuracy and effectiveness of the proposed solution algorithm. The numerical results validate the reliability of the proposed numerical method and highlight the advantages of triangular meshes.

The analysis of traffic data of wildfire evacuation: the case study of the 2020 Glass Fire

Evacuation is a crucial policy to mitigate wildfire impacts. Understanding traffic dynamics during a wildfire evacuation can help authorities to improve in improving emergency management plans, thus improving life safety. In this study, we developed a methodology to extract historical traffic data from vehicle detector stations and automate the analysis of traffic dynamics for actual wildfire evacuations. This has been implemented in an open-access tool called Traffic Dynamic Analyser (TDA) which generates speed-density and flow-density relationships from data using both commonly used macroscopic traffic models as well as machine learning techniques (e.g., support vector regression). The use of the methodology is demonstrated with a case study of the 2020 Glass Fire in California, USA. The results from TDA showed a slight reduction in speeds and flows on US Highway 101 during the evacuation scenario, compared with the routine scenario. Moreover, background traffic has been shown to play a key role in the 2020 Glass Fire compared with previous wildfire evacuation scenarios (e.g., the 2019 Kincade fire). The case study showed that the methodology implemented in the TDA can be used to understand traffic evacuation dynamics in wildfire scenarios and to validate evacuation models.

Highway Managed Lane Usage and Tolling for Mixed Traffic Flows with Connected Automated Vehicles and High-Occupancy Vehicles

This paper investigates managed lane toll setting and its effect under mixed traffic of connected automated vehicles (CAVs), high-occupancy vehicles (HOVs), and human-driven vehicles (HDVs), with the goal of avoiding flow breakdown and minimizing total social cost. A mesoscopic finite difference traffic simulation model considers the flow–density relationship at different CAV market penetration rates, lane-changing behaviors, and multiple entries/exits, interacting with a reactive toll setting mechanism. The results of Monte Carlo simulation suggest an optimal policy of untolled HOV/CAV use with tolled HDVs in particular scenarios of limited CAV market penetration. Small and targeted tolling avoids flow breakdown in managed lanes while prioritizing HOVs and other vehicles with high values of time. Extensions of the formulation and sensitivity analysis quantify the benefits of converting high-occupancy HDVs to CAVs. The optimal tolling regime combines traffic science notions of flow stability and the economics of resource allocation.

Two-Level Full Factorial Design Approach for the Analysis of Multi-Lane Highway Section under Saturated and Unsaturated Traffic Flow Conditions

Oversaturation of highways occurs due to their inadequate assessment and design. In this paper, we propose both a mathematical queuing model and a Discrete-Event Simulation (DES) framework based on Newell’s triangular flow-density relationship for the performance analysis of a multi-lane highway section. The proposed framework is a finite capacity queuing system, which captures an increase in the flow with the vehicle density up to the capacity of the section in an unsaturated condition and a decrease in the flow in the case of a saturated condition, depicting the actual traffic conditions on the highway section. First, the Birth–Death Process is used to build the mathematical queuing model (BDP), and the average number of vehicles (average queue length) and blocking probability on the highway section are estimated. Then, the accuracy of the mathematical queuing model is verified by the proposed DES framework. The “significance and effects” of different design factors are evaluated using the two-level full factorial design technique. The analysis of the experimental results reveals that the length of the highway section and the number of lanes are the most significant factors affecting the average queue length and blocking probability, while the jam density only has a significant effect on the average queue length and does not affect the blocking probability. In case of a two-way interaction, the combined effect of the “length-lanes” significantly affects the average queue length. In the end, a multiple-factor linear regression model is also developed for the prediction of the average number of vehicles on the highway section based on the design factors.

The Impact of Automated Vehicles on Road and Intersection Capacity

With the rapid development of autonomous driving technology, future road traffic must be composed of autonomous vehicles and artificial vehicles. Although autonomous vehicles have greatly improved road capacity, few studies have involved capacity at signal-controlled intersections, and most of the studies are based on experimental simulation. As such, there is a need to more scientifically analyze the impact of autonomous vehicles on road and intersection capacity. Based on three theories of flow-density relationships, traffic flow equilibrium analysis, and the following model, this paper firstly deduces the flow-density relationship of different vehicle types in a single environment. Secondly, flow-density relationships under different proportions of self-driving vehicles are derived. Through the derivation of these two models, the basic road saturation flow rates under different permeabilities of self-driving vehicles, can be obtained. Based on these results, a revised calculation model for the capacity of signalized intersections with different proportions of autonomous vehicles is proposed, which is essentially to revise the basic saturation flow rate under different permeabilities of autonomous vehicles. By using SUMO 1.15.0 traffic simulation software, the theoretical models are individually tested. The results show that the error rate between the theoretical calculation results and the SUMO simulation results, is less than 16%. This study can provide a basis for the calculation of basic capacity of roads and intersections in a future man-machine hybrid driving environment, and provide theoretical guidance for traffic management and control.

Operational Assessment of Speed Control Measures on Capacity of Urban Roads in Metropolitan Area of Barranquilla, Colombia

With the aim to reduce accident rates and assist in pedestrian traffic, in the city of Barranquilla (Colombia), electronic devices for speed control (speed cameras) have been installed on roads of high vehicular flow with difficulty in pedestrian transit. Traffic Calming Measures are described in the literature as a set of measures to regulate traffic on roads, ensure mobility, and reduce accidents. This article focuses its study on the impact of Traffic Calming Measures through electronic devices on urban or inner cities roads (capacity and operating speed on them). This study centers in the evaluation of the operational impact of speed cameras and the data of vehicular speed and flow collected through them in four main roads of the city, with the purpose of estimating models that establish relationships of speed-flow, and analyzing the impact of these Traffic Calming Measures on the vehicular capacity of urban roads. In the estimation of road capacity, a Greenshields model was applied, with which via speed–density and flow-density relationships, we aim to determine how operational transit is affected by the use of speed cameras. Conclusions show that operational transit is affected by diminishing road capacity between 7 to 41 percent and these Traffic Calming measures are ineffective to ease pedestrian traffic as is expected.

Modified State-Dependent Queuing Model for the Capacity Analysis of Metro Rail Transit Station Corridor during COVID-19

The COVID-19 pandemic policies have had a significant impact on the daily commuter flow at the metro rail transit stations. In this study, we propose a modified state-dependent M(n)/G(n)/C/C queuing model for the analysis of commuter flow in the corridor of metro rail transit stations in the COVID-19 situation in order to ensure safe social distance. The proposed model is a finite capacity queuing system with state-dependent commuter arrivals and state-dependent service rates based on the flow–density relationship. First, a mathematical queuing model is developed by using the birth–death process (BDP) and the expected number of commuters, and average area occupied per commuter and blocking probabilities are computed. Then, the accuracy of the proposed model is verified by a discrete-event simulation (DES) framework. (1) The proposed model’s results are compared to those of the existing M/G(n)/C/C model. The proposed modified model’s sensitivity analysis revealed that the anticipated number of commuters in the corridor remains smaller when the arrival rate is state-dependent. (2) In accordance with COVID-19 protocol, when the facility is congested, commuters are discouraged from entering and a safe social distance is maintained between them. (3) No commuters are impeded, and adequate throughput is ensured from the corridor. The proposed model will assist the metro rail transit station operators in making intelligent decisions regarding the operations in the COVID-19 situation.

Assessing the Network-Wide Impacts of Dedicated Lanes for Connected Autonomous Vehicles

Dedicated lanes for autonomous vehicles (AV) are introduced as an effective strategy to improve mobility in mixed traffic of AVs and heterogeneous drivers. However, adding a new lane is costly, and dedicating an existing lane may increase traffic congestion in other lanes. Previous studies investigated the impacts of AV dedicated lanes on throughput at the segment level and/or assumed a fixed route choice. However, AV dedicated lanes change route choice behavior, which affects traffic distribution over the network. Therefore, this study explores the impacts of AV dedicated lanes on the traffic performance of large-scale networks by conducting a network-level cost–benefit analysis on the implementation of AV dedicated lanes under different demand and AV market penetration rate (MPR) scenarios. This study examines the impacts of implementing AV dedicated lanes to freeway links on traffic congestion at corridor and network levels. Various factors are explored: changes in aggregate flow–density relationships, throughput, and average travel times. To this end, DYNASMART-P software is updated to consider AV dedicated lanes and is used to simulate the mixed traffic. Traffic simulation analyses on the large-scale network of Chicago indicate that these impacts depend on the AV MPR, demand level, and AV dedicated lane implementation approach. Dedicated lanes for AVs are beneficial for a high demand level scenario at all AV MPRs. However, for the base demand scenario, (760,000 vehicles during a.m. peak), deploying such lanes is only justified for low AV MPRs. Furthermore, the impacts of AV dedicated lanes on traffic at the network level are different from those on single segments.

Impact of lane function layout on the performance of urban traffic system

In urban traffic systems, alleviating traffic congestion originated from left-turn/U-turn lanes at highly demanded intersections plays an important role in enhancing the traffic efficiency. For a typical intersection, the layout with double left-turn (DLT) lanes is a relatively common geometric feature. However, the effect of DLT scheme on the performance of the entire traffic system is not clear. This paper compares the performance of urban traffic system with different lane function schemes with extended cellular automaton (CA) model considering the car-following, lane-changing, intersection-traversing and route-choice behavior of vehicles. Using the flow-density relation, three macroscopic parameters (average trip completion rate, average traffic flow and average velocity) are examined to evaluate the system performance. With the wide application of DLT scheme, the vehicles can be distributed more evenly among the lanes. As a result, the traffic flow and the trip completion rate will be higher, especially at high density. Case studies of realistic Hefei city network and Sioux Falls networks are also carried out. The results show that the performance of traffic network will be better if more intersections adopt the DLT scheme.

Improve urban traffic efficiency with prohibiting left-turn policy

Enhancing traffic flow plays an important role in the traffic management of urban arterial networks. The policy of prohibiting left-turn (PLT) at selected highly demanded intersections has been adopted as an attempt to increase the efficiency at these intersections. In this paper, we study the impact of PLT by mathematical analysis and simulations based on the cellular automaton model. Using the flow-density relation, three system performance indexes are examined: the average trip completion rate, the average traffic flow, and the average velocity of vehicles. Different route guidance strategies, including the shortest path and the quickest path, are investigated. We show that when left turn is prohibited, vehicles are distributed more homogeneously in the road network, and the system performs better and reaches a higher capacity. We also derive a critical length of link, above which the benefit of PLT will decrease.

Congestion in cities: evidence on the form and causality of the macroscopic fundamental relationship of traffic flow

This paper quantifies the technology driving congestion in urban road networks. To do so, we estimate macroscopic fundamental relationships for homogeneously congested sub-networks (reservoirs) in thirty-four cities worldwide. We adopt a causal approach based on non-parametric instrumental variables to estimate the form of the reservoir-level flow-density relationship using large-scale traffic sensor data. Specifically, we apply a Bayesian non-parametric spline-based regression model with instrumental variables to adjust for potential confounding/endogeneity biases due to simultaneity and omitted variables such as vehicle interactions and traffic controls. Our estimates suggest that the provision of vehicular travel in cities is subject to decreasing returns to density and network size. Importantly, we find that increasing road network capacity is not an efficient solution to manage congestion because average travel speed does not change substantially with increase in capacity. As a by-product of the estimation, we also deliver estimates of important traffic control inputs such as capacity and critical occupancy for these reservoirs. Our results have implications for traffic flow modelling used by both economists and traffic engineers.

Research on the impact of ship traffic flow on the restricted channel segment of the middle Yangtze River based on traffic wave theory

On the basis of the influence of dry season on ship traffic flow, the gathering and dissipating process of ship traffic flow was researched with Greenshields linear flow—density relationship model, the intrinsic relationship between the ship traffic congestion state and traffic wave in the unclosed restricted channel segment was emphatically explored when the ship traffic flow in a tributary channel inflows, and the influence law of multiple traffic waves on the ship traffic flow characteristics in unclosed restricted segment is revealed. On this basis, the expressions of traffic wave speed and direction, dissipation time of queued ships and the number of ships affected were provided, and combined with Monte Carlo method, the ship traffic flow simulation model in the restricted channel segment was built. The simulation results show that in closed restricted channel segment the dissipation time of ships queued is mainly related to the ship traffic flow rate of segments A and C, and the total number of ships affected to the ship traffic flow rate of segment A. And in unclosed restricted channel segment, the dissipation time and the total number of ships affected are also determined by the meeting time of the traffic waves in addition to the ship traffic flow rate of segments. The research results can provide the theoretical support for further studying the ship traffic flow in unclosed restricted channel segment with multiple tributariesArticle HighlightsThe inflow of tributaries' ship traffic flows has an obvious impact on the traffic conditions in the unenclosed restrictedchannel segment.The interaction and influence between multiple ship traffic waves and the mechanism of generating new traffic waves are explained.The expression of both dissipation time of queued ships and the total number of ships affected in the closed and unclosed restricted channel segment are given.

A Continuum model for pedestrian flow with explicit consideration of crowd force and panic effects

This paper proposes a second-order pedestrian model that comprises two types of equations: continuity equation and a set of transport equations. To complete the model, we develop Eikonal equations to explicitly consider the effects of the collective decisions of individuals and crowd pressure on pedestrian dynamics. Then, the crowd movement is simulated using a set of partial differential equations under appropriate initial and boundary conditions. Based on the stability requirements derived by performing a standard linear stability analysis, suitable parameters are selected to test the model in a numerical example. The proposed second-order system is then solved using the characteristic-wise third-order weighted essentially non-oscillatory (WENO3) scheme, and the Eikonal equations are solved using the fast sweeping method. The numerical results indicate the effectiveness of the model because the derived local flow-density relationship produces a second peak in the high-density region, which is consistent with previous empirical studies. Besides, the applicability of the model to an unstable condition is verified through the simulation of complex phenomena such as stop-and-go waves. Furthermore, the estimate of crowd pressure in the simulation results can be used as a risk-level indicator for crowd management and control.

Experimental study on properties of lightly congested flow

Congested traffic flow exhibits rather complex properties of spatiotemporal evolution, e.g. stop-and-go waves, phantom jams, wide scattering of flow-density data, etc. To the best of our knowledge, previous car-following experiments on circular tracks only studied heavily congested flow. The experimental study concerning lightly congested flow, which exhibits different stability characteristic from heavily congested flow, is still lacking. To fill this gap, very recently we conducted a large-scale car-following experiment on an 812-meter circumference circuit with densities of 0.049, 0.043 and 0.037 m−1. The flow-density relation of the experiment is highly consistent with the empirical one, which indicates that the driving behavior in the experiment is similar to that in real traffic. By comparing the macroscopic spatiotemporal evolution diagrams in different runs under the same global density, we found that the traffic flow exhibits stochastic features with even the same initial conditions. We analyzed the car-following behavior and its relation to the macroscopic traffic flow evolution. It was found that the indifference region, the sensitivity of drivers, and the strength of stochasticity all contribute to the macroscopic traffic flow evolution pattern. Specifically, the traffic flow becomes more homogeneous in time-space with the expansion of indifference region, the overall decrease of sensitivity and the decrease of stochasticity strength. The roles of stochasticity and speed adaptation were further investigated. It was found that stronger traffic oscillations correspond to stronger stochasticity and speed adaptation. The analysis for competition between the two effects indicated that the ratio between speed adaptation effect and stochastic effect grows with the increase of the oscillation amplitude. The stochastic effect plays a dominant role in car following when the oscillation amplitude is small; its influence wanes and the speed adaptation effect grows to outweigh the influence of the stochastic effect as oscillation amplitude grows. These findings are expected to improve our understanding of the underlying mechanism that produces the spatiotemporal evolution patterns of lightly congested flow and the data collected in this study can be used to test traffic models and theories.

Multi-class traffic flow model based on three dimensional flow–concentration surface

This paper proposes a continuum model based on a three-dimensional flow-concentration surface for multi-class traffic. The model assumes that the flow of any vehicle class is a function of the class density as well as the fraction of road area occupied by other vehicle classes. By considering occupancy of road area instead of lane occupancy, the model effectively describes traffic flow that does not follow lane discipline. The propagation speed of small disturbance (PSSD), conventionally defined from the two-dimensional flow–density relationship, is reformulated for each class using a three-dimensional flow–concentration surface. Using the proposed PSSD and a speed–area occupancy (speed-AO) relationship, a second-order continuum model for multi-class traffic is formulated. The speed–AO relationship captures class-specific congestion and replicates the gap-filling behaviour commonly observed in lane-indisciplined traffic. Properties of the proposed model are validated theoretically where possible, and through numerical simulation when theoretical derivations are cumbersome. Numerical simulation of the proposed multi-class traffic model replicates field-observed phenomena such as shockwaves and rarefaction waves, local cluster effect, and gap-filling behaviour. Finally, the model is calibrated using field traffic data collected on a road section with bottleneck, and is found to replicate class-wise vehicle flows and speeds, and stop-and-go phenomena.