🚗Transportation Systems Engineering Unit 3 – Traffic Flow Theory

Key Concepts and Definitions

• Traffic flow theory studies the interactions and behavior of vehicles, drivers, and infrastructure in transportation systems
• Fundamental variables of traffic flow include flow rate (vehicles per unit time), density (vehicles per unit distance), and speed (distance traveled per unit time)
• Capacity represents the maximum sustainable flow rate under prevailing conditions
  • Influenced by factors such as road geometry, weather, and traffic composition
• Level of service (LOS) qualitatively measures traffic conditions based on factors like speed, travel time, freedom to maneuver, and safety
  • Ranges from A (best) to F (worst)
• Queuing theory analyzes the formation, growth, and dissipation of queues or waiting lines in traffic systems
• Shockwaves describe the propagation of disturbances (sudden changes in traffic conditions) through a traffic stream
• Traffic stream models represent the relationships between flow, density, and speed
  • Include macroscopic (flow as a whole), microscopic (individual vehicle interactions), and mesoscopic (hybrid) models

Fundamentals of Traffic Flow

• Flow rate ($q$) measures the number of vehicles passing a point per unit time (vehicles per hour)
• Density ($k$) represents the number of vehicles occupying a unit length of roadway at a given instant (vehicles per kilometer)
• Speed ($u$) is the distance traveled by a vehicle per unit time (kilometers per hour)
  • Can be measured as time mean speed (average of individual vehicle speeds) or space mean speed (average speed based on travel time over a fixed distance)
• The fundamental relationship between flow, density, and speed is given by $q = ku$
• Jam density ($k_j$) is the maximum density achieved when vehicles are stopped bumper-to-bumper
• Free-flow speed ($u_f$) occurs when density approaches zero and vehicles travel at their desired speed
• Capacity ($q_m$) is reached at the critical density ($k_c$) and critical speed ($u_c$)
  • Represents the maximum sustainable flow rate under prevailing conditions

Traffic Stream Models

• Greenshields model assumes a linear relationship between speed and density
  • $u = u_f(1 - \frac{k}{k_j})$, where $u_f$ is free-flow speed and $k_j$ is jam density
• Greenberg model uses a logarithmic relationship between speed and density
  • $u = u_0\ln(\frac{k_j}{k})$, where $u_0$ is the optimum speed at maximum flow
• Underwood model proposes an exponential relationship between speed and density
  • $u = u_f e^{-\frac{k}{k_0}}$, where $k_0$ is the density at which speed is $\frac{1}{e}$ of the free-flow speed
• Pipe's car-following model describes the behavior of a following vehicle based on the speed and spacing of the lead vehicle
  • Assumes a minimum safe following distance that increases with speed
• General Motors (GM) car-following models relate the acceleration of a following vehicle to the spacing and relative velocity between vehicles
  • Includes the Gazis-Herman-Rothery (GHR) model and the Optimal Velocity (OV) model
• Cellular automata models discretize space and time, representing traffic flow as the interaction of individual cells following simple rules
  • Nagel-Schreckenberg model is a well-known example

Queuing Theory in Traffic

• Queuing theory analyzes the formation, growth, and dissipation of queues in traffic systems
• Queues form when the arrival rate of vehicles exceeds the service rate (capacity) of a facility
• Basic queuing model components include arrival process, service process, number of servers, queue discipline, and system capacity
• Arrival process describes the pattern of vehicle arrivals over time
  • Often modeled using Poisson distribution for random arrivals
• Service process represents the time required to serve each vehicle
  • Commonly modeled using exponential or deterministic distributions
• Little's Law relates the average number of vehicles in the system ($L$) to the average arrival rate ($\lambda$) and average time spent in the system ($W$)
  • $L = \lambda W$
• Queuing models help predict queue lengths, waiting times, and system performance measures
  • M/M/1 model assumes Poisson arrivals, exponential service times, and a single server
  • M/G/1 model allows for general service time distributions

Shockwave Analysis

• Shockwaves describe the propagation of disturbances (sudden changes in traffic conditions) through a traffic stream
• Occur when there is a change in the flow-density relationship, such as at bottlenecks or incidents
• Shockwave speed ($\omega$) is the speed at which the disturbance propagates, given by $\omega = \frac{q_2 - q_1}{k_2 - k_1}$, where $q_1$, $k_1$ and $q_2$, $k_2$ are the flow and density before and after the disturbance
• Types of shockwaves include compression (density increase), expansion (density decrease), and stationary (no change in density)
• Shockwave analysis helps predict the location and duration of congestion, queue lengths, and travel times
• Graphical methods, such as the time-space diagram, visualize the propagation of shockwaves
  • Trajectories of individual vehicles appear as lines, with the slope representing speed
• Queuing shockwaves form at bottlenecks when demand exceeds capacity
  • Consist of a backward-moving compression wave and a forward-moving recovery wave

Capacity and Level of Service

• Capacity is the maximum sustainable flow rate under prevailing conditions
  • Influenced by factors such as road geometry, weather, traffic composition, and driver behavior
• Highway Capacity Manual (HCM) provides methods for estimating capacity and level of service for various facility types
  • Freeway segments, weaving sections, ramp junctions, signalized and unsignalized intersections, etc.
• Level of service (LOS) qualitatively measures traffic conditions based on factors like speed, travel time, freedom to maneuver, and safety
  • Ranges from A (best) to F (worst)
• LOS criteria vary by facility type and are based on measures such as density, speed, delay, and volume-to-capacity ratio
• Capacity analysis helps identify bottlenecks, evaluate design alternatives, and assess the impact of traffic management strategies
• Volume-to-capacity (v/c) ratio compares the actual flow rate to the capacity of a facility
  • Values greater than 1.0 indicate oversaturated conditions and queuing
• Breakdown flow is the flow rate at which traffic transitions from stable to unstable flow, often lower than the theoretical capacity

Traffic Flow Measurement Techniques

• Traffic volume studies measure the number of vehicles passing a point over a given time period
  • Manual counts, automatic traffic recorders (ATR), and video-based systems
• Spot speed studies measure the instantaneous speeds of individual vehicles at a specific location
  • Radar guns, laser devices, and loop detectors
• Travel time and delay studies assess the time taken to traverse a route and the delays experienced
  • Floating car technique, license plate matching, and Bluetooth/Wi-Fi detection
• Origin-destination (O-D) studies determine the start and end points of trips within a study area
  • Roadside interviews, license plate surveys, and GPS-based methods
• Vehicle occupancy studies count the number of people in each vehicle
  • Observational surveys and automated occupancy detection systems
• Classification studies categorize vehicles by type, such as passenger cars, trucks, and buses
  • Manual observation, axle-based classification, and length-based classification
• Weigh-in-motion (WIM) systems measure the weight and axle configuration of vehicles while in motion
  • Piezoelectric sensors, bending plates, and load cells

Applications and Case Studies

• Traffic impact studies evaluate the effects of proposed developments on the surrounding transportation network
  • Estimate trip generation, distribution, and assignment to assess capacity and LOS impacts
• Congestion pricing schemes use variable tolls to manage demand and maintain optimal traffic flow
  • Singapore's Electronic Road Pricing (ERP) system and London's Congestion Charge
• Ramp metering regulates the flow of vehicles entering a freeway to maintain optimal mainline conditions
  • Minnesota's I-35W and Seattle's I-5 ramp metering systems
• Intelligent transportation systems (ITS) apply advanced technologies to improve safety, efficiency, and sustainability
  • Variable message signs, adaptive traffic signals, and incident management systems
• Work zone traffic management plans minimize disruptions and ensure safety during construction and maintenance activities
  • Temporary traffic control devices, reduced speed limits, and real-time traveler information
• Evacuation planning and modeling help optimize routes and strategies for emergency situations
  • Hurricane evacuation studies and wildfire evacuation simulations
• Autonomous and connected vehicles (AV/CV) have the potential to revolutionize traffic flow and safety
  • Platooning, cooperative adaptive cruise control (CACC), and vehicle-to-infrastructure (V2I) communication