Unit
- Traffic flow variables: The three primary macroscopic variables are flow (vehicles per hour), density (vehicles per mile), and speed (miles per hour). These are related by the fundamental equation flow = density × speed. Understanding these relationships is essential for analyzing traffic operations and designing control strategies.
- Measurement techniques: Flow is typically measured using induction loops or video detection at fixed points over a time interval. Density can be estimated from occupancy or measured via aerial photography. Speed is obtained from radar guns or inductance-based vehicle detectors. Each method has limitations regarding accuracy and coverage.
- Space-mean vs. time-mean speed: Time-mean speed is the arithmetic average of spot speeds at a location, while space-mean speed is the harmonic average based on travel time over a segment. Space-mean speed is more appropriate for flow-density relationships because it reflects the actual travel time across a section.
- Fundamental diagram: The relationship between flow and density is typically concave, with a maximum flow point called capacity. The speed-flow curve is decreasing, and the speed-density curve is linear in the uncongested regime. The critical density corresponds to the density at capacity.
- Uncongested vs. congested regimes: In uncongested flow, speed is relatively constant and flow increases with density. In congested flow, both speed and flow decrease as density increases. The transition occurs at capacity and beyond, where stop-and-go waves form.
- Greenshields model: A linear speed-density model proposed by Greenshields: speed = free-flow speed × (1 - density/jam density). This yields a parabolic flow-density relationship. While simplistic, it provides a good approximation for many traffic conditions.
- Other traffic flow models: Greenberg proposed a logarithmic model, Underwood an exponential model, and Pipes a linear car-following model. These models account for different driving behaviors and are used in microsimulation software like VISSIM and Aimsun.
- Practical applications: Traffic engineers use these relationships to determine level of service (LOS) from flow and speed data. LOS A to F classification depends on density and speed thresholds. The data helps in designing lane additions, signal timing, and ramp metering.
Code Example
Python
# Calculate flow rate from count interval
count = 150 # vehicles observed in 5 minutes
time_minutes = 5
flow_rate_veh_per_hour = (count / time_minutes) * 60
print('Flow rate: {} veh/hr'.format(flow_rate_veh_per_hour))
# Using Greenshields model to compute speed
v_f = 60 # free-flow speed mph
k_j = 200 # jam density veh/mile
k = 50 # current density
speed = v_f * (1 - k / k_j)
print('Speed: {:.2f} mph'.format(speed))
# Compute flow from speed and density
flow = speed * k
print('Flow: {:.2f} veh/hr'.format(flow))This Python code demonstrates basic traffic flow calculations. The first part computes hourly flow rate from a short count. The second part applies Greenshields linear model to estimate speed given density, then computes flow. These are typical operations in traffic engineering analysis.
Quick Revision
- Traffic flow variables are flow, density, and speed, related by flow = density × speed.
- Measurement methods include induction loops, radar, and video detection.
- Space-mean speed is preferred for flow-density relationships.
- The fundamental diagram shows a concave flow-density curve with capacity at critical density.
- Greenshields model is a linear speed-density relationship used for analysis.
- Level of service classification uses density and speed thresholds.