9.1 Traffic Engineering and Capacity Analysis

Traffic Engineering and Capacity Analysis

Traffic Flow Parameters

Traffic flow theory forms the mathematical and physical foundation of traffic engineering. A clear understanding of traffic stream characteristics is essential for analyzing the performance of any roadway facility. The traffic stream is characterized by three primary macroscopic variables: volume (or flow rate), speed, and density.

Volume and Flow Rate

• Traffic volume ($V$) is defined as the actual number of vehicles that pass a given point or transverse a specific section of a lane or roadway during a specified time interval (typically one hour or one day). It is represented in units of vehicles per hour (veh/h) or vehicles per day (veh/day).
• Flow rate ($q$) is the equivalent hourly rate at which vehicles pass a point on a lane or roadway during a time period of less than one hour (most commonly 15 minutes). The conversion between hourly volume and the peak flow rate is dictated by the Peak Hour Factor (PHF), which accounts for the short-term fluctuations of traffic within the peak hour.

Speed

Speed is defined as the rate of motion of traffic, typically expressed in miles per hour (mph) or kilometers per hour (km/h). In traffic capacity analysis, speed must be defined carefully because vehicles travel at different speeds. There are two primary measures of average speed: time mean speed and space mean speed.

• Time mean speed ($u_t$) is the arithmetic average of the speeds of all vehicles passing a specific point on a roadway during a specified time interval. It is a point-measure of speed, typically captured by spot sensors such as radar guns, loop detectors, or tube counters. $u_t = \frac{1}{N} \sum_{i=1}^N u_i$
• Space mean speed ($u_s$) is the harmonic average of the speeds of all vehicles passing a specific point on a roadway during a specified time interval. It represents the average speed of all vehicles occupying a given section of a roadway. Space mean speed is the only speed measure that satisfies the fundamental traffic flow relationship. $u_s = \frac{N}{\sum_{i=1}^N \frac{1}{u_i}}$
• Mathematical Relationship: Time mean speed is mathematically guaranteed to be greater than or equal to space mean speed. The difference between the two measures increases as the variance of individual vehicle speeds increases. They are related by the variance of the space distribution of speeds ($\sigma_s^2$): $u_t = u_s + \frac{\sigma_s^2}{u_s}$ On the PE Civil exam, a common trick is calculating these speeds from a small sample of vehicle speeds. Always remember that space mean speed is the harmonic mean and will be lower than the arithmetic mean (time mean speed).

Density

• Density ($k$ or $D$) represents the concentration of vehicles, defined as the number of vehicles occupying a given length of a roadway segment (typically expressed in vehicles per mile, veh/mi, or passenger cars per mile per lane, pc/mi/ln). Density is the primary measure of traffic density and directly reflects driver comfort, freedom to maneuver, and proximity to other vehicles. It is difficult to measure directly in the field (requiring aerial photography or double-loop detectors) but is easily computed from flow and speed.

Headway

Microscopic parameters describe the relationship between individual, successive vehicles in a traffic stream:

• Time headway ($h_t$) is the elapsed time between the arrival of consecutive vehicles at a point, measured from a common reference point (e.g., front bumper to front bumper). The average time headway ($\bar{h}_t$) in seconds is related to the hourly flow rate ($q$) in veh/h by: $\bar{h}_t = \frac{3600}{q}$
• Space headway ($h_s$) is the physical distance between consecutive vehicles, measured from front bumper to front bumper. The average space headway ($\bar{h}_s$) in feet is related to density ($k$) in veh/mi by: $\bar{h}_s = \frac{5280}{k}$

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The Fundamental Relationship of Traffic Flow

The three macroscopic parameters are linked by the fundamental identity of traffic flow: $q = u_s \cdot k$ Where:

• $q$ = flow rate (veh/h or pc/h/ln)
• $u_s$ = space mean speed (mph)
• $k$ = density (veh/mi or pc/mi/ln)

This equation is analogous to the continuity equation in fluid mechanics ($Q = A \cdot V$). It shows that flow rate is a product of speed and density. When density is zero, flow rate is zero because there are no vehicles on the road. When density reaches its maximum limit (the road is completely full), flow rate is also zero because all vehicles are stopped. Between these two extremes, flow rate rises to a peak capacity and then falls back to zero.

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Greenshields Linear Speed-Density Model

In 1935, Bruce Greenshields proposed the simplest model of traffic flow, assuming a linear relationship between speed and density: $u(k) = u_f \left( 1 - \frac{k}{k_j} \right)$ Where:

• $u_f$ = free-flow speed (the speed of the traffic stream when density is zero; representing the speed a driver chooses when unimpeded by other traffic).
• $k_j$ = jam density (the density when speed is zero; representing bumper-to-bumper traffic where vehicles cannot move).

Substituting this speed-density function into the fundamental relationship ($q = u \cdot k$) yields the parabolic flow-density relationship: $q = u_f \left( k - \frac{k^2}{k_j} \right)$

To find the maximum flow (also known as the capacity of the roadway, $q_{max}$), we take the derivative of flow with respect to density and set it to zero: $\frac{dq}{dk} = u_f \left( 1 - \frac{2k}{k_j} \right) = 0 \implies k_p = \frac{k_j}{2}$ Substituting $k_p$ back into the linear speed-density equation yields the speed at capacity ($u_p$): $u_p = u_f \left( 1 - \frac{k_j / 2}{k_j} \right) = \frac{u_f}{2}$ Therefore, the maximum flow rate is: $q_{max} = u_p \cdot k_p = \frac{u_f \cdot k_j}{4}$

Uncongested vs. Congested Flow

The parabolic relationship between flow and density (or speed and flow) divides traffic operations into two distinct regimes:

1. Uncongested flow: Occurs when density is below the optimal density ($k < k_p$) and speed is above the optimal speed ($u > u_p$). In this region, drivers are able to maintain higher speeds, and an increase in density leads to an increase in total flow rate.
2. Congested flow: Occurs when density is above the optimal density ($k > k_p$) and speed is below the optimal speed ($u < u_p$). In this region, queues begin to form, and any further increase in density restricts vehicle movements, causing the speed and total flow rate to drop.

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Level of Service (LOS) Concepts

The Level of Service (LOS) is a qualitative ranking system defined by the Highway Capacity Manual (HCM) to describe operational conditions within a traffic stream. It represents factors such as speed, travel time, freedom to maneuver, traffic interruptions, comfort, and convenience.

LOS is categorized into six grades, from LOS A (best) to LOS F (worst):

• LOS A: Free-flow operations. Vehicles are completely unimpeded in their ability to maneuver within the traffic stream.
• LOS B: Reasonably free flow. The presence of other vehicles becomes noticeable, but control is still high.
• LOS C: Stable flow. Maneuvering requires vigilance. Minor disruptions can cause local queues.
• LOS D: Approaching unstable flow. Comfort and convenience are low. Freedom to maneuver is severely restricted.
• LOS E: Unstable flow operating at or near capacity. Any minor disruption will cause the system to break down.
• LOS F: Forced or breakdown flow. Queues form behind breakdown points. Demand exceeds capacity.

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HCM Capacity Analysis for Basic Freeway Segments

A basic freeway segment is a portion of a freeway facility that is outside the physical influence of ramp merges, diverges, or weaving areas. The HCM analysis procedure determines the Level of Service by comparing the calculated traffic density against standard thresholds.

Step 1: Calculate Free-Flow Speed (FFS)

The free-flow speed represents the average speed of passenger cars measured under low-to-moderate flow conditions (less than 1,000 pc/h/ln). FFS is estimated using the following formula: $FFS = BFFS - f_{LW} - f_{LC} - f_{TRD} - f_{ID}$ Where:

• $BFFS$ = Base Free-Flow Speed (typically 75.4 mph for rural freeways or 70.0 mph for urban freeways).
• $f_{LW}$ = Adjustment for lane width (mph). Standard lane width is 12 ft. Reducing lane width to 11 ft or 10 ft reduces speed.
• $f_{LC}$ = Adjustment for right lateral clearance (mph). Standard right-side lateral clearance is 6 ft or greater. Encroachments reduce FFS.
• $f_{TRD}$ = Adjustment for total ramp density (ramps per mile, representing the number of on-ramps and off-ramps within 3 miles upstream and downstream of the segment).
• $f_{ID}$ = Interchange density adjustment (interchanges per mile).

Step 2: Adjust Demand Flow Rate ($v_p$)

To analyze capacity, raw traffic volumes must be converted into a standardized passenger-car equivalent flow rate under equivalent base conditions ($v_p$), measured in passenger cars per hour per lane (pc/h/ln): $v_p = \frac{V}{PHF \cdot N \cdot f_{HV} \cdot f_p}$ Where:

• $V$ = Hourly traffic volume (veh/h).
• $PHF$ = Peak Hour Factor ($PHF = \frac{V}{4 \cdot V_{15}}$).
• $N$ = Number of lanes in one direction.
• $f_{HV}$ = heavy vehicle adjustment factor (dimensionless).
• $f_p$ = driver population factor (reflects driver familiarity with the roadway; typically 1.0 for weekday commuters, but ranges from 0.85 to 0.99 for recreational areas or weekend traffic).

Step 3: Compute Heavy-Vehicle Adjustment Factor ($f_{HV}$)

Heavy vehicles (trucks, buses, RVs) occupy more space and have lower acceleration and climbing capabilities than passenger cars. The adjustment factor converts these vehicles into equivalent passenger cars: $f_{HV} = \frac{1}{1 + P_T(E_T - 1) + P_R(E_R - 1)}$ Where:

• $P_T$ = proportion of trucks and buses in the traffic stream (expressed as a decimal).
• $P_R$ = proportion of recreational vehicles (RVs) in the traffic stream (expressed as a decimal).
• $E_T$ = passenger car equivalent for trucks and buses.
• $E_R$ = passenger car equivalent for RVs.

For basic freeway analysis, equivalents depend on the terrain classification:

• Level terrain: General highway segments with short, gentle grades (less than 2%). Equivalents are: $E_T = 2.0$, $E_R = 1.5$.
• Rolling terrain: Segments with short-to-medium grades that cause trucks to reduce speed significantly. Equivalents are: $E_T = 3.0$, $E_R = 2.0$.
• Specific grades: For steep, long grades (greater than 3% for more than 0.5 miles), specific equivalents from HCM tables must be used, which depend on the exact grade percentage, length of grade, and percentage of heavy vehicles.

Step 4: Determine Density ($D$) and Level of Service (LOS)

The average passenger-car speed ($S$) is calculated using the FFS and demand flow rate ($v_p$). If the demand flow rate is below the breakpoint (which ranges from 1,000 to 1,400 pc/h/ln depending on the FFS), speed remains constant at the FFS ($S = FFS$). Above this flow rate, speed decreases. Density ($D$) is computed as: $D = \frac{v_p}{S}$ The calculated density (in pc/mi/ln) is then mapped to the LOS criteria:

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HCM Capacity Analysis for Multilane Highways

Multilane highways differ from freeways in that they do not have full access control (they may have at-grade intersections, driveways, and median cross-overs) and typically have lower design speeds.

Free-Flow Speed (FFS) for Multilane Highways

The FFS for a multilane highway is calculated as: $FFS = BFFS - f_{LW} - f_{LC} - f_M - f_A$ Where:

• $BFFS$ = Base Free-Flow Speed (typically 60 mph if unknown).
• $f_M$ = Adjustment for median type (mph). Divided highways have a median adjustment of 0.0, while undivided highways have a median adjustment of 1.6 mph (due to interference from left turns and oncoming traffic).
• $f_A$ = Adjustment for access-point density (mph), based on the number of driveways and unsignalized intersections per mile on the right side of the highway.

Level of Service for Multilane Highways

The density thresholds for LOS A through D are identical to those of freeways. However, because multilane highways operate at lower speeds and have lower capacity, the boundary for LOS E (capacity) is smaller:

• LOS E density range: $35.0 < D \le 40.0$ pc/mi/ln.
• LOS F occurs when density exceeds 40.0 pc/mi/ln or when demand exceeds capacity (capacity is 2,200 pc/h/ln for a multilane highway with a 60 mph FFS).