Traffic Capacity and Flow Theory for the FE Civil Exam

NCEES FE Civil spec: Transportation Engineering, C. Traffic capacity and flow theory. FE Reference Handbook: Civil Engineering chapter, Transportation section (Traffic Flow Relationships, Basic Freeway Segment Highway Capacity).

What the FE Civil Spec Covers

Section C is traffic capacity and flow theory. Expect the fundamental relationship between flow, speed and density, the Greenshields linear speed-density model, peak hour factor, headway and spacing, volume-to-capacity ratios and a basic freeway segment level of service (LOS) calculation using the Highway Capacity Manual method summarized in the handbook.

Key Concepts

Flow $q$ (veh/h), density $k$ (veh/mi), space-mean speed $v$ (mi/h).
Headway $h$ is time between successive vehicles; spacing $s$ is distance between them.
Free-flow speed $v_{f}$ occurs at near-zero density; jam density $k_{j}$ is where traffic stops.
Peak hour factor (PHF) measures how uneven traffic is within the peak hour. PHF of 1.0 means perfectly uniform flow.
Level of service for a basic freeway segment is set by density in passenger cars per mile per lane.

Core Formulas

q = v k q = \frac{3600}{h} k = \frac{5280}{s}

Greenshields model:

v = v_{f} (1 - \frac{k}{k _{j}}) q = v_{f} k - \frac{v _{f}}{k _{j}} k^{2} q_{ma x} = \frac{v _{f} k _{j}}{4}

Peak hour factor and freeway analysis flow rate:

P H F = \frac{V}{4 V _{15}} v_{p} = \frac{V}{P H F \cdot N \cdot f _{H V}} f_{H V} = \frac{1}{1 + P _{T} ( E _{T} - 1 )}

D = \frac{v _{p}}{S}

Freeway LOS thresholds by density (pc/mi/ln): A up to 11, B up to 18, C up to 26, D up to 35, E up to 45, F above 45 or whenever demand exceeds capacity.

Time-mean versus space-mean speed

Time-mean speed is the arithmetic average of spot speeds measured at a point. Space-mean speed is the harmonic mean of those spot speeds, which is the same as total distance divided by total travel time. For two vehicles at 50 and 60 mph, time-mean speed is 55 mph while space-mean speed is $2/ (1/50 + 1/60) = 54.5$ mph. Only space-mean speed belongs in $q = v k$ .

Queues at a bottleneck

A toll plaza or a single-lane work zone behaves like a single server queue. For random arrivals and service (M/M/1) with arrival rate $λ$ and service rate $μ$ :

ρ = \frac{λ}{μ} L_{q} = \frac{ρ ^{2}}{1 - ρ} W = \frac{1}{μ - λ}

With 500 arrivals per hour and 600 served per hour, $ρ = 0.833$ , about 4.2 vehicles wait in line on average, and each vehicle spends $1/100$ h = 36 s in the system. Notice how fast delay grows as $ρ$ approaches 1.

Peak hour factor from counts

Fifteen-minute counts of 420, 510, 540 and 480 vehicles give an hourly volume of 1,950 vehicles. The busiest interval is 540, so $P H F = 1950/ (4 \times 540) = 0.90$ and the peak flow rate used for design is $4 \times 540 = 2160$ veh/h. Design for the flow rate, not the hourly volume, or the road will be over capacity during its busiest 15 minutes.

Where It Is in the FE Reference Handbook

The Transportation section of the Civil Engineering chapter has the peak hour factor, a Basic Freeway Segment Highway Capacity part (speed-flow parameters, lane width and lateral clearance adjustments, heavy vehicle factor, density and the LOS table), and a Traffic Flow Relationships part with the Greenshields model and its diagrams. Queueing models are in the Industrial and Systems Engineering chapter.

Worked Example: Freeway Level of Service

A three-lane (one direction) freeway segment on level terrain carries a peak hour volume of 4,500 veh/h with PHF = 0.90 and 8% trucks ( $E_{T} = 2.0$ ). The average passenger car speed is 70 mph. Find the flow rate per lane, density and LOS.

Step 1: heavy vehicle factor.

f_{H V} = \frac{1}{1 + 0.08 ( 2.0 - 1 )} = \frac{1}{1.08} = 0.926

Step 2: flow rate in passenger cars per hour per lane.

v_{p} = \frac{4500}{0.90 ( 3 ) ( 0.926 )} = 1800 pc/h/ln

Step 3: density.

D = \frac{1800}{70} = 25.7 pc/mi/ln

Step 4: level of service. 25.7 falls in the range above 18 and up to 26, so the segment operates at LOS C, close to the D boundary.

Greenshields check on a different road: with $v_{f} = 60$ mph and $k_{j} = 200$ veh/mi/ln, capacity is $60 (200) /4 = 3000$ veh/h/ln at $k = 100$ . At $k = 50$ , speed is $60 (1 - 50/200) = 45$ mph and flow is $45 (50) = 2250$ veh/h/ln.

Common Mistakes

Multiplying by PHF instead of dividing. Peak flow rate is always higher than the hourly volume.
Forgetting to divide by the number of lanes before computing density.
Using percent trucks as 8 instead of 0.08 in the heavy vehicle factor.
Reading LOS from flow instead of density for a basic freeway segment.
Assuming any flow below capacity is uncongested: in Greenshields every flow below the maximum occurs at two densities.

Practice Problems

Try these from the free PolyCalc FE Civil problem bank. Each one has a full worked solution.

A traffic stream has density 30 vehicles/km and space-mean speed 60 km/h. What is its flow rate? See the worked solution or practice the Transportation deck
The average time headway in a lane is 2 s per vehicle. What is the approximate lane flow rate? See the worked solution or practice the Transportation deck
Traffic demand is 900 vehicles/h and capacity is 1,200 vehicles/h. What is the volume-to-capacity ratio? See the worked solution or practice the Transportation deck
Fifteen-minute traffic counts during the peak hour on an arterial are 420, 510, 540 and 480 vehicles. The peak hour factor is most nearly: See the worked solution or practice the Transportation deck
Vehicles arrive at a toll booth at 500 per hour and are served at 600 per hour, both Poisson distributed. Modelled as an M/M/1 queue, the average time a vehicle spends in the system is most nearly: See the worked solution or practice the Transportation deck

Related Calculators and Examples

Frequently Asked Questions

What is the difference between volume and flow rate?

Volume is the actual count of vehicles in an hour. Flow rate is an hourly rate based on a shorter peak period, usually the busiest 15 minutes. Dividing volume by the peak hour factor converts volume to peak flow rate.

At what density does maximum flow occur in the Greenshields model?

At half the jam density. Speed there is half the free-flow speed, and the maximum flow equals free-flow speed times jam density divided by 4.

Which speed goes in q = vk?

The space-mean speed. Time-mean speed, the simple average of spot speeds, is always equal to or higher than space-mean speed.

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