Traffic Flow Calculator — Greenshields Model, PHF, Design Hour Volume & Speeds
The traffic-stream fundamentals in one place. The Greenshields model gives capacity, critical density and speed, the speed and flow at any density, and both densities that carry a given flow. Below it are the peak hour factor and 15-minute flow rate, design hour volume from AADT with K and D factors, and time-mean versus space-mean speed from a spot speed study.
Greenshields speed–flow–density model
Flow–density parabola. The slope of the dashed ray from the origin to the operating point is the space-mean speed.
Peak hour factor
Design hour volume
Spot speed study
Flow, density and speed
Three measures describe a traffic stream. Flow q is vehicles passing a point per hour. Density k is vehicles occupying a mile of lane at an instant. Space-mean speed vs is the average speed of those vehicles over the length. They are tied together by the identity q = k·vs, which holds for any stream. A model is needed only to say how speed changes with density; Greenshields' straight line is the simplest, and it is the one most textbook and exam problems use.
From the identity, average headway is h = 3,600/q seconds and average spacing is s = 5,280/k feet. At capacity in the default example (2,700 veh/h/ln, 90 veh/mi/ln) that is a 1.33-s headway and 59-ft spacing.
| Parameter | Typical range | Notes |
|---|---|---|
| Jam density, kj | 165–265 veh/mi/ln | about 20–32 ft per stopped vehicle |
| Freeway capacity (HCM) | 2,250–2,400 pc/h/ln | depends on free-flow speed (55–75 mi/h) |
| Density at freeway capacity (HCM) | 45 pc/mi/ln | LOS E/F boundary |
| Peak hour factor | 0.80–0.98 | HCM default 0.94 urban, 0.88 rural |
| K factor | 0.08–0.12 urban; 0.12–0.18 rural/recreational | share of AADT in the design hour |
| D factor | 0.50–0.60 urban; 0.60–0.70 rural | peak-direction share |
| Saturation headway at signals | 1.8–2.2 s | about 1,600–2,000 veh/h/ln |
Peak hour factor and design volumes
Traffic is not uniform within the peak hour. HCM analyses use the flow rate in the worst 15 minutes, v = V/PHF. A PHF of 1.00 means perfectly uniform flow; the theoretical minimum is 0.25, when the entire hour's traffic arrives in one 15-minute period. Design volumes for a new or widened road start from AADT: DHV = K·AADT for both directions together, and DDHV = K·D·AADT for the heavier direction, which is what sets the number of lanes.
Worked examples
Example 1 — Greenshields
Example 2 — Spot speeds
Limits of this tool
- Greenshields' linear speed–density line is a teaching and first-estimate model. It puts capacity at half the free-flow speed, which is lower than observed on modern freeways; use the HCM 7 speed–flow curves for level-of-service work.
- The 85th-percentile speed from a small sample is rough — speed studies normally collect at least 50–100 observations.
- Space-mean speed from spot speeds assumes each vehicle holds its speed over the section.
Frequently asked questions
What is the Greenshields model?
Greenshields (1935) observed that speed falls roughly linearly with density: v = vf(1 − k/kj). Combined with the fundamental relation q = k·v, flow becomes a parabola in density, q = vf(k − k²/kj). Flow is zero at k = 0 (no cars) and at k = kj (jammed), and peaks at capacity qmax = vf·kj/4 when k = kj/2 and v = vf/2.
How do you calculate capacity from free-flow speed and jam density?
Under Greenshields, qmax = vf·kj/4. For vf = 60 mi/h and kj = 180 veh/mi/ln, capacity is 60 × 180 / 4 = 2,700 veh/h/ln, reached at 90 veh/mi/ln and 30 mi/h. Real freeways reach capacity at higher speeds (about 50 mi/h in the HCM speed-flow curves), so use Greenshields for exam-style problems and the HCM for design.
How do you calculate the peak hour factor?
PHF = V / (4 × V15), where V is the hourly volume and V15 is the volume in the busiest 15 minutes of that hour. If 1,800 vehicles pass in an hour and 500 in the peak 15 minutes, PHF = 1,800 / 2,000 = 0.90. Divide hourly volume by PHF to get the peak 15-minute flow rate used in HCM analyses.
What is the difference between time-mean speed and space-mean speed?
Time-mean speed is the arithmetic average of spot speeds measured at a point. Space-mean speed is the average speed of the vehicles occupying a length of road at an instant; from spot speeds it is the harmonic mean, n / Σ(1/vᵢ). Space-mean speed is always less than or equal to time-mean speed and is the speed that belongs in q = k·v.
What are the K and D factors?
K is the fraction of AADT that occurs in the design hour (often the 30th highest hour of the year), commonly 0.08–0.12 for urban roads and higher on rural recreational routes. D is the fraction of the design-hour traffic in the heavier direction, typically 0.55–0.70. DDHV = AADT × K × D.
Why does one flow correspond to two densities?
Below capacity, the same flow can be carried by few fast vehicles (uncongested) or many slow ones (congested, after breakdown). The calculator solves the quadratic for both roots. Which one applies depends on whether a queue has formed upstream of a bottleneck.
References: Greenshields, B.D. (1935), "A study of traffic capacity," Proc. Highway Research Board 14, 448–477. Transportation Research Board, Highway Capacity Manual, 7th ed. (2022), ch. 4 (traffic flow and capacity concepts). Mannering, F.L. and Washburn, S.S., Principles of Highway Engineering and Traffic Analysis, 7th ed., Wiley (2020), ch. 5. Garber, N.J. and Hoel, L.A., Traffic and Highway Engineering, 5th ed., Cengage (2015), ch. 4 and 6.
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