Relate traffic speed, density, and flow with the Greenshields model. Find capacity and compare congested and uncongested density solutions.
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Worked examples
These are the solver's own reference problems — the answers come from published sources or independent hand computation, never from the solver itself. The solution below is the live solver output for each.
Example 1
flow from density (the motivating exam problem)
Given
free flow speed
80 km/h
jam density
120
density
30
Assumptions
Greenshields' linear speed–density relation; densities are per km; flows are vehicles per hour on one lane.
Solution steps
Apply the linear model
Speed falls linearly from vf at empty road to zero at jam density; flow is density times speed.
v = vf·(1 − k/kj); q = k·v
k = 30 veh per kilometre; v = 60 km/h; q = 1800 veh/h
= 1800
Capacity check
The model's maximum flow occurs at half the jam density and half the free-flow speed.
qmax = vf·kj/4
qmax = 2400 veh/h
Results
Traffic flow q (veh/h)
1800
Space-mean speed v
16.67m/s
Density k (veh/km)
30
Capacity qmax (veh/h)
2400
Where this answer was checked
source
v = vf(1 − k/kj); q = k·v — vf = 80 km/h, kj = 120 veh/km, k = 30