SME
Transportation Engineering
Traffic flow
Use this for “A lane has free-flow speed 100 km/h and jam density 120 pc/km/ln. At 40 pc/km/ln, find the speed, the flow and the capacity.”
Chart
q(40) ≈ 2,667
q(k) = chord(k)
The curves cross at k = 0 and k = 40.
This is an example. Type your own values to start a new problem.
v (km/h)
=
vf (km/h)
×
(1
−
k (pc/km/ln)
÷
kⱼ (pc/km/ln)
)
q (pc/h/ln)
=
k (pc/km/ln)
×
v (km/h)
kc
Density at capacity
calculated · pc/km/ln
qₘₐₓ
Capacity
calculated · pc/h/ln
Formulas
v
=
vf
×
(1
−
k
kⱼ
)
66.6667
=
100
×
(1
−
40
120
)
q
=
k
×
v
2,666.6667 = 40 × 66.6667
qₘₐₓ
=
vf
×
kⱼ
4
3,000
=
100
×
120
4
kc
=
kⱼ
2
60
=
120
2
Assumptions
•
Greenshields’ model: speed falls in a straight line from vf when empty to 0 at jam density kⱼ.
•
Values are per lane; each vehicle counts as one passenger car (pc).
•
Flow is uncongested while k < kc = kⱼ ÷ 2; past kc, flow falls as density rises.
Step-by-step
WE KNOW
vf = 100 km/h, kⱼ = 120 pc/km/ln, k = 40 pc/km/ln
FIND
speed (v), flow (q), capacity (qₘₐₓ), density at capacity (kc)
Step 1 · Find speed (v)
Use v = vf × (1 − k ÷ kⱼ)
Speed falls from vf in a straight line as the density rises, reaching 0 at kⱼ.
v
=
100
×
(1
−
40
120
)
v
=
100
×
(1
−
0.3333)
v
=
100
×
0.6667
v
=
66.66667
km/h
Step 2 · Find flow (q)
Use q = k × v
Vehicles per kilometre times kilometres an hour gives vehicles an hour.
q
=
40
×
66.66667
q
=
2,666.667
pc/h/ln
Step 3 · Find capacity (qₘₐₓ)
Use qₘₐₓ = vf × kⱼ ÷ 4
At
half
the
jam
density
the
speed
is
half
vf,
so
q
=
(
kⱼ
2
)(
vf
2
).
qₘₐₓ
=
100
×
120
4
qₘₐₓ
=
12,000
4
qₘₐₓ = 3,000 pc/h/ln
Step 4 · Find density at capacity (kc)
Use kc = kⱼ ÷ 2
The flow curve is a parabola with zeros at 0 and kⱼ; its top is halfway.
kc
=
120
2
kc = 60 pc/km/ln
Check
Put the numbers back in. Both sides should match.
66.66667
=
100
×
(1
−
40
120
)
✓
2,666.667 = 40 × 66.66667 ✓
3,000
=
100
×
120
4
✓
60
=
120
2
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
More problem types
Shock wave at a queue
Use this for “Traffic at 40 pc/km/ln and 2667 pc/h/ln meets a queue at 100 pc/km/ln and 1667 pc/h/ln. How fast does the back of the queue move?”
Headway and spacing
Use this for “A lane carries 2667 pc/h/ln at a density of 40 pc/km/ln. Find the mean headway, the mean spacing and the speed.”