This document discusses traffic stream flow models and the variables used to describe traffic flow conditions. It introduces concepts like traffic flow, concentration, speed, spacing, headway, and gap. Flow-density relationships are presented, including the Greenshield linear model that shows the relationship between density, flow, and speed in traffic. Car following rules are discussed as the basis for models that determine safe spacing between vehicles based on speed.

1. Dr. Lina Shbeeb 1
Traffic stream flow models
Transportation engineering

2. Dr. Lina Shbeeb 2
Traffic stream flow models
• When we analyse traffic flow we are concern with the
interaction between different vehicles in the traffic stream
• Traffic condition varies from almost free flow (relatively
few vehicles are occupying the roadway to highly
congested conditions( roadway is jammed with slow
vehicles)
• The determinant of traffic flow models is the car-following
rule adopted by drivers in an attempt to maximize their
speed while maintaining an acceptable level of safety.
• Basic variables that describe the prevailing condition
within traffic stream are
– Traffic flow
– Traffic concentration
– Traffic speed

3. Dr. Lina Shbeeb 3
Notation used in relationship among speed,
spacing and acceleration
• V= initial speed of the two vehicles
• dl= deceleration rate of the leading vehicle
• df= deceleration rate of the following vehicle
• =perception reaction time
• x○= safety margin after stop
• L=length of vehicle
• N= number of vehicle in train (N=1 for cars

4. Dr. Lina Shbeeb 4
Vehicle flowing concepts
2 1
v
Direction of travel
L Spacing S
2 1 2 1
L
v
v
=perception reaction time
v2
2df x○
v2
2dl

5. Dr. Lina Shbeeb 5
Vehicular stream models
• The braking distance of the leading vehicle is
• If the perception reaction time and braking distance of the following
vehicle is included, then the total distance covered by the following
vehicle is
• In terms of the initial spacing, length of vehicle and safety margin
and xl
• By equating the last two equations, the spacing is
estimated by
l
l
d
v
x
2
2

f
f
d
v
vx
2
2
 
olf
xNLxsx 

 xNL
d
v
d
v
vs
lf
22
22


6. Dr. Lina Shbeeb 6
Traffic flow types
• Uninterrupted flow (Freeway)
• Interrupted flow (Arterials with traffic light signal)
• Vehicles in uninterrupted flow conditions are spaced so to provide
ample time and distance for a following vehicle to perceive and react
to decelerate safely without colliding with a leading vehicle that
suddenly decelerate and stop.
• The choice of the spacing between vehicle as shown in the next
slide is function of the deceleration that took place.
• There are three values of deceleration that are relevant to the
operation’s safety level
– dn= normal or comfortable deceleration (safest condition operation)
– de = emergency deceleration (low level of safety if the spacing is
selected so that the following vehicle need to apply emergency braking)
– = instantaneous or stonewall stop
• Combination of leading-following vehicle deceleration are give in
Table 3.2.1 and their relation to the spacing versus speed are given
in Figure 3.2.2

7. Dr. Lina Shbeeb 7

8. Dr. Lina Shbeeb 8
Stream variables
• Flow (q)
The equivalent hourly rate at which vehicles pass a point
on a highway during a time period less than 1 hour
q = (n x 3600)
T
Where;
n=# of vehicles passing a point in T seconds
q=equivalent hourly flow rate (veh/hour)
 Density (k) – veh/mi
The number of vehicles traveling over a unit length (usually
1 mile) of a highway at an instant in time
 Speed, u (mph or fps): Distance traveled by a vehicle
during a unit of time. Speed at anytime t is the slope of the
time-space diagram

9. Dr. Lina Shbeeb 9
Density (k)
• Concentration
• Number of vehicles traveling
over a unit length of highway
at an instant in time
• Usually veh/mile or vpmpl
• Example:
– 4 vehicles over 600 feet of roadway
– Over a mile
– k = 4 veh. x 5280 feet = 35.2 veh/mi
600 ft mile

10. Dr. Lina Shbeeb 10
Speed (u)
• Time mean speed ( )
Arithmetic mean of the speeds of vehicles
passing a point on a highway during an
interval of time (radar gun or road tube study)
tu


n
i
it
u
n
u
1
1
Where;
n = # of vehicles
ui = speed (ft/sec or mi/hr)

11. Dr. Lina Shbeeb 11
Speed (u)
• Space mean speed ( )
Harmonic mean of the speeds of vehicles passing
a point on a highway during an interval of time
(total distance traveled by 2 or more vehicles dived
by time required to travel that distance)
su

 n
i
i
s
t
nL
u
1
Where;
n = # of vehicles
ti = time to cross section of highway (sec)
L=length (ft)
(ft/sec)

12. Dr. Lina Shbeeb 12
Space vs. Time Mean Speed
Example
For a 500-foot section, the following were measured
Vehicle Measured Time to Travel 500 ft
(sec)
Measured Velocity (mph)
1 6.0 63
2 6.5 58
3 5.3 60
4 5.8 65
5 5.9 64
6 6.1 61
7 5.7 66
8 5.2 72
9 5.5 68
10 5.4 69

13. Dr. Lina Shbeeb 13
Space vs. Time Mean Speed
Example
Calculate time mean speed and space mean
speed.


n
i
it
u
n
u
1
1
ut = 63+58+60+65+64+61+66+72+68+69 = 64.5 mph
10

14. Dr. Lina Shbeeb 14
Space vs. Time Mean Speed
Example
Calculate space mean speed.
us = (500 ft) ( 10 ) = 85.61ft/sec=58.4 mph
6.0+6.5+6.3+5.8+5.9+6.1+5.7+5.2+5.5+5.4
us < ut (always)

 n
i
i
s
t
nL
u
1

15. Dr. Lina Shbeeb 15
Time Headway (h)
• The difference between the time the front of a
vehicle crosses a point on the highway and the time
the front of the next vehicle crosses the same point
(seconds)
t1
t2
h = t2 – t1

16. Dr. Lina Shbeeb 16
Space Headway (s)
• The distance between the front of a
vehicle and the front of the following
vehicle (ft)
(s)

17. Dr. Lina Shbeeb 17
Gap
• The distance between the back of a
vehicle and the front of the following
vehicle (ft)
(d)

18. Dr. Lina Shbeeb 18
Line A-A:
Stationary
observers
whose location
does not
change with
time
Line B-B: Arial photograph of the
stream at a given instant

19. Dr. Lina Shbeeb 19
Flow-Density Relationships
 q = k us
 us = q s
 s = 1/k
 k = q t
 h = t s
• q = flow
• k = density
• us = Space mean speed
• s = Average space headway
• h = Average time headway
• t = Avg. travel time for unit distance

20. Dr. Lina Shbeeb 20
Flow-Density Example
If the spacing between vehicles is 500 feet what is the
density?
s = 1/k k = 1/s = 1 veh/500 feet
= 0.002 vehicles/foot = 10.6 veh/mile
If the space mean speed is 45.6 mph, what is the flow rate?
q = kus = (10.6 veh/mile)(45.6 mph) = 481.5 veh/hr

21. Dr. Lina Shbeeb 21
Traffic Flow Diagrams
• Explains the relationship between density
(k), flow (q), and speed (u)
• Density is the number of vehicles
physically occupying the roadway, flow is
vehicles moving past a point per unit time
• So a number of vehicles can occupy the
roadway and have a low flow rate

22. Dr. Lina Shbeeb 22
Greenshield Linear Model
km
speed(mph)
Concentration (veh/mi)
kj
0
0
um
uf









j
f
k
k
uu 1

23. Dr. Lina Shbeeb 23
Greenshield’s Linear Model
Continued
km
kj
Flow(veh/hr)
0
qm
0
Concentration (veh/mi)

24. Dr. Lina Shbeeb 24
Greenshield’s Linear Model
Continued
km
kj
Flow(veh/hr)
0
qm
0
Concentration (veh/mi)
Congested
flow
Uncongested
flow

25. Dr. Lina Shbeeb 25
0
0 qm
um
uf
speed(mph)
Flow (veh/hr)
Uncongested
Flow
Congested
Flow

26. Dr. Lina Shbeeb 26
General rule
• Car following rule: Traffic experts suggest
that keeping a distance of one car length
for each 10mph increment of speed. This
result distance is the safe gap that should
be at least provided between vehicles

27. Dr. Lina Shbeeb 27
Example
• A driver that follow the car following rule. The car length
he/she uses is 15 ft. develop the equations of stream flow.
• Solution:
– The safe spacing is function of speed to determine the
length of the gap plus the car length.
u
x
q
bewillqthenkofinsteaduuseweIf
kukq
kuk
us
k
vehmi
u
L
u
Ls















10
350010
3500
103500
103500
10
35001
/
5280
5.115
10