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Dr. Lina Shbeeb 1
Traffic stream flow models
Transportation engineering
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
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
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
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
vx
2
2
f
f
d
vvx
2
2
olfxNLxsx
xNL
d
v
d
vvs
lf22
22
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
Dr. Lina Shbeeb 7
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
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
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
iit
un
u1
1
Where;
n = # of vehicles
ui = speed (ft/sec or mi/hr)
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
ii
s
t
nLu
1
Where;
n = # of vehicles
ti = time to cross section of highway (sec)
L=length (ft)
(ft/sec)
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
Dr. Lina Shbeeb 13
Space vs. Time Mean Speed
Example
Calculate time mean speed and space mean
speed.
n
iit
un
u1
1
ut = 63+58+60+65+64+61+66+72+68+69 = 64.5 mph
10
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
ii
s
t
nLu
1
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
Dr. Lina Shbeeb 16
Space Headway (s)
• The distance between the front of a
vehicle and the front of the following
vehicle (ft)
(s)
Dr. Lina Shbeeb 17
Gap
• The distance between the back of a
vehicle and the front of the following
vehicle (ft)
(d)
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
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
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
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
Dr. Lina Shbeeb 22
Greenshield Linear Model
km
sp
ee
d (
mp
h)
Concentration (veh/mi)
kj
0
0
um
uf
j
f
k
kuu 1
Dr. Lina Shbeeb 23
Greenshield’s Linear Model
Continued
km
kj
Flo
w (
ve
h/h
r)
0
qm
0
Concentration (veh/mi)
Dr. Lina Shbeeb 24
Greenshield’s Linear Model
Continued
km
kj
Flo
w (
ve
h/h
r)
0
qm
0
Concentration (veh/mi)
Congeste
d
flo
w
Uncongeste
d
flow
Dr. Lina Shbeeb 25
0
0 qm
um
uf
sp
ee
d (
mp
h)
Flow (veh/hr)
Uncongested Flow
Congested Flow
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
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
xq
bewillqthenkofinsteaduuseweIf
kukq
kukus
k
vehmiu
Lu
Ls
10
3500103500
103500
10350010
35001
/5280
5.115
10