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

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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

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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)

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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)

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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

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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

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Space Headway (s)

• The distance between the front of a

vehicle and the front of the following

vehicle (ft)

(s)

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Gap

• The distance between the back of a

vehicle and the front of the following

vehicle (ft)

(d)

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Line A-A:

Stationary

observers

whose location

does not

change with

time

Line B-B: Arial photograph of the

stream at a given instant

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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

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Greenshield Linear Model

km

sp

ee

d (

mp

h)

Concentration (veh/mi)

kj

0

0

um

uf

j

f

k

kuu 1

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Greenshield’s Linear Model

Continued

km

kj

Flo

w (

ve

h/h

r)

0

qm

0

Concentration (veh/mi)

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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

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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

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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