lec 10 traffic stream models (transportation engineering dr.lina shbeeb)

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

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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

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