chapter-1 traffic engineering Traffic-flow-theory

Traffic
Engineering
Romharsh Oli
(MSc in Transportation Engineering)
Lecturer
Everest Engineering College
Traffic Engineering 1

Traffic Engineering 2
1.2.4 Traffic Flow Theory

Traffic flow theory 3
Traffic flow theory
• The theory of traffic flow can be defined
as a mathematical study
over
of
the road
net-
movement of vehicles
work.
• The subject is a mathematical approach
to define, characterize and describe
different aspects of vehicular traffic.

Parameters connected with traffic flow
• There are eight basic variables:
1. speed (v)
2. volume (q)
3. density (k)
4. headway(h)
5. spacing (s)
6. occupancy(R)
7. clearance (c)
8. gap(g)

Speed
• Speed is defined as a rate of motion, as distance per
unit time, generally in meter per second.
There are two mean speeds:
1.Space mean speed (vs).
• It is called “space” mean speed because the use of
average travel time essentially weights the average
according to the length of time each vehicle spends in
space.
2.Time mean speed (vt).
• This is the arithmetic mean of the measured speeds of
all vehicles passing; say a fixed roadside point during a
given interval of time, in which case, the individual
speeds are known as “spot” speeds.

• If travel times t1,t2,…….tn are observed for n
vehicles traversing a segment of length L, the
average travel speed is:
– average travel speed or space mean speed (m/sec)
– length of the highway segment (m)
–Travel time of the ith vehicle to traverse the section
(sec) n- number of vehicles observed
- spot speed (m/sec)

• It can be shown that whereas the time mean
speed is the arithmetic mean of the spot
speeds, the space mean speed is their
harmonic mean.
• Time mean speed always greater than space
mean speed, except in the situation where all
vehicles travel at the same speed.
• It can be shown that an approximate
relationship between the two mean speeds is:

Relationship between Vs and Vt
•

2
 t
vt

2

s
vs
also, vs  vt
vt  vs
2
s
 = variance of the space mean speeds.

Volume and rate of flow
• Volume and rate of flow are two different
measures.
• Volume is the actual number of vehicles
observed during a given time interval.
• The rate of flow represent the number of
vehicles passing a point during a time interval
less than one hour , but expressed as an
equivalent hourly rate.

Density or concentration
• It is defined as the number of vehicles occupying a
given length of lane or roadway, usually expressed
as vehicles per km (veh/km).
• Direct measurement of density can be obtained
through aerial photography, but more commonly it
is calculated from equation, if speed and rate of
flow are known.
q  v  k
– q= rate of flow (veh/hr)
– v=average travel speed (km/hr)
– k= average density (veh/km)

Spacing and headway
• Spacing and headway are two additional
characteristics of traffic streams.
• Spacing (s) is defined as the distance
between successive vehicles in a traffic
stream as measured from front bumper to
front bumper.
• Headway (h) is the corresponding time
between successive vehicles as they pass a
point on a roadway.
• Both spacing and headway are related to
speed, flow rate and density.

Relationship between k,h,q,s & v
• Spacing of vehicles in a traffic lane can generally be
observed from aerial photographs.
• Headways of vehicles can be measured using
stopwatch observations as vehicles pass a point on a
lane.
; veh /
hour
average headway (h) ,
sec
q

average speed (v),
m/sec
3600
average spacing (s), m
, sec
h

; veh /
km
k
 average spacing (s),
m
100
0

Lane occupancy (R)
• Lane occupancy (R) is a measure used in
freeway surveillance.
• If one could measure the lengths of
vehicles on a given roadway section and
compute the ratio:
L
i
R  
sumof lengthsof vehicles
lengthof road way section
D

Clearance (c) and gap (g)
• Clearance (c) and gap (g) are related to the
spacing parameter and headway.
• These four measurements are shown in
figure below.
L, m
Clearance
(m) gap
(sec)
Spacing (m)
or headway
(sec)
Figure:

• The difference between spacing and clearance is
obviously the average length of a vehicle in m.
• Similarly the difference between headway and
gap is the time equivalence of average length of a
vehicle (L/v)
•
•
•
•
•
g is the gap, sec;
L is the mean length of vehicle, m;
c is the mean clearance, m;
h is the mean headway, sec;
v is the mean speed, m/sec
g  h 
L
v
and, c  g
 v

Example
• Four vehicles 6m, 6.5m, 6.75m and 6.9m long,
are distributed over a length of roadway
200m long.
What is the occupancy and density?

Analysis of Speed, Flow, and Density
Relationship
• It has been assumed that a linear relationship
exists between the speed of traffic on
an
uninterrupted traffic lane and the traffic
density.
• Mathematically it is represented by:
v  A  Bk....................(1)
– v is the mean speed of the vehicle.
– k is the average density of vehicles veh/km.
– A and B are empirically determined parameters

v2
(v  A)v A
q  kv 
 B

B
v 
B
.......... .......... .
(3)
q  kv  Ak  Bk 2
.......... .......... ...(2)
We know;

Speed-flow-density curves
M
e
a
n
s
p
e
e
d
,
k
m
/
h
F
l
ow,
ve
h
/
h
M
e
a
n
s
p
e
e
d
,
k
m
/
h
A
A/
B
A2
/4B
V=A-
Bk
A/
2B
A/
B
A
A/
2
Density,
veh/km
a)
Density,
veh/km b)
Flow,
veh/h
c)
Speed -Flow-Density
curves
A2/4
B

Relationship between q and k
• The theoretical relationship between flow (q) and
density (k) on a highway lane, represented by a
parabola (see fig. below).
• As the flow increases, so does the density, until
the capacity of the highway lane is reached.
• The point of maximum flow (qmax)
corresponds to “optimal” density (k0).
• From this point onward to the right, the flow
decreases as the density increases.
• At jam density (kj), the flow is almost zero.
• On a freeway lane, this point may be likened to
the traffic coming to a halt, where the lane
appears to look like a parking lot.

Density, veh/km
Flow,
veh/h
vf
v
o
qmax
Ko Kj
Flow-Density Curve
If rays are drawn from zero through any point on the curve, the
slope of the rays represents the corresponding space mean speed.

Relationship between V and k
• The theoretical relationship between speed and
density represented by a straight line; (see Fig.
below):
• Flows can be calculated simply by multiplying
coordinates of speeds and densities for any point on
the straight line.
STrapffeiceflodw- 23
k
A
P
kj
kx
vx
vf
0

Relationship v and q
STpraeffiec fdlo- 24
q
v
qmax
Curve
v f
v 0
• The theoretical relationship between speed and flow
shows fig. below: Rays drawn from zero to any point
on the curve have slopes whose inverse is equal to
the density.
1/k

Speed-flow-density curves
Density, k Flow, q Density, k
Speed -Flow-Density curves (curves showing the connections between mean speed, density and flow)
Mean
speed,
v
Mean
speed,v
A
B
C
D
Flow,
q
B
A
C
D
B
C
A D

• At point A, density is closed to zero, and there are
only a very few vehicles on the road; the volume
is also close to zero and these few vehicles on the
road can choose their own individual speeds. Or
change lanes with no restrictions.
• At point B, the number of vehicles has increased
but the conditions are of “free flow” and there
are hardly any restrictions.
• From B to C, the flow conditions may be called
“normal”, but as density increases, drivers
experience significant lack of freedom to
maneuver their vehicles to the speed and lane of
their choice.

• Around point C traffic conditions begin to show
signs of instability, and speeds and densities
fluctuate with small change in volume.
• Point C is the point of maximum volume, and
further increases in density reduce speeds
considerably. Such behavior is called forced flow.
Flow near point D is known as jam density.
• A driver perceive excellent
moderately
driving
good
conditions
conditions
would
from
from
A. to B,
B. to C, but Increasingly
deteriorating conditions from C to D.

Linear relationship between speed
and concentration
• It has been indicated above that a linear
relationship between speed and concentration
is one of the possibilities.
• Greenshields found a linear relationship
between speed and concentration based on
empirical studies.

Greenshields
 space mean speed for free flow
conditions
K j  jamming concentration
K  concentration
vs  space mean speed
)K
 (
vsf
vs  vsf
vsf
K j

.......... .......... .....
(ii)
or, Q  vs K j
 vs )K jvs
 (vsf
.......... ........
(i )
2
2
 vsf
 K j

vs
vs
Weknow that Q  vs  K or, K 
Q

 K j

 



 vsf 
Q
Then,vs  vsf    or, Qvsf

 K j



vsf
Then, Q  vsf K  
K
Greenshields

• Differentiating the equation (i) with respect to
concentration, we can get the value of
concentration corresponding to the maximum
flow.
.......... .......... .......(ii
i)
2
max
K j
Then,K  K 
dK K j
 2vsf 
0
dQ
 vsf
K
Greenshields

• To obtain the speed corresponding to the
maximum flow, the equation (ii) is
differentiated with respect to vs.
.......... .......... ......
(iv)
2

0

2K jvs
s max 
vsf
s
or, v 
v
sf
j
s v
dQ
 K
dv
Greenshields

4
vsf K j
 .......... ..........
(v)
2 2
Therefore: Q(max imum)  vs(max imum)  K(maximum)
K j
vsf
 
Greenshields

Greenberg’s model
• Greenberg developed a model for speed, flow
and density measurement as follow;
• ……………… (1)
• Where C is a constant
• Substituting q/k for vs

• Differentiating q with respect to k, we obtain;
• And for maximum q;
• Therefore,

• Or
• Substituting in equation (1)
• So, C is the speed at maximum flow.

Example 1
• The speed density relationship of traffic on a
section of a freeway lane was estimated to be
• What is the maximum flow, speed, and
density at this flow?
• What is the jam density?

• Given kj =130veh/mi and
• k=30veh/mi when
Vs=30mph.
• Find qmax.
Example 2

Example 3
• Assuming a linear speed-density relationship, the mean
free speed is observed to be 84 km/h near zero density,
and the corresponding jam density is 140veh/km.
Assume that, the average length of vehicles is 6m.
a. Write down the speed-density and flow-density
equations.
b. Draw the v-k, v-q and q-k diagrams indicating critical
values.
c. Compute speed and density corresponding to flow
of 1000 veh/h.
d. Compute the average headways, spacing, clearances
and gaps when the flow is maximum.

Example 4
• Assuming a linear speed-density relationship, the mean
free speed is observed to be 85 km/h near zero density,
and the corresponding jam density is 140veh/km.
Assume that, the average length of vehicles is 6m.
a. Write down the speed-density and flow-density
equations.
b. Draw the v-k, v-q and q-k diagrams indicating critical
values.
c. Compute speed and density corresponding to flow of
1000 veh/h.
d. Compute the average headways, spacing, clearances
and gaps when the flow is maximum.

chapter-1 traffic engineering Traffic-flow-theory

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