University of Duhok
College of Engineering
Civil Department
Fourth Year Students
2020-2021
Traffic Engineering
Lecture 3
Lecturer: Dr. Nasreen A. Hussein
1
Traffic Flow Theory
Objectives:
Understand the fundamental relationships among traffic parameters
Estimating traffic parameters using the fundamental relationship
Types of flow
Traffic flow is usually classified as:
a) Uninterrupted flow: A vehicle traversing a section of lane or roadway is not
required to stop by any cause external to the traffic stream (Ex: Freeways)
b) Interrupted flow: A vehicle traversing a section of a lane or roadway is required
to stop by a cause outside the traffic stream, such as signs or signals at
intersections or junctions (Ex: Urban Arterials).
Note:
Stoppage of vehicles by a cause internal to the traffic stream does not constitute
interrupted flow.
2
Relationship among Macroscopic Traffic Flow Parameters
Traffic flow theory involves the development of mathematical relationships among the
primary elements of a traffic stream. The relationship between speed (space mean
speed), flow (number of vehicles passing a given point per unit time), and density (number
of vehicles per unit length of highway) is called the fundamental relations of traffic flow.
The three basic macroscopic parameters of a traffic stream (flow, speed and density) are
related to each other as follows:
The traffic flow, q, a measure of the volume of traffic on a highway, is defined as the
number of vehicles, n, passing some given point on the highway in a given time interval,
t, i.e.:
In general terms, q is expressed in vehicles per unit time.
The number of vehicles on a given section of highway can also be computed in terms of
the density or concentration of traffic as follows:
Where the traffic density, k, is a measure of the number of vehicles, n, occupying a length
of roadway, x.
For a given section of road containing n vehicles per unit length x, the average speed of
the n vehicles is termed the space mean speed v (the average speed for all vehicles in a
given space at a given discrete point in time).
1
( ) ∑𝑛𝑖 𝑥𝑖
𝑣= 𝑛
𝑡
Where xi is the length of road used for measuring the speed of the ith vehicle.
3
It can be seen that if the expression for q is divided by the expression for k, the expression
for v is obtained:
𝑛
𝑞 𝑛 𝑥 𝑥
= 𝑛𝑡 = ∗ = = 𝑣
𝑘 𝑡 𝑛 𝑡
𝑥
Thus, the three parameters v, k and q are directly related under stable traffic conditions:
𝑞 = 𝑣𝑘
Or
Flow= speed (space mean speed) *density
This constitutes the basic relationship between traffic flow, space mean speed and
density.
Collecting information about traffic flow characteristics is necessary for planning,
management and solving traffic problems. Also they are necessary to optimize the
operation of existing traffic system and to design future facilities.
Speed-Density Relationship
To begin, consider a section of highway with only a single vehicle on it. Under these
conditions, the density (veh/km) will be very low and the driver will be able to travel freely
at a speed close to the design speed of the highway.
When speed is maximum, it refers to the free flow speed, and when the density is
maximum, the speed will be zero. The simplest assumption is that this variation of speed
with density is linear as shown by the solid line in the figure.
Corresponding to the zero density, vehicles will be flowing with their desire speed, or free
flow speed. When the density is jam density, the speed of the vehicles becomes zero. It
is also possible to have non-linear relationships.
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Where
Vf = free flow speed,
Kj = Jam density
If density is 0 (No vehicle), Maximum speed is available
As density increases from 0, speed decreases initially
If density is maximum, speed is 0
Flow- Density Relationship
Some characteristics of an ideal flow-density relationship is listed below:
1. When the density is zero, flow will also be zero, since there is no vehicles on the road.
2. When the number of vehicles gradually increases the density as well as flow increases.
3. When more and more vehicles are added, it reaches a situation where vehicles can’t
move. This is referred to as the jam density or the maximum density. At jam density, flow
will be zero because the vehicles are not moving.
4. There will be some density between zero density and jam density, when the flow is
maximum. The relationship is normally represented by a parabolic curve as shown in the
figure
5. For each value of q, there are two values of k.
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Where
Vf = free flow speed,
Kj = Jam density
If density is 0, flow is 0 (No vehicle).
As density increases from 0, flow increases initially.
After the max flow point (qmax), flow decreases as density increases.
If flow is 0 (Traffic Jam), Maximum jam density (kj).
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An empty road An jammed road
Speed-Flow Relationship
The relationship between the speed and flow can be assumed as follows.
1. The flow is zero either because there is no vehicles or there are too many vehicles so
that they cannot move.
2. At maximum flow, the speed will be in between zero and free flow speed. This
relationship is shown in the figure.
3. The maximum flow qmax occurs at speed v. It is possible to have two different speeds
for a given flow.
Where
Vf = free flow speed,
Kj = Jam density
If flow is 0 (No vehicle), Maximum speed is available.
As flow increases from 0, speed decreases initially (Uncongested flow).
After the max flow point (qmax), speed decreases as flow decreases
(Congested flow).
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Greenshields Model: Greenshields carried out one of the earliest recorded works in
which he studied the relationship between speed and density. He hypothesized that a
linear relationship existed between speed and density that is expressed as:
k
vs vf 1 ………… (1)
kj
This model is simple to use and several investigators have found good correlation
between the model and field data.
Since q = k *vs, substitute q/vs for k in equation (1)
q
vs vf 1 vs Multiply by vs
k
j
vf
vs 2 vf * vs *q …………(2)
kj
Also substituting q/k for vs
q vf
vf * k
k kj
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vf
q vf * k *k2 ……….(3)
kj
Differentiating q with respect to vs, Eq. 2 we obtain
vf
vs 2 vf * vs *q
kj
vf dq
2vs vf *
kj dvs
dq kj kj
vf 2vs
dvs vf vf
dq kj
kj 2vs
dvs vf
For maximum flow,
dq
0,
dvs
kj
0 kj 2vs
vf
kj
kj 2vs
vf
vf
vs
2
Consider equation 3,
vf
q vf * k *k2
kj
Differentiating q with respect to k, we obtain:
dq vf
vf 2k
dk kj
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For maximum flow,
dq
0,
dk
vf
0 vf 2k
kj
vf
vf 2k
kj
kj
k
2
The maximum flow for the Greenshields relationship can therefore be obtained:
kj vf
q max *
2 2
kj * vf
q max
4
Fundamental Diagrams of Traffic Flow
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So if a 1-mile of a roadway contains 20 vehicles, and the mean speed of the 20
vehicles is 40 mile/h.
After 1 hour, 800 vehicles (40 x 20) would have passed.
The value of the flow (q) or traffic volume in this case would be equal to 800 v/hr.
Example (1):
A section of highway is known to have free-flow speed 55 mi/h and a capacity of 3300
veh/h. In a given hour, 2100 vehicles were counted at a specified point along this highway
section. If the linear speed-density relationship applied what you estimate the space
mean speed of these 2100 vehicles to be?
Solution
kj * vf kj * 55
q max , 3300
4 4
kj 240veh / mi
Using equation 2
vf
vs 2 vf * vs *q
kj
Rearranging the above equation we obtain
kj 2
vs kj * vs q 0
vf
240 2
vs 240 * vs 2100 0
55
Either
vs =44.09 mi/hr or
vs=10.92 mi/hr
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Example (2)
Free-flow speed of vehicles on a road section is 88 km/h. Jammed density is 228 veh/km.
Estimate the traffic density on the road if traffic flow is at a maximum level. Estimate also
the possible maximum flow for the road section and the average speed of vehicles at that
maximum traffic volume.
Solution
Density when traffic flow reaches the maximum volume,
kcap = kj/2 = 228/2 = 114 veh/km.
Expected maximum flow,
kj * vf
q max
4
228 * 88
q max 5016 veh/hr
4
Average speed at qcap,
v = vf/2 = 88/2 = 44 km/hr
Example (3)
A highway section has an average spacing of 25ft under jam conditions and a free-flow
speed of 55mph. Assuming that the relationship between speed and density is linear,
determine the jam density, the maximum flow, the density at maximum flow, and the
speed at maximum flow.
Solution
1
kj
spacing
1
kj 212.7 213 Veh/mile
25 / 5280
kj * vf
q max
4
12
213 * 55
q max 2929 veh/hr
4
At maximum flow:
kj
k
2
213
k 106.5 107 veh/mile
2
At maximum flow:
vf
vs
2
55
vs 27.5 mile/hr
2
Example (4)
A road has capacity 4000 veh/hr, and a free-flow speed of 50 mi/hr. If the density is 100
veh/mi, what is the speed and flow?
Solution
kj * vf
q max
4
kj * 50
4000
4
kj 320 veh/mile
Speed-density relationship
k
vs vf 1
kj
100
vs 501
320
vs 34.37 mile/hr
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Flow-density relationship
vf
q vf * k *k2
kj
50
q 50 *100 *100 2
320
q 3437 .5 3438 veh/hr
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