Fundamental Principles of
Traffic Flow
Chapter 6
Dr. TALEB AL-ROUSAN
Introduction
 Traffic flow theory involves the development
of mathematical relationships among the
primary elements of a traffic stream>
 Flow
 Density
 Speed
 These relationships help traffic engineer in
planning, designing, and evaluating the
effectiveness of implementing traffic
engineering measures on a highway system.
Applications of Traffic Flow Theory
 Traffic flow theory is used in design to determine:
 Adequate lane length for storing left-turn vehicles on
separate left-turn lanes.
 Average delay at intersections and freeway ramp
merging areas.
 Changes in the level of freeway performance due to
the installation of improved vehicular control devices
on ramps.
 Traffic flow theory is used in simulation:
 Mathematical algorithms are used to study the
complex interrelationship between elements of traffic
stream.
 Estimate the effect of changes in traffic flow on factors
such as accidents, travel time, air pollution, and
gasoline consumption.
Traffic Flow Elements
 Time-space diagram: A graph the describes
the relationship between the location of
vehicles in a traffic stream and the time as
the vehicles progress along the highway.
 See Figure 6.1.
 (X,Y) = (Time, Distance)
 Primary elements are: flow, density, &
speed.
 Another element, associated with density, is
the gap or headway between vehicles in
traffic stream
Traffic Flow Elements Definitions
 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] ……veh/h
n = number of vehicles passing a point in the roadway in
(T) seconds.
 Density (k) : ,also referred to as concentration,
number of vehicles traveling over a unit length
(usually one mile) of highway at an instant in time.
Unit of density is (vehicle/mi)
Traffic Flow Elements Definitions
Cont.
 Speed (u) : the distance traveled by
a vehicle during a unit of time (mi/h)
or (km/h) or (ft/sec),
 The speed of a vehicle at any time (t)
is the slope of the time-space
diagram for that vehicle at time (t).
 Two types of mean speed:
 Time mean speed
 Space mean speed
Traffic Flow Elements Definitions
Cont.
 Time Mean Speed (ūt) :the
arithmetic mean of the speeds of
vehicles passing a point on a
highway.
ūt = [(1/n) (Sum (ui)) ] …. i = 1 to n
n= number of vehicles passing a point
on a highway.
ui = speed of the ith vehicle (ft/sec)
Traffic Flow Elements Definitions



Space Mean Speed (ūs) :the harmonic mean of the
speeds of vehicles passing a point on a highway during an
interval of time.
Obtained by dividing the total distance traveled by two or
more vehicles on a section of a highway by the total time
required by these vehicles to travel that distance.
Space mean speed is the one involved in flow-density
relationship.
ūs = [n/ (Sum (1/ui)) ] …. i = 1 to n
ūs = [nL / (Sum (ti)) ] …. i = 1 to n
n= number of vehicles passing a point on a highway.
ui = speed of the ith vehicle (ft/sec).
ti = the time it takes the ith vehicle to travel a cross a section
of highway (sec).
L= length of section of highway (ft).
Traffic Flow Elements Definitions
Cont.
 The time mean speed is always
higher than the space mean speed.
 The difference between these speeds
tends to decrease as the absolute
values of speeds increase.
ūt = ūs + (s2/ ūs)
Traffic Flow Elements Definitions
Cont.
 Time Headway (h): is the difference
between the time the front of a vehicle
arrives at a point on the highway and the
time the front of the next vehicle arrives at
that same point, expressed in (seconds)..
 Can be found from time-space diagram at
specified distance.
 Space Headway (d): is the distance
between the front of a vehicle and the front
of the following vehicle, expressed in (feet).
 Can be found from time-space diagram at
specified time (t).
 See Example 6.1
Flow-Density Relationships
 The general equation relating flow, density,
and space mean speed is given as:
 Flow = density x space mean speed
q= k ūs
 Each of variable depends on several other
factors, including:




Characteristics of the roadway.
Characteristics of the vehicle
Characteristics of the driver
Environmental factors (e.g. weather)
Flow-Density Relationships Cont.
 Other relationships exist among traffic flow variables>
 Space mean speed = (flow) x (average space headway)
ūs = q đ
average space headway = đ = (1/k)
 Density = (flow) x (average travel time for unit distance)
K= q Ť
 Average space headway = (space mean speed) x
(average time headway)
đ = ūs Ћ
 Average time headway = (average travel time for unit
distance) x (average space headway)
Ћ=Ťđ
Fundamental Diagram of Traffic
Flow
 See Figure 6.4
 Flow vs. Density:
 When there are no vehicles on the highway, the
density is zero and flow is also zero.
 As density increase flow also increase.
 When density reaches max. (jam density Kj), the flow
must be zero because vehicles will tend to line up end
to end.
 It follows that as density increases from zero, the flow
will also initially increase from zero to a max. value.
 Further continuous increase in density will then result
in continuous reduction of flow, which will be zero
when density is equal to the jam density.
 See Figure 6.4a
 Some controversy exist regarding the exact shape of
the curve.
Fundamental Diagram of Traffic
Flow Cont.
 Space Mean Speed vs. Flow:
 When flow is very low, there is little interaction between vehicles,
therefore drivers are free to travel at max possible speed.
 The absolute max speed is obtained as the flow tends to zero
(Mean Free Speed uf ).
 Magnitude of (uf) depends on the physical characteristics
of the highway.
 Continuous increase in flow will result in a continuous
decrease in speed.
 A point will be reached when further addition of vehicles
will result in the reduction in the actual number of
vehicles that pass a point on the highway (reduction of
flow).
 At this point congestion is reached and eventually both
speed and flow become zero.
 See Figure 6.4 c.
Fundamental Diagram of Traffic
Flow Cont.
 Space Mean Speed vs. Density:
 When there are no vehicles on the highway,
the density is zero.
 When density is zero there will be little or no
interaction between vehicles, therefore drivers are
free to travel at max possible speed.
 Further continuous increase in density will
then result in continuous reduction of
speed, which will be zero when density is
equal to the jam density
 See Figure 6.4 b.
Fundamental Diagram of Traffic
Flow Cont.
 knowing that: [ūs = q/k ] means
that slopes of lines (OB, OC, OE) in Figure
6.4a represent the space mean speeds at
densities (Kb, kc, ke) respectively.
 Slope of OA = the speed at density tends to
zero = mean free speed (uf) = max speed
that can be attained on the highway.
 Slope of OE = the speed for max flow =
capacity of the highway.
 It is desirable for highways to operate at
densities not greater than that required for
maximum flow.
Mathematical Relationships
Describing Traffic Flow
 Macroscopic approach: considers flow-density relationship.
1- Greenshields Model: used for light or dense traffic (satisfies boundary
conditions when density approach zero or jam density).
ūs = uf – ((uf /kj)k)
q = (uf k )– ((uf /kj) k2)
2- Greenberg model: used only for dense traffic (satisfies boundary
conditions when density approach jam density).
ūs = c ln (kj/k)
q = ūs k = c k ln (kj/k)

Microscopic approach: (referred to as car-following theory or follow-theleader theory) considers spacing between vehicles and speeds of
individual vehicles.
Gap & Gap Acceptance
 Another important aspect of traffic flow is
the interaction of vehicles as they:
Join a traffic stream : ramp vehicles merging
into an expressway stream.
 Leave a traffic stream : freeway vehicles leaving
the freeway onto frontage roads.
 Cross a traffic stream: changing of lanes by
vehicles on a multilane highway.

 the most important factor a driver consider
in making any of these maneuvers is the
availability of a gap between two vehicles
that, in drivers judgment, is adequate for
him/her to complete the maneuver.
Important Measures In Concept of
Gap Acceptance
 Merging: the process by which a vehicle in
one traffic stream joins another traffic
stream in the same direction.
 Diverging: the process by which a vehicle
in a traffic stream leaves the traffic stream.
 Weaving: the process by which a vehicle
first merges into a stream of traffic
obliquely crosses that stream, and them
merges into a second stream moving in the
same direction.
Important Measures In Concept of
Gap Acceptance
 Gap: the headway in a major stream, which is
evaluated by a vehicle driver in the minor stream who
wishes to merge into the major stream. Expressed in
units of (time or distance)
 Time Lag: the difference between the time a vehicle
that merges into a main traffic stream reaches a point
on the highway in the area of merge and the time a
vehicle in the main stream reaches the same point.
 Space lag: the difference, at an instant of time,
between the distance a merging vehicle is a way from
a reference point in the area of merge and the
distance a vehicle in the main stream is a way from
the same point.
Gap Acceptance