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Intersection B: Traffic in = 350 + 125.
Traffic out = x1 x4 . Thus x1 x4 475 .
PROJECT FOR SECTION 8.2
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Intersection C: Traffic in = x3 x4.
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Traffic
outOR
= 600
+ 300. Thus x3 x4 900 .
Traffic Flow
Intersection D: Traffic in = 800 + 250.
Traffic out = x2 x3 . Thus x2 x3 1050 .
Gareth Williams, Ph.D.
Mathematics and Computer
Science
Department,
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& Bartlett
Learning, LLC
© Jones
& Bartlett
LLC
These constraints on the traffic
are described
by the Learning,
folStetson University NOT FOR SALE OR DISTRIBUTION
lowing system of linear equations:
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625
x1 x2
Network analysis, as we saw in the discussion of
x4 475
x1
Kirchhoff’s point and loop rules in Section 8.2, plays an
x3 x4 900
important
in electrical
engineering.
In recent
years,
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Jones
& Bartlett
Learning,
LLC
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Learning,
LLC
x3
1050
the concepts and tools of network analysis have been
NOT
FOR
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OR
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NOT
FOR
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OR
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found to be useful in many other fields, such as informaThe method of Gauss-Jordan elimination can be used
tion theory and the study of transportation systems. The
to solve this system of equations. The augmented matrix
following analysis of traffic flow through a road network
and reduced row-echelon form of the above system are
during the peak period illustrates how systems of linear
as follows:
equations
with many
solutions LLC
can arise in practice.
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& Bartlett
Learning,
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Consider the typical road network of Figure 1. It rep1 0 0
1 475
1 1 SALE
0 0 OR
625 DISTRIBUTION
NOT FOR
SALE OR DISTRIBUTION
NOT FOR
resents an area of downtown Jacksonville, Florida. The
Row
operations
0 1 0 1 150
1 0 0 1 475
≤.
±
≤ operations ±
streets are all one-way, with the arrows indicating the di1
1 900
0 0 1
0 0 1 1 900
rection of traffic flow. The flow of traffic in and out of
0 0 0
0
0
0 1 1 0 1050
the network is measured in terms of vehicles per hour
(vph). The figures given©here
are based
on midweek
Jones
& Bartlett
Learning, LLC
© Jones & Bartlett Learning, LLC
The system of equations that corresponds to this reduced
peak traffic hours, 7 A.M. to 9 A.M. and 4 P.M. to 6 P.M.
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row-echelon form is
An increase of 2 percent in the overall flow should be allowed for during the Friday evening traffic flow. Let us
x1
x4 475
construct a mathematical model that can be used to anax
x4 150
2
lyze this network.
x3 x4 900.
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350 vph
Duval Street
x1
A
Hogan Street
N
x
2
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Learning, LLC
800 vph
Monroe Street
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x
C
ing variable, we get
125 vph
B
Laura Street
225 vph
400 vph
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Expressing
eachFOR
leading
variable
in terms
of the remainNOT
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OR
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x4
300 vph
x1 x4 475
x2 x4 150
Learning,
x3 x4 LLC
900.
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NOT FOR SALE OR DISTRIBUTION
As perhaps might be expected, the system of equations
has many solutions—many traffic flows are possi250 vph
600 vph
ble. A driver does have a certain amount of choice at
Figure 1 Downtown Jacksonville, Florida
intersections. Let us now use this mathematical model to
develop more information
about the
traffic flow.
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& Bartlett
Learning, LLC
Assume that the following traffic law applies:
Suppose
it
becomes
necessary
to
perform
road
NOT FOR SALEwork
OR on
DISTRIBUTION
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the stretch DC of Monroe Street. It is desirable to have
All traffic entering an intersection must leave that
as small a traffic flow x3 as possible along this stretch of
intersection.
road. The flows can be controlled along various branches by means of traffic lights. What is the minimum value
This conservation of flow constraint (compare it to
of x3 along
DC that would
not leadLearning,
to traffic congestion?
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& Bartlett
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& Bartlett
LLC
Kirchhoff’s
point rule)
leads to a Learning,
system of linear
equaWe
use
the
preceding
system
of
equations
to answer this
tions: NOT FOR SALE OR DISTRIBUTION
NOT FOR SALE OR DISTRIBUTION
question.
Intersection A: Traffic in = x1 x2.
All traffic flows must be nonnegative (a negative
Traffic out = 400 + 225. Thus x1 x2 625 .
flow would be interpreted as traffic moving in the wrong
3
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xx
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PROJECT FOR SECTION 8.2 Traffic Flow
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4591X_PROJ_Zill.qxd
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7:24 PM
Page xxi
direction on a one-way street). The third equation in the
possible along the branch BC? What are the other flows
system tells us that x3 will be a minimum when x4 is as
at that time? (The units of flow are vehicles per hour.)
large
as
possible,
as
long
as
it
does
not
exceed
900.
The
© Jones3.&Figure
Bartlett
Learning,
LLCentering and leaving an© Jones & Bartlett Learning, LLC
4 represents
the traffic
largest value x4 can be without causing negative values NOT FOR SALE
other
type
of
roundabout
road junction in continental
OR
DISTRIBUTION
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of x1 or x2 is 475. Thus the smallest value of x3 is 475
Europe. Such roundabouts ensure the continuous
+ 900, or 425. Any road work on Monroe Street should
smooth flow of traffic at road junctions. Construct linallow for traffic volume of at least 425 vph.
ear equations that describe the flow of traffic along the
In practice, networks are much vaster than the one
various branches. Use these equations to determine the
discussed here, leading©toJones
larger systems
of linearLearning,
equa& Bartlett
LLC minimum flow possible
© Jones
Bartlett Learning, LLC
along x&
1. What are the other
tions that are handled NOT
on computers.
VariousOR
values
of
flows at that time? (It
is notFOR
necessary
to compute
the
NOT
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OR DISTRIBUTION
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DISTRIBUTION
variables can be entered into a computer to create differreduced row-echelon form. Use the fact that the traffic
ent scenarios.
flow cannot be negative.)
Related Problems
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1. Construct a mathematical model that describes the trafSALE
OR depicted
DISTRIBUTION
ficNOT
flow inFOR
the road
network
in Figure 2. All
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Learning, LLC
100
90
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x3
streets are one-way streets in the directions indicated.
The units are vehicles per hour. Give two distinct possible flows of traffic. What is the minimum possible flow
that can be expected along branch AB?
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200
100
100
x1
A
D
C
x3
x1
155
130
x8
x4
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110
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DISTRIBUTION
x5
150
x6
120
x7
75
80
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Figure 4
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x2
x4
100
B
x2
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Traffic flow in Problem 3
4. Figure 5 describes a flow of traffic, with the units being
50
vehicles per hour.
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50
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(a) Construct a system of linear equations that deNOT
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scribesFOR
this flow.
Figure 2 Traffic flow in Problem 1
(b) The total time it takes the vehicles to travel any
stretch of road is proportional to the traffic along
2. Figure 3 represents the traffic entering and leaving a
that stretch. For example, the total time it takes x1
“roundabout” (rotary) road junction. Such junctions are
vehiclesLearning,
to traverse AB
is kx1 minutes. Assuming
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Bartlett
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LLC
common
in Europe.
Construct a mathematical© Jones & Bartlett
that
the
constant
is
the
same
for all sections of road,
model that
the flow of traffic along the vari-NOT FOR SALE OR DISTRIBUTION
NOT FOR SALE
ORdescribes
DISTRIBUTION
the
total
time
for
all
200
vehicles
to be in this netous branches. What is the minimum flow theoretically
work is kx1 2kx2 kx3 2kx4 kx5. What is
this total time if k = 4? Give the average time for
50
each car.
x4
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B
100
A
C
x1
x2
C
x1
150
200
B
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x4
A
D
x3
200
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x5
2
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F
200
Figure 3 Traffic flow in Problem 2
x4
x2
E
Figure 5 Traffic flow in Problem 4
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PROJECT FOR SECTION 8.2 Traffic Flow
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xxi