Traffic Modeling
MAT/220 Version 1
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University of Phoenix Material
Traffic Modeling
Figure 1 shows the intersections of five one-way streets and the number of cars that enter each
intersection from both directions. For example, I1 shows that 400 cars per hour enter from the top and that
450 cars per hour enter from the left. See the Applications section in Section 6.2 of College Algebra as a
reference.
For this assignment, use Figure 1 to answer the questions following the figure and to prepare a Microsoft ®
PowerPoint® presentation.
Figure 1. The intersections of five one-way streets
The letters a, b, c, d, e, f, and g represent the number of cars moving between the intersections. To keep
the traffic moving smoothly, the number of cars entering the intersection per hour must equal the number
of cars leaving per hour.
1. Describe the situation.
2. Create a system of linear equations using a, b, c, d, e, f, and g that models continually flowing
traffic.
3. Solve the system of equations. Variables f and g should turn out to be independent.
4. Answer the following questions:
Traffic Modeling
MAT/220 Version 1
a. List acceptable traffic flows for two different values of the independent variables.
b. The traffic flow on Maple Street between I5 and I6 must be greater than what value to keep
traffic moving?
c. If g = 100, what is the maximum value for f?
d. If g = 100, the flows represented by b, c, and d must be greater than what values? In this
situation, what are the minimum values for a and e?
e. This model has five one-way streets. What would happen if the model had five two-way
streets?
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Traffic Modeling
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1. This system can be modeled by assuming that traffic flow is maintained at each intersection, namely
that the number of cars into an intersection must equal the number out in some period of time. We use
the labels as shown in the diagram below to indicate the number of cars per hour traveling on each
section of the road.
2. With traffic flowing in the directions indicated above, we have
-a - f = -850
a - b + g = 350
b - c = 150
c - d = -200
(1)
d - e - g = -350
e + f = 900
3. With seven variables, we do not have enough information in this system for a unique solution. Instead,
we can let any two variables be independent, and then solve for the other four in terms of these two
independent variables. Choosing f and g to be the independent variables, we find
a = 850 - f
b = 500 + g - f
c = 350 + g - f
d = 550 + g - f
e = 900 - f
(2)
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4.a. Here are three possible scenarios:
f =g=0
f = g = 100
f = 0, g = 500
a = 850
b = 500
a = 750
b = 500
a = 850
b = 1000
c = 350
c = 350
c = 850
d = 550
d = 550
d = 1050
e = 900
e = 800
e = 900
4.b. From (2) we have e = 900 - f . Since all traffic must be positive (one way streets and traffic can’t
flow backwards), we also know that a = 850 - f ³ 0 , or 0 £ f £ 850 . Therefore e ³ 50. So the traffic flow
on Maple Street between I5 and I6 must be at least 50 cars per hour.
4.c. Suppose g = 100 cars per hour. Then since c = 350 + g - f ³ 0 (Eq. 2), the maximum value of f is
450 cars per hour.
4.d. If g = 100, then
b = 600 - f ³ 0 ³
³
c = 450 - f ³ 0 ³ ® 0 ³ f ³ 450
d = 650 - f ³ 0 ³
³
So in this case, we see that f can be no greater than 450 (lest c become negative). This means b, c and
d must follow
b ³ 150
c³0
d ³ 200
The flows a and e are, in this case, constrained to the following ranges:
a = 850 - f ® 400 £ a £ 850
e = 900 - f ® 500 £ e £ 900
4.e. If these were two-way streets, each direction would (in general) be independent of each other. The
result would be fourteen variables instead of seven, with the same number of boundary conditions and
intersections. The solutions to this would be much more complicated, involving many more free
parameters.
Traffic Modeling
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General discussion
It is worth noting that our solution in Equation (2) indicates that flows a, f and e are bounded:
0 £ a £ 850
0 £ f £ 850
150 £ e £ 900
but that b, c, d and g can grow indefinitely:
b ³ 500
c³0
d ³ 200
(
g ³ max 0, f - 350
)
How can this be, with fixed amounts of cars entering and leaving? The answer is that g can grows as
large as you like, and this simply represents a set of cars performing a closed loop around the right block
(bcdg). These cars are neither entering nor leaving the region, but just circling the block, so are
determined by the initial condition of the system.