Team # 4094 Page 3 of 15
1 Introduction
This paper is concerned with modeling and optimizing traffic flow in a traffic circle. As in
past studies of traffic flow (i.e. [1], [3], [4], [7]), we rely heavily on the following conservation
law for cars:
ρ
t
+ f(ρ)
x
= 0, (1)
Where ρ : R
+
× [a, b] → [0, ρ
max
] is the density of cars (number of cars per unit distance),
and f : [0, ρ
max
] → R is the traffic flow or flux across a boundary (number of cars per
unit time). The flux and density are related by the following rule: f(ρ) = ρv(ρ). This
conservation assumption has been used in almost every macroscopic model for traffic flow
so far, and it makes for a very useful description of the dynamics involved. Here it is
appli ed to the idea of a traffic circle, which is an a lternative to a traditional traffic light
junction. We note here that no distinction is made in the paper between a traffic circle
and a roundabout, even though there are slight technical differences. The rest of our
assumptions we list here:
• No cars are created or destroyed, as described by (1)
• Traffic density within the roundabout is uniform.
• Traffic flux into the circle is constant within a sufficiently small time interval.
• Al l drivers that enter the traffic circle wil l exit without going around the loop multiple
times.
• The velocity is a function of ρ and it is zero at ρ
max
.
• The traffic from incoming roads is distributed on outgoing roads according to time-
varying coefficients.
• Drivers attempt to ma ximize flux when possible.
Assuming traffic flux to be constant during small time intervals may seem like a bit of a
stretch, but really it is just a condition that ensures rough continuity of incoming traffic
density. In fact, our model allows for a very large number of jump discontinuities, so
that the traffic control device can handle a piecewise constant wave of incoming traffic
density. The condition on the velocity is one given in most traffic models, and it tests out
empirically. We will b e employing a velocity equation derived empirically from [8] [3]:
v(ρ) =
v
s
, 0 ≤ ρ ≤ σ
β(
1
ρ
−
1
ρ
max
) , ρ > σ
(2)
Here, v
s
is the velocity a car would go if it was by itself, in other words it represents the
speed limit. The σ is what’s known as the “capacity” of a system; it is the density at which
the flux is maximum, and the β is a constant chosen to ma ke the function continuous.
Note that the function for velocity makes intuitive sense as well; as the density increases,
the velocity decreases, until it hits the maximum density, where it becomes zero.
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