Graphics Programs Reference
In-Depth Information
the “sensitivity coefficient” λ may depend on u n 1 ( t ) u n ( t ), the spacing
between cars, and/or u n ( t ), the speed of the n th car. The idea behind this
equation is this. Drivers will tend to decelerate if they are going faster than
the car in front of them, or if they are close to the car in front of them, and
will tend to accelerate if they are going slower than the car in front of them.
In addition, drivers (especially in light traffic) may tend to speed up or slow
down depending on whether they are going slower or faster (respectively)
than a “reasonable” speed for the road (often, but not always, equal to the
posted speed limit). Since our road is circular, in this equation u 0 is
interpreted as u N , where N is the total number of cars.
The simplest version of the model is the one where the “sensitivity
coefficient” λ is a (positive) constant. Then we have a homogeneous linear
differential-difference equation withconstant coefficients for the velocities
u n ( t ). Obviously there is a “steady state” solution when all the velocities are
equal and constant (i.e., traffic is flowing at a uniform speed), but what we
are interested in is the stability of the flow, or the question of what effect is
produced by small differences in the velocities of the cars. The solution of (*)
will be a superposition of exponential solutions of the form u n ( t ) = exp( α t ) v n ,
where the v n s and α are (complex) constants, and the system will be
unstable if the velocities are unbounded; that is, there are any solutions
where the real part of α is positive. Using vector notation, we have
u ( t ) = exp( α t ) v, u ( t + T ) = α exp( α T )exp( α t ) v.
Substituting back in (*), we get the equation
α exp( α T )exp( α t ) v = λ ( S I )exp( α t ) v,
where
0
01
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10 ·
0
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01 ·
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S =
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10 ·
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010
is the “shift” matrix that, when it multiplies a vector on the left, cyclically
permutes the entries of the vector. We can cancel the exp( α t ) on eachside to
get
α exp( α T ) v = λ ( S I ) v, or { S [1 + ( α/λ )exp( α T )] I } v = 0 ,
( ∗∗ )
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