Applications - A Guide to MATLAB for Beginners and Experienced Users - page 196

Solution at t = 4

25

24

23

22

21

20

19

18

17

16

15

-5 -4 -3 -2 -1 0 1 2 3 4 5

x

Again the results are very similar to those obtained before.

A Model of TrafficFlow

Everyone has had the experience of sitting in a traffic jam, or of seeing cars
bunchup on a road for no apparent good reason. MATLAB and SIMULINK
are good tools for studying models of such behavior. Our analysis here will be
based on “follow-the-leader” theories of traffic flow, about which you can read
more in Kinetic Theory of Vehicular Traffic , by Ilya Prigogine and Robert
Herman, Elsevier, New York, 1971 or in The Theory of Road Traffic Flow ,by
Winifred Ashton, Methuen, London, 1966. We will analyze here an extremely
simple model that already exhibits quite complicated behavior. We consider a
one-lane, one-way, circular road witha number of cars on it (a very primitive
model of, say, the Inner Loop of the Capital Beltway around Washington, DC,
since in very dense traffic, it is hard to change lanes and each lane behaves
like a one-lane road). Each driver slows down or speeds up on the basis of his
or her own speed, the speed of the car directly ahead, and the distance to the
car ahead. But human drivers have a finite reaction time . In other words, it
takes them a certain amount of time (usually about a second) to observe
what is going on around them and to press the gas pedal or the brake, as
appropriate. The standard “follow-the-leader” theory supposes that

( ∗ ) u n ( t + T ) = λ ( u n − 1 ( t ) − u n ( t )) ,

where t is time; T is the reaction time; u n is the position of the n thcar; and