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Details on The Mathematical Model Used

Read about the Competition this simulation was originally designed for

Each road has two lanes, one each direction. The lanes are each 4 metres wide (so each road is 8 metres wide). All cars travel straight ahead; none try to turn. For convenience, we've numbered the lanes from 1 to 10 and labelled the lights A through F.

We assume traffic moves according to a simple mathematical
model (follow the link for more information). The relevant
parameters are the number of cars per minute *d*[1], . . . , *d*[10]
trying to use each lane, the speed limit *c*, two numbers *l* and *L*
that control how cars tend to space themselves out, and the timings
of the traffic lights. In this simulation, you can set the values of
any or all of these parameters and watch what happens. As well as a
graphical display, you'll see what the average speed of the cars is
as a percentage of the speed limit: a measure of how effective your
light timings are at keeping the traffic moving!

You can run the simulation in any one of the following ways:

- Work with a pre-defined traffic density scenario. You can program the lights any way you like. See if you can keep the traffic flowing smoothly and minimize the waiting time of the cars. The computer will keep track of high scores, so you can compare your efforts against those of others! This form of the simulation goes hand in hand with Part III of the competition this simulation was designed to accompany.
- Choose a pre-programmed scenario that lets you watch what happens as a line of cars starts moving when a light turns green. This form of the simulation goes hand in hand with Part II of the competition, which is a challenge to figure out mathematically how long it takes the line to get moving and whether or not slow acceleration is to blame when you're kept waiting at the back of the line. With the simulation, you can test out your mathematical predictions for different traffic behaviour.
- Completely customize the simulation. You can decide on the traffic patterns as well as the light timings, and watch what happens. This can be a useful test of your reasoning in Part I of the competition.

In each of the scenarios, the basic traffic behaviour is the same:
*c*=40, *l*=4, and *L*=20 (see the
mathematical model for what these mean),
and there's no limit to how fast cars can accelerate. The difference
is in the numbers *d*[1], . . . , *d*[10]: the number of cars per minute
trying to use each lane.

Scenario 1.(d[1], . . . ,d[10) = (10, 5, 5, 10, 15, 10, 8, 8, 5, 10)Scenario 2.(d[1], . . . ,d[10) = (5, 15, 2, 15, 2, 10, 1, 8, 10, 5)Scenario 3.(d[1], . . . ,d[10]) = (7, 20, 15, 15, 12, 6, 10, 10, 2, 12)

You may also want to make the lights turn
green at different times, so that a lane with heavy traffic can
encounter all green lights. The scenario begins just as light A
starts its east-west green cycle.
To stagger the lights, you can specify how far into the scenario
lights B through F begin their east-west green cycle. We'll call these numbers
*B*0 through *F*0. For example, setting *B*0 = 15 means that light B will
begin an east-west green cycle 15 seconds after the start of the scenario.
Setting *B*0 = -15 means that light B started an east-west green cycle
15 seconds before the start of the scenario.

Choose the scenario you want from the popup menu above, and enter the
values for *Aew*, *Ans*, . . . , *Few*,
*Fns*, *B*0, . . . , *F*0 that you think will lead to the smallest average
waiting time (greatest average speed) for cars
over a 10-minute period. (You may want to refer back to the
map to see where each of the ten lanes is located
relative to the six lights).
Then press "Start Simulation". You'll see empty
roads at the start of the ten-minute period, and you can advance the
time by amounts you choose. Once you've advanced the time all the way to
ten minutes, you will see what the average speed of the cars was under
your timings and how that compares to the previous high score. Can you
set a new high score?

In this simulation, you can choose the values for the three basic
parameters *c*, *l*, and *L* in the mathematical
model we are assuming the traffic follows. You can also
decide how many seconds *S* it takes for the lead car to accelerate to
full speed; leave this at zero if you want the lead car to be able to
accelerate infinitely fast. You'll probably be surprised at how long
it takes the line to get moving even when this number is set to zero
(and less inclined to blame the reaction time of the cars in front of
you next time you're kept waiting!)

Once you've chosen value for *c*, *l*, *L*, and *S* (or
you can just leave them at their defaults if you like),
press "Start
Simulation". You'll see empty roads at the start of the simulation.
Then select "advance simulation by 120 seconds"; during this
period, eastbound cars will line up at red light B, and red light B
will just have turned green. Then keep advancing the simulation at
1-second intervals, seeing how long it takes cars far back in the line
to get moving.

This part of the simulation was designed to accompany Part II of the competition, which is a challenge to find mathematical formulas for how long it takes cars far back in the line to get going. You can use the simulation to see how accurate your mathematical predictions are.

This page last updated: April 12, 1997

Original Web Site Creator / Mathematical Content Developer: Philip Spencer

Current Network Coordinator and Contact Person: Joel Chan - mathnet@math.toronto.edu