0:30

So we already learned in the previous module what's the basic idea of

Â cellular automaton.

Â So it's a discrete system, and you can use it to describe in a very simplified way,

Â but still a relevant one, the motion of cars on a lane.

Â So it turns out to be a [COUGH] one-dimensional two-state cell automata,

Â which is part of those at Wolfram Classified.

Â Turns out to be the rule numbered as 184 and what does it, it's a very simple idea.

Â So it means that a black dot which represents a car can only move

Â to the right in that case if the next cell Is empty.

Â So, for instance this car here see that in front there's another car,

Â so, at the next time stop, he will not move.

Â But this one sees an empty spot so, it knows it can move and

Â it will actually move.

Â This one will move, this one will move, but

Â this one will not, this one will not, and this one will move.

Â So you only move if you have enough space between you and car in front of you,

Â which is actually exactly what you do in real traffic.

Â You never really get attached to the car in front of you, you first let it move and

Â when the space is big enough, then you start moving yourself, okay?

Â So we can capture this interesting behavior with just a simple cell route.

Â Here we have an example where you have cells which are occupied or

Â not occupied by a car, so to make the connection with real traffic a cell is

Â typically about seven to ten meters wide.

Â So of course this is very simple situation but

Â out of this very simple model, you can build much more complex systems,

Â where you can connect lanes to round about or rotaries okay.

Â Which are all one D system, which are based the same rule

Â except maybe at the intersection which you should give some priority and

Â in that case it's clear that the car who is in the roundabout has priority.

Â But it's still the nearest neighbor rule on maybe

Â 3:07

So, what you can also do is try to connect the observation,

Â you can do on your cellular automata with

Â quantity that traffic engineer are more used to.

Â So, for instance, who can define the density of car which we call ro and it's

Â just the number of cars in a given segment divided by the lengths of this segments.

Â So it's a very simple and intuitive definition.

Â We can also define the speed of this collection of cars by

Â simply counting how many cars will be able to move

Â 4:13

traffic control is the flux, which is the product of

Â the average density times the average speed of the car.

Â And that's something you want to maximize,

Â of course to increase the throughput of your traffic network.

Â So, in this very simple traffic model,

Â you can actually compute the relation between the car density and

Â the flow, this product of density times velocity.

Â And you get this interesting curve, this triangle which means that

Â 4:52

when you are at low density your flow just increases with the number of car.

Â Then you reach a maximum point and then you start decreasing, so

Â we can understand that very easily.

Â So when you have only a few cars, actually less than half of the segment is filled

Â with cars, then you can manage to have all the car with the free space in front and

Â then you all move at the maximum speed.

Â So you can never been stopped so your velocity is one and

Â your density is how many cars you have.

Â So it's why you are going linearly until you have reached the fact that now half

Â of the system is full of car and then some,

Â if you keep increasing the density, some will be blocked by another car.

Â So now in this spot of the flow diagram,

Â the flow is limited by the number of free space.

Â So each time you have a free space then you have a car that can occupy it.

Â So then you will start decreasing slowly the flow until you reach zero or

Â the segment is full and nobody has free space to move, okay?

Â So this is a bit simplified diagram in the real world you have exactly this

Â type of shape with a maximum somewhere, it might be asymmetric, it can be less clean

Â than this one, but of course, the key point is to find this area.

Â So if you have high traffic, you would like, of course, to not go over that

Â point, because then the capacity of your road network will decrease.

Â 6:21

So, I was telling you that we can consider a situation a bit more complex

Â you can actually build a city like a Manhattan type of city,

Â with a vertical and horizontal roads.

Â So this is illustrated in these two picture here, okay,

Â you see the street and the avenues.

Â And of course you have a crossing, and the crossing you can

Â 6:48

model it as a simple roundabout on these four cells

Â with all the priority that goes with a normal roundabout.

Â Or you can model it with traffic light, if you prefer, meaning that for

Â instance, you leave all the car in the horizontal direction to go first.

Â And then you alternate with the other car going in vertical direction, and so

Â on and so on.

Â So if you do so, you get, I will just comment this flow diagram here,

Â you see that the flow diagram is the same shape as before.

Â So according to car density the traffic flow first increases,

Â because you put more car so there's more movement but

Â if you reach some maximum density then you start decreasing the situation.

Â So, in this example, we compare several situations.

Â So, actually the curve here corresponds to

Â situation where I manage my network of road with

Â traffic light, so, if I use a roundabout,

Â I can see from this simulation that I get better flow so

Â I can have more cars going through the network.

Â So it looks like a better management, and here this last curve is more for fun.

Â It's just a strategy where you have a round about

Â that you always do the opposite than the car that came before you.

Â So if the car in front of you turn left,

Â then you turn right, that means that you spread very nicely, the flow traffic

Â across all the segments, and of course you will improve the flow.

Â But this is of course a very artificial situation,

Â usually you don't do the opposite of the car in front of you.

Â You just go where you have to go, but here you see that even this simple model you

Â can see that [COUGH] at least for the parameter we chose in the simulation,

Â the roundabout of a flow, then the traffic light.

Â And I wanna explain you that out of this simulation

Â where you see the car moving on this street network.

Â And what you observed rather quickly is that in front of the junction

Â you have a line of car waiting, okay because here you have a traffic light.

Â So then you blocked alternatively horizontal and vertical movement, and

Â then as a consequence you create this line and

Â of course this line concentrate the car in some area.

Â So now if you take the other strategy, which is using the roundabout,

Â you see that actually the same number of cars,

Â they occupy the much more uniformly, the area and

Â so they much less congestion.

Â 10:06

So you can also wanna go to a more ambitious situation,

Â which is modeling the traffic in the real city.

Â For instance, we try this idea on the city of Geneva so

Â the road network that you can see here was described as 1,066 junction,

Â [COUGH] 3,145 segments, some of them we can see.

Â And all this was divided in small cells of about 8 meters wide,

Â so it gives that about 560,000 cells and

Â on this network and this number of cars,

Â 85,000 cars during the rush hour, okay?

Â And for these cars we could know from [COUGH] traffic engineering in Geneva

Â what their origin and destination so

Â they could we know where they start from and where they wanna go to.

Â And then what we decided was to test this model on different paths, okay?

Â Some going into the center of town, some going around,

Â where you have a bigger road and so on, and so on.

Â And, of course,

Â living in Geneva we could also test if the prediction are correct or not.

Â And here you see the result of the simulation where

Â on the left you see as a function of the duration of the rush hour,

Â the proportion of cars that leave from their origin, of

Â course this is something we put ad hoc in our model because the data was not known.

Â But here you see are the function or your departure time, the duration of your trip,

Â along this is trip number 2, and you see that unless

Â you start here at the beginning of the rush hour, let's just start a bit later.

Â Your time to go to your destination is very much constant, okay?

Â So it means that it's an area where you don't have much traffic,

Â if you start a bit earlier you see that there are some fluctuations.

Â So this yellow part shows the standard deviation of the time to reach your

Â destination, but it's still very small I would say.

Â But you should take trip number one which goes through the center of the city you

Â see that the things are really bad, that you can have a large fluctuation and

Â basically it means that you cannot predict your

Â time to destination better than about ten minutes, okay?

Â So if you really want to be at work at a given time, well,

Â it's going to be difficult unless you are out of the rush hour.

Â