Hello everybody. So we'll continue our course on simulation and modeling of natural phenomenon with the module on traffic simulation by cellular automata.

So we already learned in the previous module what's the basic idea of cellular automaton. So it's a discrete system, and you can use it to describe in a very simplified way, but still a relevant one, the motion of cars on a lane. So it turns out to be a [COUGH] one-dimensional two-state cell automata, which is part of those at Wolfram Classified. Turns out to be the rule numbered as 184 and what does it, it's a very simple idea. So it means that a black dot which represents a car can only move to the right in that case if the next cell Is empty. So, for instance this car here see that in front there's another car, so, at the next time stop, he will not move. But this one sees an empty spot so, it knows it can move and it will actually move. This one will move, this one will move, but this one will not, this one will not, and this one will move. So you only move if you have enough space between you and car in front of you, which is actually exactly what you do in real traffic. You never really get attached to the car in front of you, you first let it move and when the space is big enough, then you start moving yourself, okay? So we can capture this interesting behavior with just a simple cell route. Here we have an example where you have cells which are occupied or not occupied by a car, so to make the connection with real traffic a cell is typically about seven to ten meters wide. So of course this is very simple situation but out of this very simple model, you can build much more complex systems, where you can connect lanes to round about or rotaries okay. Which are all one D system, which are based the same rule except maybe at the intersection which you should give some priority and in that case it's clear that the car who is in the roundabout has priority. But it's still the nearest neighbor rule on maybe

a combination of regular and irregular topologies. So you can build a real

traffic network out of this simple construct and we'll see an example.

So, what you can also do is try to connect the observation, you can do on your cellular automata with quantity that traffic engineer are more used to. So, for instance, who can define the density of car which we call ro and it's just the number of cars in a given segment divided by the lengths of this segments. So it's a very simple and intuitive definition. We can also define the speed of this collection of cars by simply counting how many cars will be able to move

out of the number of cars that are present, okay? And if I go back to my simple example here, I can count that I have one, two, three, four cars moving out of seven. So the average velocity of this group of cars is [COUGH] four over seven.

Okay, that's this quantity here, and now what is the most interesting quantity for

traffic control is the flux, which is the product of the average density times the average speed of the car. And that's something you want to maximize, of course to increase the throughput of your traffic network. So, in this very simple traffic model, you can actually compute the relation between the car density and the flow, this product of density times velocity. And you get this interesting curve, this triangle which means that

when you are at low density your flow just increases with the number of car. Then you reach a maximum point and then you start decreasing, so we can understand that very easily. So when you have only a few cars, actually less than half of the segment is filled with cars, then you can manage to have all the car with the free space in front and then you all move at the maximum speed. So you can never been stopped so your velocity is one and your density is how many cars you have. So it's why you are going linearly until you have reached the fact that now half of the system is full of car and then some, if you keep increasing the density, some will be blocked by another car. So now in this spot of the flow diagram, the flow is limited by the number of free space. So each time you have a free space then you have a car that can occupy it. So then you will start decreasing slowly the flow until you reach zero or the segment is full and nobody has free space to move, okay? So this is a bit simplified diagram in the real world you have exactly this type of shape with a maximum somewhere, it might be asymmetric, it can be less clean than this one, but of course, the key point is to find this area. So if you have high traffic, you would like, of course, to not go over that point, because then the capacity of your road network will decrease.

So, I was telling you that we can consider a situation a bit more complex you can actually build a city like a Manhattan type of city, with a vertical and horizontal roads. So this is illustrated in these two picture here, okay, you see the street and the avenues. And of course you have a crossing, and the crossing you can

model it as a simple roundabout on these four cells with all the priority that goes with a normal roundabout. Or you can model it with traffic light, if you prefer, meaning that for instance, you leave all the car in the horizontal direction to go first. And then you alternate with the other car going in vertical direction, and so on and so on. So if you do so, you get, I will just comment this flow diagram here, you see that the flow diagram is the same shape as before. So according to car density the traffic flow first increases, because you put more car so there's more movement but if you reach some maximum density then you start decreasing the situation. So, in this example, we compare several situations. So, actually the curve here corresponds to situation where I manage my network of road with traffic light, so, if I use a roundabout, I can see from this simulation that I get better flow so I can have more cars going through the network. So it looks like a better management, and here this last curve is more for fun. It's just a strategy where you have a round about that you always do the opposite than the car that came before you. So if the car in front of you turn left, then you turn right, that means that you spread very nicely, the flow traffic across all the segments, and of course you will improve the flow. But this is of course a very artificial situation, usually you don't do the opposite of the car in front of you. You just go where you have to go, but here you see that even this simple model you can see that [COUGH] at least for the parameter we chose in the simulation, the roundabout of a flow, then the traffic light. And I wanna explain you that out of this simulation where you see the car moving on this street network. And what you observed rather quickly is that in front of the junction you have a line of car waiting, okay because here you have a traffic light. So then you blocked alternatively horizontal and vertical movement, and then as a consequence you create this line and of course this line concentrate the car in some area. So now if you take the other strategy, which is using the roundabout, you see that actually the same number of cars, they occupy the much more uniformly, the area and so they much less congestion.

So you can also wanna go to a more ambitious situation, which is modeling the traffic in the real city. For instance, we try this idea on the city of Geneva so the road network that you can see here was described as 1,066 junction, [COUGH] 3,145 segments, some of them we can see. And all this was divided in small cells of about 8 meters wide, so it gives that about 560,000 cells and on this network and this number of cars, 85,000 cars during the rush hour, okay? And for these cars we could know from [COUGH] traffic engineering in Geneva what their origin and destination so they could we know where they start from and where they wanna go to. And then what we decided was to test this model on different paths, okay? Some going into the center of town, some going around, where you have a bigger road and so on, and so on. And, of course, living in Geneva we could also test if the prediction are correct or not. And here you see the result of the simulation where on the left you see as a function of the duration of the rush hour, the proportion of cars that leave from their origin, of course this is something we put ad hoc in our model because the data was not known. But here you see are the function or your departure time, the duration of your trip, along this is trip number 2, and you see that unless you start here at the beginning of the rush hour, let's just start a bit later. Your time to go to your destination is very much constant, okay? So it means that it's an area where you don't have much traffic, if you start a bit earlier you see that there are some fluctuations. So this yellow part shows the standard deviation of the time to reach your destination, but it's still very small I would say. But you should take trip number one which goes through the center of the city you see that the things are really bad, that you can have a large fluctuation and basically it means that you cannot predict your time to destination better than about ten minutes, okay? So if you really want to be at work at a given time, well, it's going to be difficult unless you are out of the rush hour.

So, again, this is a simulation but you could easily verify, for instance, the average time of the trip by driving yourself in the city in the morning and see if the order of magnitude is correct. This turned out to be actually a very good estimate of the real situation.

So this is the end of our module on traffic model with cellular automata, and the next module we'll discuss more deeply complex systems in the context of cellular automata. So thank you for your attention. [MUSIC]

Cellular Automata Models for Traffic

Simulation and modeling of natural processes

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