Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

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Unit 7

Making Sense of Randomness

7.8 Modern Probability


Most of the ideas that we have discussed so far in this unit were first developed in the 17th and 18th centuries. As in all of mathematics, there has been continuous further development of these ideas since then. (In fact, we’ve concentrated here on discrete probability and have not really said much regarding continuous probability, the situation in which there is a continuous possible range of outcomes, as with the height of an individual). An exciting case in point is in the modeling of theoretical traffic flows.

The Biham-Middleton-Levine (BML) Traffic Model, first proposed in 1992, provides a useful model to study how probability affects traffic flow and phase transitions, such as the transformation of liquid water into ice. To get an idea of how this model works, let’s imagine an ideal grid of city blocks.

City Blocks

To make things easier, let’s assume that the grid extends to infinity in all directions. That way we don’t have to worry about any kind of boundary conditions or effects. Let’s fill our grid with commuter cars, red ones trying to go east and blue ones trying to go north.

City Blocks

To simplify things further, assume that cars move only one space at a time and are allowed to move only as follows: Every odd second, red cars get to move if the space immediately east of them is vacant, whereas blue cars get to move every even second only if the space immediately north of them is vacant. This process goes on indefinitely.

To determine the starting configuration of cars, we can select a probability, p, that assigns whether or not a space is occupied by a car. We will be interested in the different behaviors that are associated with different values of p. If a particular space ends up being populated with a car, the car’s color, and therefore its directional orientation, is determined by a method equivalent to flipping a coin.

After the grid is populated, the simulation runs. After a period of time, patterns and structure begin to emerge.

free flow Raissa DíSouza, FREE FLOW R< R-C (1/2005). Courtesy of Raissa DíSouza.

Some initial probabilities lead to continuous flow. Cars can move freely forever. In this picture, both red and blue cars are able to move throughout the grid. If the cars were water molecules, these results would correspond to the liquid state.

fully Raissa DíSouza, FULLY JAMMED R> R-C (1/2005). Courtesy of Raissa DíSouza.

Other initial probabilities lead to traffic gridlock. Movement becomes impossible. Notice how the red and blue cars are stalemated along the center diagonal of the grid. In the water analogy, this would be ice. Note that the parts in the corners are due to the boundary conditions of the grid, so that if a car leaves the left part of the screen, it returns on the right side and vice versa. The same boundary conditions apply to the top and the bottom of the grid. Recalling a concept from our previous unit on topology, this is a flat torus.

The BML model is perplexing because, while at low initial densities traffic flows freely forever and at high initial densities traffic jams up rather quickly, the density at which this transition occurs is not known. Also, there are intermediate states of free flow mixed with periodic jams, depending on the initial population density. As of this writing, there is no detailed mathematical explanation for these behaviors, making this an area for continued exploration.

Rigorous attempts to address the issues involved in the BML Traffic Model and similar models play a huge role in modern probability. Mathematicians are truly just beginning to find ways of dealing with models that correspond to our physical world in meaningful ways. These sorts of results correspond to some of the deepest and most beautiful work in modern mathematics.

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