Timing Traffic Lights |
TASK. A stretch of a suburban road lined with shopping plazas carries heavy commuter traffic. The road has 15 traffic signals, unevenly spaced, at the intersections with cross streets and mall entrances. Figure out how to time the lights in order to maximize the flow of commuter traffic.
COMMENTARY. One approach to this task uses a very powerful geometric technique to model the situation. The technique combines the information for cars (given by two-dimensional position versus time graphs) with the information given for traffic lights (given by one-dimensional time graphs showing how long they are red, green, and yellow). Similar diagrams are used in the planning of highways, railroad schedules, and canals. This task and the techniques in the solution below provide opportunities for geometric thinking, reasoning from graphs, and connections between slope and velocity.
MATHEMATICAL ANALYSIS. Rather than a complete solution, which would require precise data about placement of lights and about typical traffic flow and consideration of many alternatives, this section discusses a solution for a simple case. This simple case can provide understanding of the geometric model and insights into the general case.1
Suppose the road travels north-south and has only three lights. Label the lights A, B, and C, and suppose that lights B and C are 0.25 and 0.38 miles north of light A, respectively. First consider only light A. Suppose that it follows a 1-minute cycle--green for 30 seconds, yellow for 5 seconds, and red for 25 seconds. The pattern of the light can be represented graphically on a line (Figure 1).
Figure 1: The green-yellow-red cycle of light A |
||
|
||
|
|
||
To show the position of cars over time, the distance from light A may be represented on a vertical axis with time represented on a horizontal axis as in Figure 1. For smooth traffic flow, cars going a moderate speed should be able to go through all three lights without stopping. Consider first only cars traveling north and assume that all cars travel at a constant speed of 30 mph. Figure 2 shows possible positions of 12 north-bound cars over the first 120 seconds after the beginning of a cycle of light A. The first car goes through light A 5 seconds after it turns green and the sixth goes through at 30 seconds, just as the light turns yellow. The white space in the middle of the graph shows that light A, when it is red, causes "spaces" in the traffic flow, if traffic entering from other roads is ignored. In this representation, the slope of a line is equal to the speed of the car it represents. Here, the slopes are 0.5 miles/60 seconds, or 30 mph. The lines are parallel because all of the cars are assumed to be traveling at the same speed.
Figure 2: Cars going north through light A when it is green |
||
|
||
|
|
||
Lights B and C are represented as horizontal lines in Figure 3. The lines have been partially dotted to show "windows"--time intervals during which the light should be green to allow unimpeded flow of the 12 cars from Figure 2. Each light should turn green shortly before the first car arrives at the light so that the car will not need to slow down before reaching the light. The diagram shows that light B should be green from about 30 until 60 seconds and again from 90 until 120 seconds. Allowing for a 5-second yellow light, this implies a 25-second red light between the green intervals. Thus, light B should follow the same cycle as light A, but the cycle is shifted in phase by 30 seconds. Light C should be green between about 45 and 75 seconds and then again at 105 seconds. Thus, this light should also follow the same cycle, but phase shifted by 45 seconds.
Figure 3: Determining when lights B and C should be green for north-bound traffic |
||
|
||
|
|
||
This is a sufficient one-directional solution for three lights. In fact, with this sort of phase-shifting of the same light pattern, any number of lights could be added to this road and still allow for unimpeded flow of traffic in the north-bound direction. Lights on one-way streets are often timed in a manner similar to this, although the expected speed may be different from the 30 mph used here.
Allowing for similarly smooth flow of traffic in two directions is much more difficult. Figure 4 shows both north-bound traffic as in Figure 2 and also some south-bound traffic that would go through light A when it is green. Note that if light B uses the same green intervals as in Figure 3, it will allow both north- and south-bound traffic to flow unimpeded. In order to allow all the north- and south-bound traffic to flow through light C, however, it must be green all the time, which would not be practical.
Figure 4: Determining when lights B and C should be green for two-way traffic |
||
|
||
|
|
||
One solution to this dilemma is to give priority to the north-bound traffic and time the lights as indicated in Figure 3. Then north-bound traffic can flow unimpeded, but the south-bound traffic will always have to stop at at least one of the lights. Alternatively, priority could be given to the south-bound traffic.
Another approach is to change the period of the light cycles. If the lights are on a 90-second rather than a 60-second cycle, then light C can be timed to accommodate both north- and south-bound traffic, as shown in Figure 5. Now, however, light B needs an excessively long green interval to accommodate both directions of traffic.
Figure 5: Two-way traffic, assuming a 90 second light cycle2 |
||
|
||
|
|
||
Neither a 60-second light cycle (Figure 4) nor a 90-second light cycle (Figure 5) can accommodate traffic in both directions at both lights B and C. Figures 4 and 5 do, however, suggest a compromise: a 75-second light cycle.
Figure 6 shows that with a 75 second light cycle, if both lights B and C are green from about 30 to about 75 seconds after the beginning of light A's cycle, most of the north- and south-bound traffic will flow unimpeded.
Figure 6: Two-way traffic, assuming a 75 second light cycle |
||
|
||
|
|
||
The above discussion has not taken into account the east-west traffic. How should these requirements be factored in? If some of the east-west roads are major roads with much traffic, these requirements should be considered in any decisions about timing the lights. On the other hand, if they are minor roads, it is probably safe to assume that traffic from these roads will not benefit from long red signals on the north-south road.
The discussion above began with positions of the lights and tried to find a reasonable way to time the lights for traffic going 30 mph. What about the reverse? Given the timing of some traffic lights, what are the speeds that must be traveled so that cars do not have to stop? As shown in Figure 7, cars can be given positions by tilting the lines to fit the "windows" represented by green lights. The trick is to find slopes that correspond to appropriate driving speeds.
Under the scenario represented by the solid line in Figure 7, it takes the car 60 seconds to travel from light A to light B. If B A represents the distance between the two lights, then the speed is (B A)/60 feet per second (fps). If the distance between A and B is 2400 feet, then the speed is 40 fps, which is about 27 mph, and that's fine. But suppose the distance between A and B is 6000 feet. Then the speed would be 100 fps, or about 70 mph, and that's not sensible. In this case, the dotted line in Figure 7 might represent a more realistic possibility. Passing through the second "window" in the line representing light B corresponds to allowing two cycles of the lights to get from A to B. This doubles the time that it takes for the car to travel from light A to light B and cuts the speed in half to about 35 mph, which might be acceptable. In general, if N is a positive integer representing a number of cycles of light B, then the speed will be (B A )/(60N) feet per second. Just pick N to get the speed into a sensible range.
What if the road carries heavy commuter traffic northbound in the mornings and southbound in the evenings? Would it make sense to have different timings of the lights for the different rush hours? Does it make sense in very heavy traffic to slow down the expected speed of travel? What needs to happen to the timing of the lights to accommodate the slower speeds?
Much of the discussion has aimed to allow cars to pass through many lights without stopping. What about maximizing the capacity of the road? Will the solution be the same or different?
This sort of diagram is used in other situations, such as train scheduling (see Figure 8). For a one-track railroad line, or a canal like the Suez Canal which is one-way much of the way, you locate sidings (bypasses) on the distance axis where traffic in opposing directions can pass, and schedule trains (convoys) to get there at the right times. The planners of traffic for the Suez Canal use a similar diagram, but with the time and distance axes interchanged for historical reasons.
1
For other ideas and approaches, see Walker (1983).
2
A 90-second light cycle would likely include a longer green portion, allowing for more than 6 cars, at 5-second intervals, to pass through the green light. Nonetheless, Figure 5 includes only 6 cars through each cycle to allow for easier comparison with the other figures.
Figure 7: Two scenarios for a car traveling from light A to light B
Figure 8: Marey's train schedule
Source: Tufte, 1983, p. 115.
References
NOTES
