Actually in most cities the important traffic routes now have synchronized light systems, particularly during rush hours. When the drivers travel at a certain speed, they are supposed to find a green light at each intersection. In heavy traffic this sequencing forces the cars into what traffic analysts call platoons. In the gaps between them the traffic on perpendicular streets crosses the main street or merges onto it. On a properly designed route the sequence of lights moves the traffic efficiently without unduly delaying the perpendicular traffic.

To examine a synchronized light system I studied the lights along Carnegie Street in Cleveland. To be sure, I could have found out much of what I wanted to know by consulting city officials, but it would not have been as much fun Besides, there is always the possibility that an independent analysis can come up with some useful ideas. You may want to try the same kind of analysis on your own local traffic system.

The lights along Carnegie Street are controlled by a preset electronic system rather than by a feedback system sensitive to the density of the traffic. The street is a major artery between downtown Cleveland and the eastern suburbs and is heavily traveled in the morning and afternoon rush hours. In the morning rush hour the westbound traffic has four lanes and the eastbound has two; in the afternoon rush hour the traffic is one-way eastbound. For the rest of the day the street has three-lane traffic in each direction.

Carnegie Street runs through an area that includes the Cleveland Play House and a variety of stores, homes and warehouses. Side streets, each three lanes wide, connect Carnegie with parallel streets that also carry rush-hour traffic. Since much traffic crosses between the major streets, the red lights on the side streets cannot stay on too long or the queues on them may grow to the point where they block the major streets.

I studied the system of traffic lights on Carnegie Street in the early stages of the afternoon rush hour. Beginning at the intersection of Carnegie and East 71st streets, I measured the distance from one intersection to the next as I walked along Carnegie in the direction of the traffic flow. At each intersection I timed the duration of the red, yellow and green lights. In addition as I proceeded I recorded the interval between the start of the green light at one intersection and the start of the green light at the next.

My tool for ascertaining times was a digital watch with a stopwatch feature and a particularly helpful split-time readout. I started the watch when the yellow light came on at an intersection. When the red or green light came on, I pushed the button for the split time. This function briefly held the display so that I could record the time in my notebook. The stopwatch was still running, however, and after about five seconds the display again recorded the passing seconds as before.

After many observations I averaged the readings for lights of each color in order to reduce the error arising from variations in the length of time it took me to react to a change of the light and to push the appropriate buttons on the watch. I also averaged my measurements of the time between the beginning of the green light at one intersection and that at the next intersection.

To measure the distances between intersections I counted paces between the stop lines at successive intersections. Later at home I walked at approximately the same pace across my yard. After measuring the distance with a meterstick I was able to convert my measurements of distance along Carnegie into units of meters. I walked the width of my yard rather than taking a single step in order to reduce the error that might result from variations in the length of my stride. Even so, I figure that the error in my measurements of the distance between the intersections is between 10 and 20 percent. Although surveying would be far more accurate, the added precision in my results would not be worth the extra effort.

The measurements are summarized in Figure 1. The intersection of Carnegie and East 71st is represented at the lower left. The flow of traffic along Carnegie is upward in the drawing of the street. The other intersections are laid out along the drawing.

The horizontal scale of the graph represents the time of the lights. For example, the green light at Carnegie and East 71st

is plotted as beginning at a time of five seconds. This is the bench mark from which that light and the ones at succeeding intersections were measured. I found that the green light at East 71st stays on for an average of 46.4 seconds. Hence the graph shows the onset of the yellow light at 51.4 seconds, the sum of 46.4 seconds and the bench mark of five seconds. The graph indicates similarly the start of the red light and the times for the next cycle of green, yellow and red.

The green light at Carnegie and East 77th, the next intersecting street, begins 29.5 seconds after the green begins at East 71st. Therefore the graph shows the start of the green phase at East 77th to be at 34.5 seconds, the sum of 29.5 seconds and the bench mark of five seconds. Since the green light there lasts for 48.6 seconds, the beginning of the yellow light is at 83.1 seconds on the graph The color cycles for the other intersections are similarly plotted with respect to the bench mark for the green light at East 71st.

During the afternoon rush hour platoons of cars form on Carnegie near East 71st or somewhat closer to the downtown section. Usually a platoon moves eastward at a nearly constant speed. A platoon leader could race from intersection to intersection, but it would be to little advantage: over an extended time and distance one cannot go faster than the sequence of green lights allows If the leader leaves one intersection at maximum acceleration and excessive speed, the car reaches the next intersection much too early for the green light.

The graph indicates the motion of a platoon leader traveling at constant speed through the system of six intersections. The leader begins to move when the light at East 71st turns green and then travels along Carnegie at a constant speed to avoid red lights. According to my measurements, the light cycles are not synchronized perfectly for the leader. If the driver is to pass through the intersection at East 79th just at the onset of the green light, he reaches East 82nd about five seconds after the green light there has come on. Perhaps the mismatch is part of the design. It would allow the traffic merging onto Carnegie from East 79th to clear the intersections at East 82nd and East 83rd before the next platoon arrives.

The graph also indicates two other hypothetical drivers moving through the system at constant speed. One of them passes through each intersection at approximately the middle of the green phase. The other driver passes through East 71st precisely at the end of the green phase and then races to pass through East 86th just as the green light there comes on. The speed of the platoon leader and the other two drivers can be measured from the graph: the speed in meters per second is the slope of the line representing each driver.

The platoon leader travels at 10.3 meters per second, which is about 23 miles per hour. The driver passing through each intersection at the middle of the green phase travels at 10.8 meters per second, which works out to approximately 24 m.p.h. The driver who races through the system in the least time moves at 26.6 meters per second, which is about 60 m.p.h. (The speed limit on Carnegie Street is 25 m.p.h.)

Compared with what happens in real traffic my law-abiding drivers are somewhat slow. When I drive through the rush-hour system on my way home every day, my platoon usually moves at about 28 m.p.h. Three factors contribute to the discrepancy between this speed and the speed compute from the graph. If the traffic flow is to be steady, a platoon leader must have a green light one or two seconds before arriving at an intersection, otherwise caution makes him slow down. The graph also ignores the time it takes the platoon leader to respond to the onset of the green light at East 71st. I figure that the reaction time is about a second.

The third factor is that the graph over looks the time required to accelerate a car from rest to the constant speed that takes the platoon through the system. When I am the platoon leader, I need about three seconds to reach cruising speed. These three factors effectively reduce the time a platoon leader has to travel between intersections. Therefore he can drive at a constant speed of somewhat more than 23 m.p.h. without having to stop at any intersection.

At times I have driven through the system after Carnegie has become one-way but before the volume of traffic has built up. If I pass through East 71st near the end of the green phase, I can go faster than 50 m.p.h. through several of the intersections but must stop for the red light at the intersection past East 86th. Almost every time I have exceeded the speed limit I have been passed by another driver. Hence it is possible to travel through the system at a constant speed of about 60 m.p.h.

During a rush hour platoons of cars move along Carnegie about every 75 seconds, which is the interval between successive green lights at East 71st. Although the lights along the route are not synchronized perfectly, the system works well enough so that the platoons are not too large for the distance between successive intersections. In fact, the green phase for most of the intersections could be decreased by 10 or 15 seconds without delaying the average platoon. The green phases are apparently kept as long as they are in case something goes wrong. The extra time is needed if an accident clogs one of the lanes of traffic or the rush-hour traffic is notably heavy.

Most of the traffic on Carnegie in the early stage of the afternoon rush hour tends to stay toward the right side of the street, although many drivers avoid the rightmost lane because of the possibility of coming on a car that is disabled or illegally parked. As the volume of traffic builds up, the lanes become almost uniformly filled.

Until that time a platoon can normally maneuver around an obstacle in one of the lanes. When a lane of traffic slows, drivers at the rear of the queue pull into other lanes. An obstacle tends to force the traffic into a more uniform distribution among the lanes. Trouble develops when the other lanes are already full Then the queue that forms behind an obstacle can disrupt the scheme designed to move platoons of cars through the system of lights.

The plan of the light system on Carnegie seems to assume that when a platoon leader approaches an intersection, the preceding platoon of cars will have moved on. If part of that platoon still blocks the way, the leader of the new one must slow down or stop. The entire timetable of lights can be thrown off. The delay may mean that part of the new platoon will not pass through the intersection before the next one arrives. If this happens, the problem becomes progressively worse.

The illustration in Figure 2 shows two platoons on Carnegie Street. The leading platoon is stopped at a red light at East 77th. The rear platoon has just got a green light at East 71st. First the leaders of the rear platoon begin to, move into the intersection at East 71st. Then the cars behind the leaders move. This start-up motion travels the length of the platoon as a wave. Finally the last cars of the rear platoon move forward.

Once the leading platoon gets a green light it too will have a start-up wave propagating to its last cars. If they begin to move before the leaders of the rear platoon arrive, the light system works fine and all the cars move through it without unnecessary delay. Suppose, however, the rear platoon arrives too soon. Since its members must stop, the queue of cars gets longer and may eventually block the intersection at East 71st during the next change of lights there. The situation then has the makings of a traffic jam.

I have been caught in several traffic jams on Carnegie. Once it took me almost two hours to get through the six intersections from East 71st to East 86th. A heavy snowfall had delayed the early platoons of cars traveling through the system. Parts of the platoons were halted by red lights. Because of the slow driving on the snowy streets, the queues at the intersections lengthened as platoons from the rear arrived before the preceding platoons had moved on. The density of cars quickly became maximum: cars were bumper to bumper for miles along Carnegie. Since the intersections were blocked, the traffic on the perpendicular streets also became jammed. If traffic flow is considered to be an example of hydrodynamics, the traffic on that miserable winter afternoon was frozen solid.

For the situation depicted in the illustration in Figure 2, what length of the leading platoon causes the rear one to be delayed? The critical length can be determined in terms of the distance between the intersections; it depends on the rate at which the start-up wave travels the length of the leading platoon. If the wave travels much slower than the speed at which the rear platoon approaches, even a short leading platoon can cause trouble.

To measure the speed of the start-up wave I stationed myself at intersections along Carnegie during a period of heavy

traffic. I chose intersections that normally have queues of 1 O cars or more. When the light for a queue turned green, I started my stopwatch. I stopped it when the last car in the queue began to move. I estimated the total distance between the front of the first car and the rear of the last. Dividing that distance by the time elapsed on the stopwatch, I estimated the speed of the wave. After many observations I found that the average speed was about five meters per second, or about half the designed cruise speed.

This finding means that the critical length for the leading platoon is only a small fraction of the total distance between East 71st and East 77th. If that platoon is more than about three car lengths long, the leaders of the rear platoon will be delayed. If both platoons are much longer than that, the rear platoon will not clear the intersection at East 71st before the next red light there.

Traffic jams along Carnegie might be avoided if the sequence of lights could be altered when the queues become too long for the normal rush-hour sequencing. The longer the leading platoon at East 77th is, the less head start the rear platoon should be given. If the length of the leading platoon is about a third of the distance between the two intersections, the green lights should come on simultaneously. If the leading platoon is even longer, the light at East 77th should come on before the one at East 71st, an order that is the reverse of the normal rush-hour sequencing. The head start given the leading platoon then allows the start-up wave to reach the last cars before the leaders of the rear platoon arrive.

In Figure 5 is an equation bearing on the platoon length at which the normal sequence must be changed. The length of the platoon (x) is expressed as a fraction of the distance between intersections (d). It depends on the speed of the start-up wave (v₁) and the normal cruise speed (v) for the system. When the platoons are shorter than the switchover value, the rear platoon should get the head start. When they are longer, the leading platoon should be given the head start. If the platoons are as long as the switchover value, they should get simultaneous green lights.

Some systems of traffic lights can be changed in sequence when the queues become too long. The change could be preprogrammed into the system if long queues can be expected. In some systems the light sequence is affected by devices laid in the roadway to sense traffic flow. If Carnegie had such a system, my fellow prisoners and I might have been spared the ordeal of traveling only five blocks in two hours.

In the late morning and early afternoon Carnegie has three lanes of traffic in both directions. The sequencing of the lights should be different from what it is in the afternoon rush hour so that the westbound traffic will not be stopped too many times. I repeated my measurements of the light cycles at about noon. The results are given in Figure 3, which shows the green light at East 71st starting at a time of 10 seconds. Thus 10 seconds is the bench mark for the color cycles of the intersections.

At each intersection the duration of the three colors of light is the same as it is during the rush hour, but the time between the onset of green lights at successive intersections is not the same. For example, during the rush hour the green light at East 79th comes on 15.4 seconds before the green light at East 82nd. The delay allows a platoon enough time to reach East 82nd before the green appears. Far two-way traffic the system must be synchronized to minimize the delay for both directions of flow without forcing the perpendicular traffic to wait too long. At noon the green at East 82nd and East 83rd is already on when the green starts at East 79th. This system allows the-westward traffic to travel uninterrupted through the intersections.

The illustration also shows "through bands," which are often put in such a graph of distance v. time. Here a through band indicates how a platoon traveling at a constant speed can pass through a system of intersections without pause. Although in light traffic cars could proceed at various speeds, the through bands are instructive to someone deciding how to synchronize the lights for most of the traffic. The illustration has two sets of through bands, one for eastward flow (the bands running upward to the right) and another for westward flow (the bands running downward to the right).

A through band consists of two parallel lines drawn as far apart as the graph for a traffic route allows. One of the eastward through bands in the illustration begins with the onset of the green light at East 71st. A straight line passes through the green phases of the other intersections up through East 86th. (The line is straight because it represents traffic moving at a constant speed.) Since the traffic is to flow along' the route as quickly as it can without breaking the speed limit, the-line on the graph should be as steep as possible. The line also must avoid any of the red phases because it represents uninterrupted traffic. Hence it skims across the end of the red phases at East 82nd and East 83rd.

The right side of the through band is parallel to the left side because it represents the rear cars of the platoon, all of whose members are traveling at the same speed. This line also is drawn to avoid any of the red phases. In the illustration the right side of the eastward through band skims the onset of red at East 86th.

Suppose the eastward traffic along Carnegie at noon consists only of platoons of cars. The cruise speed of a platoon can be computed from the slope of the lines of the eastward through bands. If the traffic is to be uninterrupted, the platoon must begin at the onset of the green at East 71st and travel through the system of lights at a speed of 9.6 meters per second (equivalent to 21.5 m.p.h.).

How much later the rear of the platoon must pass through the intersection at East 71st is indicated by the right side of the through band. Apparently the rear must be no later than about 26 seconds after the green begins at that intersection. If a car in the platoon passes through the intersection later in the green phase there, it will be stopped by a red light at East 86th if not sooner. The through band during a rush hour is much wider, encompassing the entire green phase at East 71st. Thus when the Carnegie light system changes to the sequence for rush-hour traffic, the platoon speeds are not altered but the platoons can be much longer.

The distance between adjacent platoons can also be determined from the illustration. Start at the left side of one eastward through band, say at the bench mark for the onset of the green at East 71 St. From the vertical axis read the distance to the left side of the through band between East 83rd and East 86th for that same time. The reading (about 700 meters) is the distance between the leading cars of the successive platoons. The distance between the last cars of the leading platoon and the first Gars of the platoon beginning at East 71st is about 500 meters.

The illustration also includes information about the westward through bands. The left side of a band represents the travel of the first cars of a platoon. The right side represents the travel of the platoon's last cars. The westward and eastward through bands are approximately the same size, indicating that the system of lights is designed to pass traffic equally well in those directions at this time of day.

What is a suitable duration for the yellow light? It must be short enough to not delay the traffic unduly and long enough to allow a driver either to stop properly or to pass through the intersection before the red light comes on. The appropriate duration depends partly on the speed limit of the street, which determines a driver's ability to avoid being in the intersection when the red comes on. The higher the speed limit is, the longer the yellow light should be. This relation is sometimes ignored in setting the phases of a traffic light.

Assume that a yellow light lasts for 2.5 seconds, which is about the shortest duration I have found in Cleveland. Assume further that when the yellow first comes on, the driver immediately responds by braking to a stop. The minimum distance required for the stop depends on two factors: the car's initial speed (vO) and the acceleration (a) provided by braking. (A physicist refers to any change in velocity as an acceleration; in common parlance the slowing of a car is called deceleration.) The stopping distance is computed as being the speed squared divided by twice the acceleration.

The acceleration of a braking-car can vary greatly because it depends on the weather, the type of pavement, the condition of the tires and brakes and the skill of the driver. Usually the acceleration is initially between .9 meter and 3.1 meters per second per second. In the last stage of braking the acceleration is about 3.7 meters per second per second. In an emergency the acceleration could be as high as 6.1 meters per second per second, but braking at that rate is uncomfortable and unnerving.

Assume an acceleration of 3.1 meters per second per second. Suppose the driver is traveling at 40 m.p.h., which is 17.9 meters per second. The minimum stopping distance is 52 meters. If the driver is to stop before entering the intersection, the car must be no closer to the intersection than that distance. If it is closer, the driver has no reasonable choice except to continue through the intersection with no attempt at braking. Perhaps he should even accelerate in order to avoid being in the intersection when the red light comes on.

How successful can the driver be in such an acceleration? What he can do depends not only on the car's pickup but also on the initial distance of the car from the intersection. If the intersection is too far away, the driver cannot reach the far side before the red light comes on. The formula in Figure 5 yields the maximum distance the car must be from the intersection if the attempt to get through is to be successful. The formula encompasses the duration of the yellow light (t) and the acceleration (a) and initial speed (v_o) of the car. It also requires the width of the intersection (s) because the driver must reach the far side while the yellow light is still on.

A typical car can accelerate at from one meter to 2.2 meters per second per second. A sports car can probably accelerate at twice that rate. Consider a car with an acceleration- of 2.2 meters per second per second. Assume that the driver responds instantly to the onset of the yellow light. Take the duration of the yellow light as being 2.5 seconds and the width of the intersection as being 10 meters. If the car is initially traveling at 40 m.p.h., the maximum distance between it and the intersection-should be about 42 meters; if it is more than that, the car will be in the intersection during part of the red light.

The driver may decide to go through the yellow light without accelerating beyond the speed limit. If the initial speed of 40 m.p.h. is maintained, the car should be no farther than 35 meters from the intersection.

I have chosen the above numbers in order to make a point. If the driver is to stop successfully, the car must be no closer than 52 meters from the intersection. If he is to race through at maximum acceleration, the car must be no farther than 42 meters from the intersection. Between those limits what should he do? In principle neither strategy will succeed. The dangerous region between the two limits is larger with higher initial speeds far the car. Under the same assumptions as before except for an initial speed of 55 m.p.h., the danger region is 39 meters long. If the driver decides not to accelerate, the danger region is 46 meters long.

I once drove through such an intersection on a highway that had a speed limit of 55 m.p.h. I found myself facing a yellow light with neither the space to stop nor the acceleration to race through before the red light came on. I was saved from the possibility of a collision only by a delay in the light system: the green light for the perpendicular traffic came on about a second or so after the yellow light ended.

This month's bibliography cites two articles in which teachers of physics have analyzed the dilemma of a short yellow light. You might want to examine yellow lights in your neighborhood. Please keep in mind that the values I have used for the accelerations of a car are assumed. The braking acceleration is particularly open to question. It depends on the coefficient of friction between the tires and the street. If the street is covered with ice, snow, rain or anything else that reduces friction, the coefficient can be much lower than normal. The distance required for stopping the car is then much longer. Therefore the danger region is-also longer and the possibility of a collision between a car and a vehicle in the perpendicular traffic is greater.

Bibliography

THE STOP LIGHT DILEMMA. Howard S. Seifert in American Journal of Physics, Vol. 30, pages 216-218; 1962.

TO STOP OR NOT TO STOP-KINEMATICS AND THE YELLOW LIGHT. J. Fred Watts in The Physics Teacher, Vol. 19, No. 2, pages 114-115; February, 1981.

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