Have any of you heard of the electronics engineer who is being fined $500 for criticizing the timing of yellow lights? Yes. It’s true. The Oregon State Board of Examiners for Engineering and Land Surveying has punished Mats Järlström for presenting his analysis of the timing of yellow lights while calling himself an engineer.
I’m not going to get into the legal details; those are being discussed elsewhere, including at The Volokh Conspiracy. The legal issues have also been discussed at motherboard, doubltess other places by now.
Instead of focusing on the legal issues, I’m going to discuss timing of yellow lights and Järlström analysis. Because I was interested in this, I asked Eugene Volokh if he has any documents desribing Järlström. He provided me with a one page submission which describes Järlström, but being a court submission rather than a full engineering document it is lacking in details a person might need to be absolutely certain of each item. Specifically: when reading it, I was not sure about the precise definition of a variable indicating a deceleration rate in Järlström analysis. Owing to the court case, Järlström currently finds himself in legal jeopardy if he discusses his analysis.
So I read the document, and then did my own analysis trying to account for the features he was considering in his. The main features of his analysis are to extend that of 1959 analysis but to consider the possibility a car must slow down when approaching the intersection.
The main conclusion: Järlström is right if we consider the possibility the car must slow down to some safespeed when entering the intersection, yellow lights are mistimed.
For most people: that’s enough. But for those of us who like to show students that the topic they are learning is useful: this is a 1-D kinematics problem. In fact, it is an application of the first topic covered in high school physics. While the full analysis overly for a homework problem, the skill required to do the analysis are covered around September or October in traditional physics classes. A similar problem appears in Chapter 2 of the Sixth Edition of Giancoli Physics (see problem 32, page 40.) That problem is easier for students because the question contains numerical values, and the car is not decelerating when approaching the intersection. But it applies all the same concepts.
As I know do tutor high school students, let’s first consider a driver approaching an intersection with a stop light. The general situation is illustrated below. The intersection has some width w, the car, shown by the bold black outline, has some length d. Suppose when the car is in the position shown, it is moving with speed $latex v_a $ and, owing to the driver’s desire to turn into the driveway of the grocery store immediately after the light, the car is decelerating at a rate $latex a_{s} $. Then the light suddenly turns yellow.
Local traffic law can vary, but let us suppose for this case, the law dictates that
- If the car can stop, it must
- If the car enters the intersection, it must clear the intersection before the light turns red and
- If the car enters to clear the intersection and fails to clear, the driver will be given a ticket.
It would be unfair to put drivers in the situation where those that cannot stop also cannot clear the intersection. So the engineering question is then: How long should the yellow light last before it can turn red? In particular: we might want to know the minimum time the yellow light should be on.
In these problems the logic is that one first determine the cases when a car cannot stop without entering the intersection. Once those are determined, one determines the maximum time it takes those cars to clear the intersection.
It turns out that the key situation to analyze is that of the car whose minimum stopping distance equals the distance to the stopping light, and then find the time it would take that car to clear the intersection. All cars further from the stopping distance can stop and should; all cars closer than the stopping distance will clear the light in a shorter amount of time.
This stopping distance is called the “critical distance” and will be denoted $latex x_c $.
Stop analysis: Critical Stopping Distance
Naturally, the first step is to find the critical distance traveled. For this case, I will only consider cars moving toward the light and decelerating when the light changes. I will also assume that drivers, being human, require a finite reaction time $latex t_1 $ to respond to the change in the light. If so, the car’s speed could be described as following the black trace during the time from 0 when the light turns yellow to $latex t_1 $. The initial speed is $latex v_a $, this slows to $latex v_p $ at $latex t_1 $. I am also only going to consider cars traveling on level roads (the extension to inclines is straightforward.)
To stop in the minimum distance (and time) the driver must hit the brakes and decelerate at the maximum possible value, which is indicated as $latex a_{max} $. The distance traveled is the area under the (v,t) curve and the time axis. The car will travel Δx during the first $latex t_1 $ (dark grey) and an additional distance indicated by the hatched grey region between $latex t_1 $ and $latex t_2 $. The critical distance is the sum of these two:
(1) $latex x_c = \Delta x_1 + \frac{v_p^2}{2a_{max}} $
with (1a) $latex \Delta x_1 =\frac{(v_a+v_p)}{2} t_1 $ and
(1b) $latex \Delta v_p =v_a -a_{s} t_1 $
Go analysis: Critical Stopping Distance
To avoid a system where tickets are given out unjustly, cars that cannot stop must be given sufficient time to clear the intersection. When determining this, one must consider the range of options the drivers should be permitted. We will defer discussing the “traditional” solution (which evidently was published in 1959) and instead consider the case of the driver I described above. The driver above wishes to turn into the grocery store. To do so safely, they have a target speed for the intersection which they consider the “safe” speed to pass through the intersection. We will call this $latex v_{safe} $.
Possibly the driver has also been taught that they should pass through intersections at constant speed. This prudent strategy is often advocated as one that reduced confusion for all the cars and pedestrians who might need to gauge their own strategies for traversing the intersection. Drivers are also often discouraged from accelerating to make a light, and in this case, a prudent driver may decide to minimize the total number of operations involved in getting through the intersection.
They might chose to follow the strategy shown with the blue trace below:
To clear the intersection they must travel a distance equal to the sum of the critical distance and the adjusted length of the intersection:
(2) $latex x_c + W = \Delta x_1 +\frac{v_p^2-v_{safe}^2}{2 a_{go}} + v_{safe}/(t_{y}-t_2) $
Where W is the sum of the intersection width and the length of the vehicle, and $latex a_{go} $ is the deceleration rate of the car required for the driver to achieve the safe distance just as the front of the car enters the intersection; this occurs at time $latex t_2$ indicated.
For now, we will assume the driver has sufficient skill to just do this, if so, car will travel the critical distance $latex x_c $ during time $latex t_2$. To avoid a ticket, driver will need to yellow light to remain open at least as long as the time for his travel. So for this driver the light must still be yellow at
(3) $latex t_y = t_2 + W/v_{safe} $.
For both (2) and (4) to be true, his rate of deceleration must be:
(4) $latex a_{go} = a_{max}\{1- \frac{v_{safe}}{v_{p}} \} $
Applying the relation between change in velocity and acceleration
(5) $latex t_2 = t_1 + \frac{v_p-v_{safe}}{a_{go}} $
We can use (5) along with (1b) to determine the time the yellow light must remain on for a particular driver entering at speed $latex v_a $ and slowing to $latex v_p $ that is just sufficient to stop to clear the light if that combination of corresponds to being unable to stop. That time is $latex t_y $.
Longer times are not required for other drivers because either the combination corresponds to a condition where they can stop or a condition where they clear the light more quickly.
Design of the light.
By itself equation (6) is not sufficient to determine the time the minimum time a yellow light last on any particular roadway. The designer of the yellow light must consider all legitimate combinations of $latex v_a $ and $latex v_p $ and set the length of yellow light long enough to permit them all to clear. A few more applications of kinematics are required.
To continue the analysis I’ll binge and indulge in calculus. This exercise will be seen to be utterly unnecessary; but I’m putting it here because someone might wonder about the issue. Or perhaps my motive is merely to make this an AP Physics C question, ;). After the excursion, I will continue find the time required for the driver in the scenario above to clear the intersection.
Superfluous Calculus Interlude:
The driving scenarios I discussed above consider a wider range of entering conditions than one discussed in Jälrström 1 page exhibit. My analysis considers the possibility the driver is already decelerating at a rate $latex a_s $ prior to entering the intersection; Jälrström’s analysis assumes they do not decelerate until they are able to react to the light. If the designer in charge of setting the light timings for an intersection uses my more complicated entry condition, they must also consider how the value of $latex a_s $ the time, so that they can ensure they provide enough time for all reasonable combinations of entry speed and initial deceleration.
To do this we, must find the minimum value of $latex t_y $, which can be done by taking the differentiating (3) with respect to $latex a_s $. We conclude $latex \frac{dt_y}{da_s} = \frac{dt_2}{da_s} $. Doing this as slowly and ploddingly as possible, taking the derivative of (5)
(S1) $latex \frac{dt_2}{da_s} = \frac{1}{a_{go}} \frac{dv_p}{da_s} – \frac{v_p-v_{safe}}{a_{go}^2} \frac{da_{go}}{da_s} $.
and then the derivative of (4)
(S2) $latex \frac{da_{go}}{da_s} = -a_{max} \frac{v_{safe}}{v_{p}^2} \frac{dv_p}{da_s} $
Substituting and doing a small amount of algebra:
(S4) $latex \frac{dt_y}{da_s} = (\frac{1}{a_{go}} \frac{dv_p}{da_s}) \{1 + \frac{a_{max}}{a_{go}} \frac{v_{safe}(v_p-v_{safe})}{v_{p}^2} \}$
Observing $latex \frac{dv_p}{da_s} = -t_1 $ and substituting results in
(S5) $latex \frac{dt_y}{da_s} = (\frac{ -t_1}{a_{go}} \{1 + \frac{a_{max}}{a_{go}} \frac{v_{safe}(v_p-v_{safe})}{v_{p}^2} \}$
In the driving scenario envisioned, this derivative is always negative which is likely what many people expected without resorting to calculus.
The reason the time required to clear the light decreases with the deceleration is the critical stopping distance, $latex x_c $ is smaller for a driver who enters at a speed $latex v_a $ but already decelerating before he can respond to the light. So, although those drivers would require more time to clear the intersection were they to chose to do so, they don’t need to. They should stop. The minimum time required is for those drivers whose deceleration was insufficient to permit them to stop.
So the largest required time for the yellow occurs when $latex a_s =0 $ which means they began to slow down only in response to the light changing to yellow.
The main result of the superfluous interlude is to show that Jälrström was entirely wise and reasonable to restrict his analysis to the one that mattered at the outset.
Superfluous interlude finished: Simplify.
The problem now becomes more like the one Jälrström discussed. For this case $latex v_p= v_a $ and
(6) $latex t_{2,max} = t_1 + \frac{v_a-v_{safe}}{a_{go}} $.
Equation (4) becomes
(7) $latex a_{go} = \frac{a_{max}}{v_{a}^2} \{ v_{a}^2 – v_{safe}^2 \} $
And (8) $latex t_{2,max} = t_1 + \frac{v_a}{a_{max} } \frac{v_a} {v_a+v_s } $
So, my result for the minimum time for a yellow light should remain on is:
(9) $latex t_{y,req} = t_1 + \frac{v_a}{a_{max}} \frac{v_a} {v_a+v_s} + W/v_{safe} $
where $latex v_a $ is the maximum permissible speed on the road, $latex v_{safe} $ is the maximum safe speed for clearing the intersection, $latex a_{max} $ is the maximum reasonable deceleration rate for car’s stopping when the yellow light turns on and $latex w $ is the sum of the effective width of the intersection and the length of the vehicle.
Sharp eyed readers will notice my final result differs from Jälrström’s which is
(10) $latex t_{y,req} = t_1 + \frac{2v_a-v_{safe}}{2a_{max}} + W/v_{safe} $
That my result and Jälrström’s differ may, possibly, delight the “Oregon State Board of Examiners for Engineering and Land Surveying” or anyone who thinks it is wiser to fine someone $500 to shut Jalstrom up rather than allow people to listen to Jälrström’s analysis and conclusions about traffic lights.
But if someone is delighted, their delight is unwarranted. In fact, Jälrström’s answer is just as good as mine. Arguably, his is better.
The difference in our solutions lies in our assumption about which traffic maneuvers the driver should be ‘expected’ to do when faced with a yellow light. As you recall, I said I was unable to entirely tease out the precise scenario Jälrström intended to analyze; specifically, I couldn’t determine the nature of the deceleration “a” in the one page summary I could get my hands on.
In my analysis, I assumed the driver was sufficiently skilled to pick out $latex a_2 $ such that they would slow to the safe speed, $latex v_s $, when they just reached the intersection. Jälrström’s analysis appears to assume the driver doesn’t try to finesse their deceleration this finely; based on is answer, it appears assumes the driver decelerates at $latex a_max $ until they reached $latex v_s $ and then crossed the light.
While finessing acceleration and deceleration is a fine skill to have, one might very well argue that those designing timing of yellow lights ought not to assume drivers have the skills of Indy race car drivers. Some drivers on the road may be 87 year old retirees visiting their daughters. Using Jälrström’s equation would permit these more mature drivers to clear the intersection and arrive at their daughters home without getting a ticket.
Which scenario is the correct one to use determining the minimum time for a light is precisely the sort of discussion that people ought to have regarding setting light times. That is: Do people think overly cautious drivers who were a bit to heavy on their brakes and should get tickets if they don’t clear the intersection? If not, the yellow light should be timed using Jälrström’s equation. If you think they should be more skilled than that, the light can be timed with my equation.
How do these compare the current equations for timing of traffic lights?
Evidently, currently traffic lights are timed using an equation that results from “THE PROBLEM OF THE AMBER SIGNAL LIGHT IN TRAFFIC FLOW” by published Gazis, Herman, and Maradudin in 1959 which is available here. In that paper, the authors analyzed the situation where a car moving at constant speed either brakes and stops or they maintain their constant speed and move through the intersection at their initial speed $latex v_a $. Of course this is precisely what cars do when the light turns yellow on an intersection they can safely travel through at their current speed.
In that case, the car follows the green trace superimposed on the graph shown to the right time required for them to clear is:
(11) $latex t_{y,req} = t_1 + \frac{v_a}{2a_{max} } + W/v_{a} $
This time is indicated by the final vertical green dashed line. The Gaziz, Herman and Maradudin equation always results in shorter times for the yellow light than use of either Jälrström’s or my equation. The difference arises for the same reason my equation differs from Jälrström’s. They analyzed a different driving maneuver: They assumed that if the driver could not stop, they driver would also not slow down while approaching the light. The driver would then pass through the intersection moving at the speed limit.
However, as any experienced driver knows, there are many reasons it is often not possible. One might need to slow down to turn into a grocery store, because geese decided to cross a short way down the road, because the car in ahead slowed to enter the grocery store, or, as, evidently, Jälrström noted, because one is planning to turn at the intersection. In these cases it is rather unfair to receive a ticket when there is no way to either stop in time or clear the intersection before it turns yellow.
Scenario Dependence
One might ask whether this analysis closes the door on the issue of proper timing? Not at all.
One thing that should be evident to all here is that the computation of the minimum time light depends on the range of scenarios one envisions could occur when a driver approaching a light sees the light turn yellow. I would suggest that those setting the timing think through a range of driving scenareos and make sure those whose driving was both within the law and reasonable legal when the light turned yellow and whose responses were prudent and legal should be able to clear the light.
It seems to me the scenario involving cars that must slow down is common enough that I think this means the light designers should no longer use the equation by Gazis, Herman, and Maradudin and replace it with another. Jälrström’s equation seems like a good choice.
Are there other scenarios we might expect the driver to do? Sure. I’m sure blog visitors can think of some. Many will not require any increase in time beyond that proposed by Jälrström, but perhaps one might. Discussing hypothetical scenarios and how each might require some minimum time for a yellow light seems like a sensible approach to the issue of traffic signals. It is certainly a more reasonable response to Jälrström’s criticism of light timing than having the Oregon State Board of Examiners for Engineering and Land Surveying fine him $500 for discussing the timing of traffic lights.
Especially as Jälrström is correct: If one is going to hand out tickets based on blowing these lights, then lights timed using the 1959 equation are mistimed.
