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Author Topic: Time duration for the yellow traffic light  (Read 38519 times)
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Fu-Kwun Hwang
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on: July 12, 2008, 01:59:37 pm »

A yellow light at a traffic intersection should last long enough that a car traveling at the suggested speed can either
apply the brakes and decelerate to a stop prior to reaching the front of the intersection, or
maintain the same speed and pass through the intersection before the yellow light turns red

If a driver traveling at the suggested speed cannot do either of the two options, then the traffic signal (specifically the time duration of the yellow light) is considered unsafe.

If the speed of the car is v, the friction coefficient between tires of the car and the road is \mu.
The maximum brake force F= -\mu N = -\mu m g, and F=ma
So a=- g\mu. The distance required for the car to fully stopped (after brake is activated) is s=\frac{v^2}{2a} =\frac{v^2}{2g\mu}.
Assume the reaction for the driver is \Delta t, then the total distance required to stop the car after the driver find the yellow light just turn on is: D_{min}=v \Delta t + \frac{v^2}{2g\mu}.

The car has to be D_{min} away from the intersection., for the car to be fully stopped behind the intersection. of the road.
If the distance is smaller than D_{min}, the time for the yellow should be enough for the car to pass the interaction. Assume the width of the intersection is W, and the time for the yellow is T.
Then it requires that v T\ge D_{min}+W= v \Delta t + \frac{v^2}{2g\mu}+W.
So T\ge \Delta t +\frac{v}{2 g\mu}+\frac{W}{v} This is the minimum required time for car to pass the intersection.

However, if the car need to stop before the traffic light, the minimum distance is D_{min}=v\Delta t + \frac{v^2}{2g\mu}
For the car to stop from initial velocity v and acceleration a=-g\mu, it need t_{brake}=\frac{v}{gu} from v(t)=v_0+at
So the total time required is T'_{min}=\Delta t+\frac{v}{gu}

It means that the time for the yellow light T_{yellow} need to satisfy two equations:
T_{yellow}\ge T_1= \Delta t+\frac{v}{2gu}+\frac{W}{v}
and
T_{yellow}\ge T_2= \Delta+\frac{v}{gu}
So the time for yellow should be larger that the maximum of T_1,T_2
The condition for T_2>T_1 is \frac{v}{gu}>\frac{v}{2gu}+W/v, which imply
  \frac{v}{2gu}>W/v i.e.  \frac{v^2}{2gu}>W 
The above condition is the same as stopping distance ge width of intersection which is the case for normal speed limit and traffic light.
However, if  W \ge \frac{v^2}{2gu}, then the minimum time for traffic light is \frac{v}{2gu}+W/v

For v=72km/hr=20m/s, \mu=1, \Delta t=0.8s, W=20 m.
T_1= 0.8+ \frac{20}{2*10*1}+\frac{20}{20} =2.8 s
T_2= 0.8+ \frac{20}{10*1}=2.8s
So the minimum time required is 2.8s.
However, if the width of the interaction is less than 20m, then 1.8<T_1<2.8s.
The minimum time required is still 2.8s

The following simulation let you play as a traffic light control manager:
You can change the width W of the interaction, the reaction time for the driver, the time for the green light and yellow light. (If you click the right most checkbox, the program will show suggested time for yellow light)

Code for the car:
 green: moving at constant speed.
 red: decelerate
 yellow: accelerate

*** the maximum speed and maximum acceleration for each car is randomly selected in the simulation , to make the simulation closer to the real case. I hope you can enjoy it!

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Let's apply physics principle to estimate yellow light time duration.

Suppose the reaction for the driver is RT, the speed of the car is V, the friction coefficient between tire and the road is mu, mass of the car is m, gravity is g.
 
Then the friction force Fr= - m*g*mu = m*a so the deceleration a=g*mu
The minimum stopping distance when driver saw the light term yellow is Dmin=V*RT+ V*V/(2*g*mu)
You can adjust the deceleration a directly with slider control.
he friction mu= 1.0-1.2 for normal tire. But it is a strong brake.
Normally, we did not brake the car with maximum deceleration. So the default value is set to a=0.5

The above analysis ignore the width of the car.
If the car want to stop before s/he reach the front of the interaction, the minimum distance is Dmin.
If distance is less than Dmin, the car has to pass the interaction before the end of the yellow light.
Suppose the length of the car is d, the width of the interaction is W, and the time for yellow light is YT.
V*YT >= Dmin+ W+d

For the car to pass the traffic light, the minimum time for yellow light should be
 YT_{min}= \frac{D_{min}+W+d}{V} = \frac{W+d}{V} + RT + \frac{V}{2*g*mu}

If the yellow light is too short, then some car would not be able to pass the intersection safely.
However, if the driver do not want to brake the car so quickly (want to be more comfortably), replace 2*g*mu with 2*g*mu/k. the above simulation use k=2 to estimate the time for yellow light).

If the yellow traffic light last too long, the driver might not want to stop the car, and ,when the light turn RED, s/he would not be able to fully stopped before the interaction.

If we want the car to stop before the traffic light, the minimum time for yellow light is RT+\frac{v}{g\mu}
Summary:
For very long intersection W\ge\frac{v^2}{2g\mu}-d, (i.e. width of interaction + width of car larger than stopping distance for the car),
the minimum time required is RT+ \frac{v^2}{2g\mu}+\frac{W+d}{v} : Reaction time + braking time+ time to pass intersection.

For short intersection where W\le\frac{v^2}{2g\mu}-d,
the minimum time required is RT+ \frac{v^2}{g\mu}: Reaction time + braking time *2

The extra time is required because we need to make decision ahead of time.



You can check out Tale Of The 3-Second Yellow Light, Traffic Light Logic, THE YELLOW LIGHT for more story.

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enalice
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Reply #1 on: March 16, 2009, 06:02:47 pm »

How is the approach speed normally determined? If the 85th percentile speed is used, that's probably realistic. BUT, if an arbitrarily low posted speed limit is used, then the yellow interval is likely to be unreasonably short for actual traffic conditions, resulting in a high number of UNintentional red light runners.

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Fu-Kwun Hwang
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Reply #2 on: March 16, 2009, 06:52:12 pm » posted from:Taipei,T\'ai-pei,Taiwan

Yes. It depends on the speed of the car.
You can adjust the maximum car speed (Vmax) with the slider at the lower right region.

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arnanbd
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Reply #3 on: August 16, 2009, 02:16:19 am » posted from:Ramat Gan,Tel Aviv,Israel

can you create a counter that will count the cars passing the junction?
Is the red light duration equal to the sum of the green & the yellow durations?

Thanks!
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Fu-Kwun Hwang
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Reply #4 on: August 16, 2009, 07:38:51 am » posted from:Taipei,T\'ai-pei,Taiwan

The purpose of the above simulation is to find out suitable time for yellow light.
The maximum speed and maximum acceleration for each car is randomly selected in the simulation, so the number of cars passing the juntcton is not the same even with all the same parameters.
The red light duration is set to be twice the green light duration in the above simulation.
It is not necessary that read light duration= green light duration+ yellow light duration.
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