Optimizing Traffic Flow: The Science Behind Yellow Light Timing

Introduction

The most common used transportation is car as people can travel in a longer distance and it is the same for me as my home is far from the school. During the way to school, there are serval traffic lights at the intersection. When the green light turns to the red light, there is a yellow light as transition. So I am curious that how long should it takes for the cars which is crossing the intersection or the cars that need to be continuously moving as they are too close to the intersection.

When drivers see the yellow light is on, they need to decide to stop or to cross the road. If chose to stop, they must have enough distance to stop at the line. On the other hand, choosing to keep moving, it must have enough times to cross the intersection safely and it contains the reaction time. In order to maintain the safety of the traffic, there are some factor that is relative to the time for yellow light is on — the velocity of the car, the width of the intersection, the length of the car, the mass of the car, the reaction time of the driver and the braking distance.

Braking is a complex process and to simplify the calculating I consider the process as friction. In order to investigate, I will develop a model that contains all components that affect the time of yellow light and using differential equation.

Assumption of the Model

There are many situation in the real traffic road, in order to make an objective data, all calculation will be base on the following assumption: the legal maximum velocity of the car and the velocity for a car to cross the intersection is V_0.

The reaction time for every driver to decide whether to stop when it turns in yellow light is the same. All drivers and pedestrians can obey the traffic rules. There is nor traffic jam and traffic control at the intersection. All cars can pass through the intersection normally.

Model 1

T = the time for the yellow light is on

T_1= the reaction time for the driver

T_2= the time for the car to cross the intersection

T_3= the driving time for the braking distance of the car

As when driver sees the yellow light, the time for all consideration and action should be contain in the time range for yellow light, therefore T= T_1+T_2+T_3. Therefore the purpose is calculating the minimum value of T, because driver can stop the car in the braking distance or pass the intersection safety. As T_1 is a constant, so the calculation are only focus on T_2 and T_3.

The time for the car to cross the intersection (T_2)

Let D= the width of the intersection

I= the length of the car

V_0= the legal maximum velocity of car

As the back of the car must pass the red line, so the actual

Length is D+L, therefore (D+I)/V_0 =T_2.

The driving time for the braking distance of the car (T_3.)

The process of stopping the car is when driver press the brake pedal to prove friction to slow down the speed and stop. Let m= the mass of the car, f= the coefficient of friction, the direction of friction is opposite from the moving force, therefore according to the Newton’s second law, F=ma, Friction=-fmg (g is the gravitational acceleration). When these two are equal, the car will stop. After driver press the brake, the distance of the car move is x(t), the process of braking should fulfill the following:

ma=-fmg

m dv/dt=-fmg

m (d^2 x)/(dt^2 )=-fmg

dx/dt |_(t=0)=v_0

x(0)=0

To find out the velocity based on the equation above,

v=dx/dt

∫((d^2 x)/(dt^2 )=(-fmg)/m) dt

dx/dt=-fgt+v_0

v_0 = 0, because the speed when stop of the car is 0, therefore the time for a driver press the brake until the car stop is t^'=v_0/fg . Under the situation that x(0)=0, the braking distance can calculate from:

x(t)=∫(-fgt+) v_0 dt

=(-fgt^2)/2+v_0 t

Total distance=x(t^' )=(v_0)^2/2fg

T_3=D/v_0 =v_0/2fg

Therefore, T=T_1+(D+L)/V_0 +v_0/2fg, v_0can be found by differentiate T and equal to 0, then the minimum value of T can get from:

0=dt/(dv_0 )(T_1+(D+I)/V_0 +v_0/2fg)

0=1/2fg-((D+I))/(V_0)^2

1/2fg=((D+I))/(V_0)^2

V_0=√((D+I)2fg)

T_min=((D+I))/√((D+I)2fg)+√((D+I)2fg)/2fg+T_1

=(2√((D+I)2fg))/2fg+T_1

=√((2(D+I))/fg) (+ T)_1

the reaction time for the driver (T_1)

There are numbers of factors affect the reaction time for a driver including the age of the driver and the traffic in that situation. Some accident reconstruction specialists showed that 85%of the drivers react in 1~2.5s, and the common used is 1.5s. A typical length for car is 4m~4.3m (here I used 4m), the width D of intersection is about 20~40m. The time length of the yellow light vary from different reaction time, car length and width of intersection in the following diagram:

Sample calculation:

Assume all f=0.8,g=10 ms^(-2), according to the formula √((2(D+I))/fg) (+ T)_1, when T_1=1,D=20,T=3.5. As fg=a, in the real life most of the velocity of the car would not decrease as much as 3.4 ms^(-2) in the research. Therefore the time calculated in the diagram 2 is idealized.

Model 2

In this model, as shown in diagram 3, it take consider the traffic from the south side which is much more realistic than Model. Actually, the time for the yellow light can turn off when the car from the west pass trough the point A, and the distance of stop line to the point A = L_1, these two distance are the same, therefore the total T =T_1+T_2+T_3=T_1+(I+L_1)/V_0 +V_0/2a.

In the further investigation, it can be found that T can be calculated when the car go from the west to the east pass through the point A before the car which from the south to the north pass the point A. Then the formula can change to T =T_1+T_2+T_3=T_1+(I+L_1)/V_0 +V_0/2a-t_1. T_1 in the formula therefore include the reaction time of driver from the south to the green light (( T_1)^') and the time for the car go to the point.

Assume that the velocity of the car is in a constant acceleration when it start from the south to the point A and the distance is (L_1)^'. The average velocity can be calculated from ∆V=V_0/2 and the time is (L_1)^'/(V_0/2)=(2(L_1)^')/V_0 . So that t_1=( T_1)^'+(2(L_1)^')/V_0 ,

T =T_1+T_2+T_3

=T_1+(I+L_1)/V_0 +V_0/2a-(( T_1)^'+(2(L_1)^')/V_0 )

=T_1-(T_1)^'+(I+L_1-2(L_1)^')/V_0 +V_0/2a

In addition, the standard velocity (V_0) at the intersection that mentioned in the Model 1 is impractical, the should be specified in the interval [V_0-∆V/2,V_0+∆V/2], then formula of T would change to:

T=1/∆V ∫_(V_0-∆V/2)^(V_0+∆V/2)(T(V) dV)

(=T)_1-(T_1)^'+(I+L_1-2(L_1)^')/∆V ln⁡((2V_0+∆V)/((2V)_0-∆V))+V_0/2a

According to actual situation of the intersection, the distance of L_1and (L_1)^'are not the same.

Let L_1=2/3 L_red ,(L_1)^'=1/3 L_blue, then the formula can change to

T_1-(T_1)^'+(I+2/3 ((L)_1-(L_1)^'))/∆V ln⁡((2V_0+∆V)/((2V)_0-∆V))+V_0/2a

From the equation above, it proves that there are three components to decide a suitable time range for yellow light: the difference between the reaction time of braking distance and the reaction time of starting, the difference of the distance of two opposite stop line in two direction, and the minimum legal velocity in the intersection.

Let a=3.4 ms^(-2),((L)_1-(L_1)^')=8 m,T_1-( T_1)^'=0.2 s,∆V=1/10 V_0,I=4 m,L=30m,T_1=2 s ,

Conclusion

From the value shown in the last diagram, Model 2 is more suitable to calculate the total time for yellow light on, and it is closer to the real life in my domestic area which is 3 s. Reaction time vary in different ages, health condition, mental state… Moreover, the materials used to build the road, the flatness of the surface of the road and even the air humidity may not be the same, the friction coefficient also will not the same.

Evaluation

In our real life, all drivers will take consideration of the traffic during that situation to decide pass or slow down when the yellow light turns on, and most of them are experienced drivers that they can respond quickly. However for the drivers who are not that skillful may occur situations like when the yellow light start flashing, drivers who are close to the stop line may suddenly stop to prevent running the red light. Then the cars which are behind them may not respond quickly and therefore cause rear-end accident.

Most of the traffic accidents occur at the intersection due to run the yellow light or the red light. The time limit for the yellow light need to take more consideration and resolve between local area as the traffic and population are quite different.