Mathematical modeling thesis of senior one explores the frequency of traffic lights at intersections. It's not too difficult to surf the internet in senior one.

Reasonable setting of traffic lights at crossroads

Chen Jinkang

Keywords: traffic light setting, traffic light period

First of all, the questions raised

As the baton of urban traffic, traffic lights play an important role in traffic. If the traffic lights are set unreasonably, it will not only affect the traffic order; It may also affect the safety of pedestrians and bicycles. At present, there are some unreasonable factors in setting traffic lights at many intersections in Hang Cheng. Let's take the traffic light setting at an intersection of Gudun Road (between Tianmushan Road and Wen Yuan Road) as an example. The intersection has just been opened, and the traffic control department's research on road conditions and traffic flow is not very mature, so there are some problems in the setting of traffic lights. The traffic volume at this intersection is small, especially in several directions, but the green light time is set too long, and it often happens that the intersection is empty but the vehicles have to wait for a long time; At the same time, at such intersections, it is unnecessary to turn right at a red light. In addition, there is no difference in the setting of traffic lights at different time periods at this intersection, which is obviously very unreasonable.

Let's study the reasonable scheme of setting traffic lights at this intersection. We mainly study two aspects: the setting of traffic light cycle and the time of green light in all aspects in a cycle.

Second, the establishment of the model

1, traffic light period

From the "road traffic automatic control", we can find the best cycle formula of red light and green light:

These include:

C is the cycle time.

Stage: Several business flows that start and end at the same time are called a stage.

L is the total lost time over a period of time. The loss time of each stage I= start delay time-end delay time; The total loss time of the whole cycle is the sum of the total loss time of each phase plus the interval time of each green light r (generally speaking, the start delay time is the reaction time from the driver seeing the green light to the start of the car, and the end delay time is the time from the green light going out to the last car passing. )

that is

Q is the traffic flow of the corresponding phase.

S is the saturated traffic flow of the corresponding phase. (When the vehicle passes through the intersection with a roughly stable flow, the flow is the saturated traffic flow of this phase. )

2. Time allocation of green light in north-south direction and east-west direction.

We might as well ignore the yellow light and take one cycle of traffic signal light conversion as the unit time, assuming that the traffic flow in both directions is stable and uniform, regardless of the turning situation.

Let e be the number of vehicles arriving at the intersection from east to west in unit time; S is the number of vehicles arriving at the intersection from north to south per unit time. Suppose that in a period, the east-west direction turns red and the north-south direction turns green for r, then in this period, the east-west direction turns green and the north-south direction turns red for1-r.

We need to determine the control scheme of traffic lights, that is, to determine R. The main basis for measuring the series efficiency of an intersection is the sum of the time that all vehicles stay at the intersection in unit time. Therefore, in order to determine R, it is only necessary to ensure that all vehicles stay at the intersection for the shortest total time in a cycle. The detention time of a car at an intersection usually includes two parts, one is the time for each car to wait for the red light, and the other is the time for the driver to wait for the green light after parking, which can be measured.

First, for any given r (0

Similarly, the total waiting time for all vehicles to travel in the north-south direction in a cycle is

It takes t units to start the vehicle running a red light, and this part of time must also be included in the total detention time. In a cycle, the total number of vehicles running red lights in all directions is, and the corresponding detention time is.

So the total residence time is

The total retention time of vehicles is the shortest.

Let B=V+H, indicating the total number of vehicles passing through the intersection in a period. The best simplified expression above is

Easy to see, the best control scheme.

3. Two-way straight and left turn green time allocation

Further consider the situation where the vehicle turns (assuming there is no red light when turning right). Let the sum of the green light time of going straight and turning left in the north-south direction (that is, R above) be the unit time, where the green light time of going straight is, and the green light time of turning left is; Let the number of straight vehicles per unit time be h and the number of left-turning vehicles be l.

From the above analysis, we can know that:

Iii. Survey results and data processing

1, survey data

Through field investigation and online inquiry, we also got the following data:

The direction of the time period turns to the actual traffic flow saturation, the total loss time of traffic flow and the optimal cycle green light time.

From 8: 00 to 9: 00, go straight in the north-south direction for 50 vehicles/minute, 80 vehicles/minute for 3 seconds 165 seconds 1 10 second.

Turn left 8 vehicles/minute 75 vehicles/minute 17 seconds.

Go straight in the east-west direction for 8 cars/min and 80 cars/min 17 seconds.

Turn left 9 vehicles/minute 75 vehicles/minute 2 1 second.

12: 00 ~ 13: 00, go straight in the north-south direction, 35 vehicles/minute, 80 vehicles/minute, 3 seconds, 46 seconds and 23 seconds.

Turn left 9 cars/minute 75 cars/minute 7 seconds.

Go straight in the east-west direction 10 vehicles/minute 80 vehicles/minute 8 seconds.

Turn left 10 vehicle/minute 75 vehicle/minute 8 seconds.

17: 00 ~ 18: 00 go straight north and south 45 cars/minute 80 cars/minute 3 seconds 73 seconds 45 seconds.

Turn left 9 vehicles/minute 75 vehicles/minute 10 second.

Go straight in the east-west direction for 8 vehicles/minute and 80 vehicles/minute for 9 seconds.

Turn left 8 cars/minute 75 cars/minute 9 seconds.

2. Data processing

According to the above data and the previous model, we can calculate the best period and time of green light in all directions. (See the last two columns in the table)

The above table gives the traffic light time at the intersection and the green light time in all directions at different time periods. However, in fact, the setting of traffic lights is very complicated and involves various factors. It is impossible to solve all problems with a fixed model, and it must be constantly adjusted according to the actual situation. The scheme we gave above is only an idealized and approximate scheme, but I believe it will have certain reference value and practical significance to the traffic police department.

References:

1, Tan Yongji, Yu Wen, Mathematical Modeling, Fudan University Press.

2. Ng Man Tat and Cheng Lizhi. Theory and practice of mathematical modeling, National University of Defense Technology Press.

3. Duan Liren, Automatic Road Traffic Control, China People's Public Security University Press.