4. Question four
4.1 Establishment of the model
Poisson distribution fit the condition of describing the times of random events per unit time. The distribution function is:
P is the probability of number of arriving vehicles k per unit time, parameter λ is the average probability of random event per unit time. The mathematical expectation and variance both are λ.
To estimate the duration of time of queuing up to upstream crossing, we need to have the mathematical expectation of passenger car unit accumulated at the cross sectional area per unit time. We set the estimated time as t, the distance from cross sectional area to upstream crossing as s, traffic density as K, then:
in which, ，
In fact, the cross sectional area still has some distance from downstream crossing. Facing the situation that there is only one lane for passing, all vehicles must flood to that single port, so the influence of downstream turning rate can be ignored. To verify out assumption, we calculate mathematical expectation E by two methods – whether taking turning rate of downstream crossing into consideration. 4.1.1 Method one
According to question one, the maximum traffic volume (passing capability in normal condition) is 60pcu/min, so we only calculate the Poisson distribution probability between 1pcu/min and 60pcu/min.
Assuming the probability of observing a(pcu/min) is , the average outflow traffic volume of cross sectional area is X, because when the outflow volume can digest present inflow volume, we only consider the condition when .
Then the expectation value of remaining vehicle number per minute is:
This is the passenger car unit accumulated at the cross sectional area per unit time 4.1.2 Method two
We set total volume flowing into the section every signal period as Y. We need to verify whether the inflow traffic volume obeys the Poisson distribution, and then we can make further probability statistics and calculation design. According to the question statement, . Based on attachment 3, we get the volume ratio – 0.21Y for lane one, 0.44Y for lane two, 0.35Y for lane three. From the Poisson distribution function, if Y obeys the distribution, then the inflow volume of the three lanes obeys it as well.
Based on the results above and video one, we make statistics of the traffic volume at cross sectional area when lanes are occupied due to the accident, which will be represented with ， ， in following calculation.
To simulate the practical situation, we set simulation model, added with the changing difficulty when traffic flow confluent. The lane changing difficulty is given by the traffic volume for each lane from video one, that is, the ratio of the difficulty is:
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：：
According to attachment 3, the free lane in video one is lane one. After the cars enter the section, we first allocate passing volume for each lane at ratio ： ： . To pass the cross sectional area as soon as possible, all vehicles will flood to lane one, and we can suppose if there is remaining volume in lane one, the rest part should be allocated to lane two and three at ： again; similarly, if lane one and two are both not saturated, the remaining part comes to lane three. Because the turning ratio downstream of lane two is the biggest and it is in the middle of road, the difficulty must be bigger than lane one. Therefore, it is impossible for lane two to have rest volume to allocate to the other two lanes.
Still, assuming the probability of observing a(pcu/min) is , we get mathematical expectation of remaining passenger car unit per minute based on the arithmetic method in appendix 6.
4.2 Solution to the model
First of all, we use SPSS to verify the discrete distribution of Y, and the result confirms our assumption of Poisson distribution.
We use distool in Matlab to calculate the Poisson distribution probability inflow traffic volume of the three lanes. (details are in appendix 4)
Figure 8. verification of Poisson distribution in SPSS
Figure 9. graph of Poisson distribution probability of upstream inflow volume in Matlab 15
According to the information in the video, we work out the index ， ， in method two: ， ，
The results from both methods are completely the same, the mathematical expectation is:
Thus, estimate the duration of time of queuing up to upstream crossing is:
To sum up, we estimate that the duration of time of queuing up to upstream crossing is 6.27 minutes. 4.3 Allergic analysis
First of all, the essential thoughts for both methods is to keep the biggest outflow volume at cross sectional area all the time, and the extreme situation happens at quite low probability in Poisson distribution, so the results are the same.
The outflow traffic volume at cross sectional area and traffic density used in question four are counted by eyes. Due to the unclearness of the monitor and subjective factor of man-made counting, we give the two parameter 5% error range. With the same arithmetic process, we get the fluctuated interval region of estimated time, [5.29, 7.51] (unit: min).
VI. Model evaluation
1. Question one
In the description of question one, we take the way that taking the traffic capacity before and after the accident into consideration, and use a line chart to show the change more intuitively.
We simplify the cars by using traffic flow model, whose disadvantage is ignoring many influence factors of flow internal. For example:
1. When cars ahead changing lanes, cars at the back need deceleration, which is not the same as flow
model in which cars can stop at the perfect place.
2. When two cars cutting in, it could happen that traffic flow decrease at short time, but the traffic
capacity is a constant, such differences is obscured by the average.
3. It is impossible that the cars are continuous and even. Also the fluctuation of the space between
two cars is relatively large, which is not agree with the flow model.
4. The length is a quantity that cannot be ignored in the condition that the distance is not long, but
we ideally simplify cars into point, the error made in this process could be measurement.
2. Question two
We suppose that the traffic flow coming from upstream section is constant, but in the actual situation, the traffic flow of different time are different, which has the peak and low peak. If the accident happened at the peak period, it is more possible to cause vehicle accumulation, which make the flow density too high and restrain traffic flow capacity. Such situation makes the maximum traffic volume we observing is inaccurate, which causing error in calculating traffic capacity. At the same time, there are many other factors influence the traffic capacity, such as road slope, road condition, weather etc. if those factors are different, our calculation of the traffic capacity would be inaccurate.
3. Question three
In question three, we use the concept of traffic wave and the formula derivation in fluid dynamics to convincingly give the function relation among the queue length of vehicle influenced by traffic accident,
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the actual traffic capacity of the cross section of the accident, accident duration and the traffic flow of section upstream. Also we determine the amount we can do and give an image to performance this relationship more visual.
However, in order to observe the queue, the object of observation we setting is a fixed length of road space. If the queue exceed this space and influence the flow entering from the upstream section, our model will not work. We will need a new one to solve this problem. The perfect solution of this problem is collecting multiple sets of data to do correlation analysis, regression analysis and then fitting, the relationship got in this way can be more truthfulness.
4. Question four
In the model building process, we always regard the traffic flow coming into this section as steady flow, which only make mutation and never change gradually. This simplification is obviously inconsistent with reality. According to the upstream intersection organization scheme from attachment four, traffic flow coming into this section should contains two parts: one part is relatively constant, which is turning right at the upstream intersection. Because they are not affected by limitation of traffic lights, they can be considered as constant flow. Another part is going straight at the upstream intersection. Because of the limitation of traffic lights, traffic flow will form stop wave and starting wave. So considering the boot process of cars in the intersection, the traffic flow coming in from the upstream intersection should be changing flow whose derivative is not zero. We use statistical average value to replace changing function, which means finding constants whose time integration are the same to replace this function. The length of queue forming in this way is discrete, but the true length function is continuous in time.
The disadvantages of this simplification is that the length of queue could achieve a warning value earlier, but return back in a period of time, which makes our calculation of the time error. The ideal
condition is that we can get a set of data of time and length to fit, whose result is the closest to the truth. It will be more complex to the actual situation, because it is possible for four directions of intersection to come into the section. Also emergency could affect the traffic flow coming in and make turbulent flow.
VII. Improvement and generalization of model
In the process of describing the traffic flow, we only obtain the relevant data through adding up cars in the giving video.
First, it is unavoidable to make error in these statistics. Second, the data obtaining from the video is mainly about number of cars. All the distance related quantity such as length of cars, distance between two cars, speed etc. which are characteristic quantity of traffic flow, cannot be measured. So we cannot use many existing theory of traffic flow, which makes us more difficult to describe the process. We have to do much simplification to facilitate the solution. The model would be more accurate based on more relevant data and consider more situation.
In addition, we only consider the four wheel vehicle, but there are many two wheeled vehicle in the video, which also influence traffic condition. This kind of traffic flow will be very complex and need consider more factors, but it will conform to the actual situation better.
In the real life, it is impossible that there is only one road or one intersection. The actual situation is a complex traffic network and cars circulate in the network. We need knowledge about graph theory to build a better model. The influence to the network of the ramp caused by traffic accidents happened in one section would change because of publishing different kinds of traffic information. Such network is more like a neural network which has a reaction mechanism drawing on advantages and avoiding disadvantages. The reaction mechanism will make the cars have not coming in the section changing lanes earlier which
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reduce the enter flow. Also the traffic police can let the vehicle more orderly pass without making the traffic flow capacity decline in short time caused by two cars scramble. So the actual traffic system is complete with many external monitoring measures which make the system running better.
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