2013高教社杯全国大学生数学建模竞赛
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2013高教社杯全国大学生数学建模竞赛
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The impacts of lane occupancy towards city traffic
Summary
The occupancy of the lanes in city will lead to the consequences that traffic can be badly influenced, with its vehicles bearing ability per unit time down. According to various categories of lane occupying
conditions, we have made analysis to the impacts of lane occupancy towards city traffic by traffic situations shown in the video samples, which will offer theoretical accordance to transport department with guidance, designing and so on.
The basic thoughts in this essay is applying Traffic Flow Theory, including factors like traffic volume and density, to turn vehicle number to passenger car unit to solve the problems of lane occupancy. We mainly take data from videos, and then take advantage of software such as Matlab to analyze these original data. In question NO.1, we use maximum transporting flow to describe the passing ability. Making 60 seconds as a period, we made statistics of mixed maximum transporting flow of cross sectional area of the accidents between the time when they happened and over, putting all the data in line charts. The maximum
transporting flow was restrained to below one third of its normal level, which means the occupancy of lanes has serious impact on the traffic ability. However, due to the non-saturation of traffic volume, the restraint of traffic ability was not shown apparently in changes of practical traffic volume at cross sectional area. In question NO.2, we set a certain area as affected confluence section, with which we can show the
complex behavior of the vehicles when passing the narrow section of the road. Because of the impossibility to make systematic statistics and assessment of that complex behavior, we make quantitative statistics of the traffic volume at the cross sectional area, and judge the differences of those impacts in accordance with the dropping ratio of passing ability of lanes when occupied. Based on our calculation, the passing ability of video 1 decrease to 30.9% of what it was while that in video 2 comes to 34.3%, meaning that the impact on practical passing ability at cross sectional area in video 1 is greater than video 2. The greater the ratio of the flow at the occupied lane, the worse the traffic jam caused by that accident will be.
We apply hydromechanics to work question NO.3. Assimilating the process of vehicle queuing up to traffic waves, we make equation of continuity of that to deduce the formula of wave velocity. Based on that
function and practical passing ability in question NO.2, we count the flux both at busy and leisure time in a whole period, and make transformation graph of length of queue versus continuous time of accident, which shows a zigzag increase tendency.
For question NO.4, to estimate the queuing time up to the crossing after the accident, we firstly verify the traffic volume at the crossing obeys the Poisson distribution. Calculating the Poisson distribution probability and maintaining the maximum traffic volume at the cross sectional area, we find that the average quantity of vehicles accumulated at that area is 6.6909 pcu/min by simulation in two reliable methods, and based on the distance and traffic density, we could estimate the queuing time is 6.2772 min. in addition, due to the inaccuracy of man-made statistics, we do allergic analysis to our results, and get an interval region [5.29, 7.51] min.
Key words: Traffic Flow Theory, confluence, hydromechanics, traffic waves, Poisson distribution
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I. Restatement of the problem and put forward
Lane occupied is the phenomenon that the traffic accident, the roadside parking, occupying-road construction and other factors leading to the lane or road cross-section capacity decrease in unit time. In the area where traffic current density is high, lane occupied phenomena will reduce the traffic capacity, and even cause serious problems of regional traffic jams, resulting in a larger impact on urban traffic.
According to the situation of the occupied various lane, we can analysis the degree of impact on the urban road traffic capacity of lane occupied through the traffic condition in the video sample. On the basis of the situation in video one and video two in which the traffic accident happening in the same section and the same cross section of road, we propose the following questions:
Question one: according to video one, describe the changing process of the actual traffic capacity at the cross section of the accident during the accident happened to leave.
Question two: according to the conclusions of question one, combined with video two, explain the differences in the influence of the actual traffic capacity of the cross section between accidents occupied different lanes at the same cross section.
Question three: construct mathematical model to analysis the relationship among the queue length of vehicle influenced by traffic accident, the actual traffic capacity of the cross section of the accident, accident duration and the traffic flow of section upstream in the video one.
Question four: if changing the condition of video one, which means the distance between the cross section of the accident and upstream intersection changes to 140 meters, section downstream direction demand is unchanged, sections upstream traffic flow is 1500pcu/h, the vehicle initial queue length is zero when the accident occurred and the accident lasts without leaving, then estimate after how long time from the beginning of the accident, the vehicle queue length will reach the upstream intersection.
Solving the above four problems, we can provide a theoretical basis for traffic management to do dredge, management, design etc.
II. Problem analysis
Vehicles can analogy to the gas or liquid molecules within the medium flow when running in the transportation network, which is called traffic flow. Regarding the passage of vehicles lane as flow contribute to the establishment and simplification of model. The case congestion and flow involved in the problems can correspond to the non-steady state and the steady flow state in the traffic flow theory. Thus we can use traffic flow theory to establish the model and then formulate corresponding rules of the index of the flow, which we can analysis to solve the traffic problems.
In the process of observing the video, we find that vehicles in city road can be roughly divided into sedan car, large passenger cars and two wheelers. Due to the division of city traffic trunk lane width is designed based on the four wheel vehicle and the mutual influence between the two wheelers vehicle and road traffic condition can be ignored whether in the congestion or smooth state, it is reasonable to only consider the traffic flow of four wheel and above vehicle and accumulator car.
Data collecting is one of the difficulties of the problem. It needs careful consideration to know how to extract data such as traffic flow and others required from the video in the annex.
1. Question one
The concept of traffic capacity related in the problem is the maximum traffic volume in the traffic engineering. Therefore, to describe the changing process of the actual traffic capacity at the cross section of the accident during the accident happened to leave, we make statistics on the traffic flow of the cross
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section of the traffic accident, then transform the original statistics data into the standard vehicle equivalent number, afterwards draw a line chart, and combined with a straightforward traffic conditions as the description standard.
2. Question two
Traffic flow theory is the basic theory of traffic engineering which uses the laws of physics and mathematics to describe the characteristic of traffic.
We use the thought of stream merging to compare the differences in the same section of traffic accidents accounting for the different lanes effects the actual capacity on the cross section. Considering the situation that the vehicle could the accident at the previous section area before the cross section of the accident and then take actions like changing the lane to pass the cross section, we also need to set the influence zone of confluence. Due to such complex behaviors cannot take systematically statistical evaluation, this article take numerical statistics of the traffic flow of the cross section and then distinguish the different influence on traffic lane occupied capacity according to the traffic blocking probability.
3. Question three
Fluid dynamics simulation theory analogy traffic flow and gas or liquid molecular mechanics. We can analogy the process of queuing vehicles and gather-disperse wave in fluid mechanics, then analysis the relationship between traffic volume, density, speed and traffic to make the function relation between the queue length of vehicle influenced by traffic accident, the actual traffic capacity of the cross section of the accident, accident duration and the traffic flow of section upstream.
In addition, due to that we can get the specific data of the accident section actual capacity and sections upstream traffic flow though the video, the ultimate result can be given in the form of related picture of the queue length of vehicle influenced by traffic accident and accident duration.
4. Question four
In the case of already know the road length, we need the expected value of the deposition of the vehicles to estimate the time that the queue length of vehicle reach the upstream intersection. For this, we firstly validate the discrete distribution of input traffic volume from the upstream intersection, and then based on the function relation got from question three we can estimate the mathematical expectation of the accumulation of vehicle equivalent number.
III. The model assumptions
1. Only consider the traffic flow of four wheel and above vehicle and accumulator car.
2. The vehicle is only divided into large passenger vehicles and small passenger car two classes, and all of them translate to the conversion of standard equivalent number to consider cross section flow.
3. Assume the visible part of video is authentic that can objectively reflect the traffic situation. The attachment is complete and reliable.
4. Assume the vehicle through the upstream intersection comply fully with the signal command. Don't consider the subjective factors of the drivers.
5. Assume queuing vehicles before the cross section of the accident is in accordance with three lane of three rows array.
6. Assume when lane is occupied the traffic volume reach saturation which can be equivalent to the maximum traffic volume.
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