3MCMBM.docx

1、3MCMBMIntroduction / Summary We set up a model, in which the optimal number of the tollbooths in the toll plaza can be calculated under the condition that lane number is given. First of all, we defined the optimal as the maximal number of vehicles passing through within a definite period of time, wh

2、ich can be attained, i.e., the outflow rate of a toll plaza. Considering drivers complaint, we added a constraint condition that the total waiting time of a vehicle should be less than 5 minutes when traffic comes stable. Secondly, under the extreme condition assuming that the number of tollbooths e

3、quals that of lanes, we gradually increased the number of tollbooths using computer simulation and finally obtained the optimal. We have discovered that when traffic is blocked up, the former congestion will gradually decrease and the latter congestion will gradually increase along with the increase

4、 of tollbooths，while the latter congestion will aggravate the former congestion, which is consistent with fact. The optimal number of tollbooths will emerge in the phase when the latter congestion aggravates the former congestion, from there on, the increase of the tollbooths should make the whole h

5、ighway section blocked more heavily. There are quite a few factors which can affect the optimal number, among which we take two as relatively major: charging time at the tollbooth and the speed at which vehicles leave out-pool back for highway. The result of sensitivity analysis is as below: fluctua

6、tion of charging time at the tollbooth affects target little, while that of the latter affects much. Finally, using our model, we analyzed the real highway sections vehicle flow rate (of 4 lanes) and got 7 tollbooths, which is the same as the reality. ContentContent 2Problem Restatement 3Terms and D

7、efinition 3Assumptions 4About Interstate-95 4About the weather 4About the driver 4About the vehicles 4About the tollbooths 5About the in-pool & out-pool 5Some Facts Derived from Assumptions 5Analysis and Model Design 6Overview 6Analysis 6Modeling 7Model Result Analysis 10Single Lane Case 11Multilane

8、 case 13Special (n-n) case 14The answer to the question raised in the problem 16Model Verification 17Data Acquisition 17Simulation 19Model Sensitivity 20Sensitivity of minLeavingIntervalInSingleLane 20Sensitivity of serving time of tollbooths 21Further Work 23Model Strength and Weakness 23References

9、 24The Optimal Number of TollboothsProblem Restatement Some busy toll roads are multi-lane highways with some toll plazas scattered all over. Since no one is willing to experience continuous traffic disruption, it is desirable to limit the amount of time spent on toll plazas.Commonly a toll plaza is

10、 made of tollbooths, a much larger number of which is located than the entering travel lanes so as to fit well into the roaring traffic inflow. Vehicles fan out to tollbooths to be served, and then take trouble in squeezing out of the bottleneck at the departure of toll plaza, finally go back to a n

11、ormal number of lanes. Chances are that congestion happens in the bottleneck when traffic is heavy. Congestion at the entry of toll plaza is also possible as long as the inflow rate is overwhelming, though the possibility is not so high. Make a model to determine the optimal number of tollbooths in

12、a barrier-toll plaza, given the freedom to define what the word “optimal” means in the context of paper. Pay special attention to the situation where the number of tollbooths is the same as that of travel lanes and compare that situation on effectiveness with current practice.Terms and DefinitionWe

13、assume the toll plaza as Figure 1:Figure 1: The top view of a toll plaza.A list of relevant variables, constants, and parameters is in Table 1.TermsDefinitionsIn-poolSpace for the vehicles to pass the toll before going to the tollboothsOut-poolSpace for the vehicles to leave the toll plaza when paid

14、 for the tollInflowThe traffic flow into the in-poolOutflowThe traffic flow into the out-poolIntervalThe word “Interval” can be either separation in time or in space. Here we just use it in time delay, while the word “Distance” is used for space.DistancenumOfLanesThe number of Lanes in the highway.n

15、umOfBoothsThe number of tollbooths in the plaza.The former congestionThe 50% filled state in in-pool.The latter congestionThe 50% filled state in out-pool.numOfWaitingInFormerThe number of vehicles waiting in the former congestion.numOfWaitingInLatterThe number of vehicles waiting in the latter cong

16、estionminLeavingIntervalInSingleLaneThe minimal interval between two vehicles getting out of the departure in one lane.The total waiting timeThe total time for one vehicle from entering In-pool to getting out of Out-pool.maxDriverToleranceThe maximal total waiting time that a driver can tolerate.(n-

17、m) deployment abbr. (n-m)A toll plaza configuration with numOfLanes = n & numOfBooths = mCiThe time constant used in the former congestionCoThe time constant used in the latter congestionThe second terms coefficient of time delay in out-poolThe third terms coefficient of time delay in out-poolU-V di

18、agramA diagram with x-axis labeled U and y-axis labeled VTable 1: Terms and DefinitionsAssumptionsAbout Interstate-951. According to reference 1, the AADT (Annual Average Daily Traffic) of Interstate-95 in year 2001 is 60,414.2. According to reference 1, Interstate-95 has 8 lanes and 4 lanes in each

19、 direction. About the weather1. We assume the weather is fine and has no bad effect on characteristics of the vehicles.About the driver1. Usually, the range of reaction time of the human beings is from 200ms to 300ms. We assume it is 250ms.2. maxDriverTolerance = 5min.About the vehicles1. It is assu

20、med that the arrival of vehicles is a bit stochastic, but stationary in the long hurl. As usual, we model the arrival time of vehicles as Poisson distributed random variables, with the intervals following the probability density function: With 1/the average traffic flow rate.2. Vehicles are homogene

21、ous. a) All vehicles are in the same length of 4.5m. b) The distance between two vehicles waiting paying for the toll is the 1.5m. c) Empirically, it takes a vehicle about 25sec 30sec (27sec on average) to speed up from 0km/h to 80km/h.d) Empirically, the vehicle has to slide 40m until it can brake

22、at a speed of 80km/he) The distance between two vehicles in the same lane getting out of the departure of the toll plaza is at least 10m so as to avoid collision.About the tollbooths1. All the tollbooths are independent. They provide the service for the vehicles independently, thus they have no idea

23、 of correlating the service time to avoid the congestion in the out poll which as the empirical parameter shows is 60s.About the in-pool & out-pool1. Empirically, the distance between the entry of toll plaza and each tollbooth is 200m. So does each tollbooth to the departure. (shown in Figure 1)2. T

24、he capacity of the in-pool & out-pool is based on their shapes.3. Ci = 16sec.4. Co = 20sec5. = 5sec.6. = 10sec.Some Facts Derived from Assumptions1. The acceleration and deceleration of a vehicle can be evaluated:(mentioned in Assumption: vehicles. 2. c)(mentioned in Assumption: vehicles. 2. d)2. mi

25、nLeavingIntervalInSingleLane can be evaluated:Interval = 5sec (mentioned in Assumption: vehicles. 2.e) and acceleration value)Notes. If we assume a vehicle wont start to leave until its precedent is on its way with 10m distance, then according to acceleration data, we can derive: 3. The pool size of

26、 both in-pool and out-pool satisfies the following function: Note: We can derive the expressions from increment analysis. If numOfBooths increase by one, the pool will increase by 10 slots. If doing so, the area of pool will grow by one triangle, as shown in the following figure. How many vehicles t

27、he triangle can hold? The triangle has a height of 200 meters long and a bottom, able to hold one vehicle (a tollbooth there). We assume each vehicle is 4.5 meters long and shares a free space of 5.5 meters with its neighbors. By division, 20 vehicles are able to pile in a rectangle with height of 2

28、00 meters and width of 1 vehicle, whose area is twice as much as that of triangle. Thus such a triangle can hold 10 vehicles, which is just the coefficient associated with numOfBooths. On the other hand, if numOfLanes increase by one, the process of analysis is almost the same except that another to

29、llbooth has to be added so as to prevent congestion. Thus the coefficient associated with numOfLanes is 20.Figure 2: Illustration on increment analysisAnalysis and Model DesignOverviewOur goal is to set up a model to get to know the behavior of toll plaza serving, and then try to use that model to w

30、ork out the optimal number of tollbooths, with respect to the number of lanes and the definition of “optimal” we raised.AnalysisTwo places are hot spots of congestion, the place before the tollbooths where there could be a very long line (“the former congestion”) and the place before the departure o

31、f toll plaza (“the latter congestion”). The former congestion is related to the serving time of tollbooths, the smaller the serving time is, the less probable will this congestion happen. The latter congestion is related to the leaving rate of the departure and the number of vehicles staying in that

32、 position.These two kinds of congestion are not uncorrelated. Once the latter congestion happens, the former one will happen sooner or later as the only drain of the traffic flow is blocked. However, the former congestion will not have such effects.We can image that the former congestion will happen if the arrival of the vehicles has exceeded the capacity of the toll