1、For office use onlyT1T2T3T4Team Control Number70545Problem ChosenBFor office use onlyF1F2F3F42017MCM/ICMSummary SheetControl Time Model: Wait One Second and Start YourJourneySummaryPeople around the world are complaining about the traffic jams every day. Somepeople may spend more than 3 hours on the
2、ir way from office to home everyday ontheir cars, which makes them frustrated. And our problem is to investigate the methodsto optimize the design of toll plazas on highways, especially the area after merging,which is also a possible location where vehicles can be stuck in.Before establishment of ou
3、r models, we list some assumptions to make the real lifescenario easier to model.And then we start to analyze the existing models, from which we conclude theirstrengthsandweaknesses. Byinvestigatingtheircharacteristics,wegettheinspirationtoform our two new models: Control Time Model (CTM) and Waitin
4、g Area Model(WAM).In these two new models, we introduced a way of conducting control of the departuretimeofthevehiclesatthetollbooths. Thiskindofcontrolisbasicallypreventingthevehi-cles which are merging in the same lane from leaving the booths simultaneously, whichmay be a hidden danger for traffic
5、 accident or leads to a traffic jam. And we?ll continueto calculate the size and shape of the merging area according to our control methods andsome assumptions. After that, we introduce a method for finding the optimal mergingpattern based on both mathematical proof and computer simulations.After th
6、at, we run some simulations to find out the throughput, risk and cost of dif-ferent models, which are based on some statistical laws for real life situations. Then wecompare these three models in all factors with the help of statistical hypothesis testing,and conclude that the CTM is the best in gen
7、eral. The next section is about some slightmodification under different conditions such as including the self-driving car, and thedifferent arrangements in terms of merging patterns when the proportion of differenttypes of tollbooths vary.For the following section, we test our model by investigating
8、 the sensitivity of con-struction cost and throughput (per hour) in terms of some variables included in ourmodel, in order to justify the reliability of our model from different perspectives.Finally, we end our report by the conclusion part followed by strengths and weak-nesses analysis.Team # 70545
9、 Page 1 of 25PO Box 5042,Woodbridge,NJ 07095-5042.Dear New Jersey Turnpike Authority,We are a group of students having conducted a project focusing on solving the problemof merging after tollbooths. And we are sending this letter to let you know the problemsthat may exist in current toll plaza and s
10、uggesting some possible ways to improve it.First of all, by analyzing the design of existing toll plazas on the highways, we find outthe following weaknesses of them. At most toll plazas, the vehicles go disorderedly after tollfor following reasons. For some toll plazas, there are no leading lines a
11、fter toll, so driversjust go at their own discretion, thus their routes become unpredictable. And even there aresome orientation lines, if two vehicles arrive at the merging point at the same time, only oneof them can go through at one time and it will also cause some problems. Disorder aftertoll le
12、ads to both harm to the eciency of the highway entrance and more risk for tracaccidents.Therefore, to solve the above problem, we introduced a new model called “Control TimeModel” (CTM). In our model, the releases of cars are controlled at the booth, ensuring thatthe time interval from any two relea
13、ses of cars are more that the “safe time”. Therefore, themerging becomes more orderly and safer.We run some simulations with our models to test and compare the eciency between theCTM and current existing models. The result is that, the eciency of CTM is only slightlysmaller than existing model (abou
14、t 0.6 throughput less per lane per hour), but we haventconsidered the time wasted when accidents happen, which is a lot more likely to take placein existing models theoretically.And also according to the simulation, the risk of car crashes is also limited to a lowerlevel, which is 13.66% less than c
15、urrent existing models.In terms of expenditure of construction and maintenance, as our model is much moreorderly, the total length of the merging area can be reduced.Therefore, a decrease in the total expense is expected, let alone the fact that maintenanceof plaza could be a lot easier if accident
16、rate decreases distinctively.All in all, our conclusion is that the CTM performs better than existing models in allaspects including but not limited to eciency, risk elimination as well as total cost. Therefore,it is worthwhile to consider the future construction with our model design.Thank you for
17、considering our model and wish you a bright future.Sincerely,Team#70545MCM2017Team # 70545 Page 2 of 25Control Time Model: Wait One Second and Start YourJourneyContents1 Introduction 31.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Restatement of the probl
18、em . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Assumptions 33 Notations 44 Models 44.1 Existing Model(EM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.2 Control Time Model (CTM)
19、. . . . . . . . . . . . . . . . . . . . . . . . . . . 54.3 Waiting Area Model(WAM) . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Analysis and Results 95.1 Throughput . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.1.1 Throughput of existing model . . . . . . . .
20、 . . . . . . . . . . . . . . 95.1.2 Throughput of control time model . . . . . . . . . . . . . . . . . . . 105.1.3 Throughput of waiting area model . . . . . . . . . . . . . . . . . . . 115.1.4 Comparison and hypothesis tests of throughputs . . . . . . . . . . . . 115.2 Accident prevention . . . . .
21、 . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.3 Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Possible Modification 156.1 Mixture of autonomous vehicles and human-driving vehicles . . . . . . . . . 156.2 Change in proportion of dierent types of
22、 tollbooths . . . . . . . . . . . . . 157 Sensitivity test 167.1 Sensitivity test of throughput . . . . . . . . . . . . . . . . . . . . . . . . . . 167.2 Sensitivity test of fixed cost . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 Conclusion 179 Strengths and Weaknesses 199.1 Strengths .
23、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199.2 Weaknesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Appendix 22Team # 70545 Page 3 of 251 Introduction1.1 BackgroundTraveling on highways in recent years are quite convenient ways for peop
24、le to go to anothercity. With a compared high speed limit on highway, people are able to travel a lot faster.However, building highways are quite expensive and it also takes a lot for the maintenance.Therefore, building of toll plazas along the highway is a common practice all around the world.Meanw
25、hile, there are many factors to be considered during the design and the constructionof the tollbooths. Firstly, the lands and the construction fee for the booths and roads arevery expensive, so we need to minimize the area. And as the vehicles on the highway alwayshave a high speed, it is also quite
26、 dangerous when they want to merge into a lane. Finally,the design of the booths is also supposed to guarantee the eciency, which is described bythe quantity throughout the highway for a certain time interval.1.2 Restatement of the problemAs required by the question, we are supposed to determine the
27、 shape, size and mergingpattern at the toll plaza depended on number of lanes on the highway in one direction(L)andtotalnumberoftollboothsinonedirection(B). Meanwhile, we will also justifyour approach by comparing it with the existing models in terms of risk, cost and eciency.Whats more, the impleme
28、nt of our plan in various situations (e.g. mixture of the self-drivingvehicle and dierent proportion distribution of types of tollbooths) will also be shown.1.3 Literature reviewThe optimal number of tollbooths needed to minimize the average waiting time is well-studiedand simulated based on dierent
29、 real situations (Corwen et al, 2005). Though the definitionof “optimal” varies, similar suggestions have been given. Tollbooths should be implementedconforming to encouraged behaviors, e.g. faster booths should be put on the left (Spannet al, 2005), which is incorporated in our model. Some literatu
30、re suggested that tollboothsemploy no barrier to ensure a relatively smooth flow (Kane, 2005), but we contend that theuplift of barrier takes negligible amount of time, the benefit of which cannot be comparedwith the chaos and potential risk if some vehicles go through toll plaza directly. Therefore
31、,barriers are included in our model.2 AssumptionsTo simplify the real life situation, we will make the following assumptions as a start ofconstruction of our models. Most of the drivers are rational. They will choose the path as suggested by signalsat entrance, and act as risk averters. It is reason
32、able to make this assumption, becausewithout it, its meaningless to make any rules as people wont obey them.Team # 70545 Page 4 of 25 The arrival time lapse between the first vehicle and the second vehicle inthe heavy trac follows uniform distribution. By the heavy tracassumption,the second vehicle
33、comes to tollbooth before the first vehicle leaves. All the paths of the vehicles follow either constant speed motion or constantacceleration motion. As the average speed for the vehicles around the tollbooths arerelatively low. vehicles are queuing at the entrance of the tollbooth one by one with l
34、owspeed in the heavy trac situation . And we further assume them to be stationaryin our models. The service by human-stas at the tollbooth is quite time-consuming, andthe approximate mean service time is around 15 seconds, which follows anormal distribution. This assumption is appropriate, because d
35、river need to bringthe fee to the sta and the sta may need to prepare changes for it. And the timevaries from people and dierent situations. The length of vehicles is omitted during the calculation for the size andshape for the merging area. It is valid because the magnitude of the length of tollpla
36、za is a lot larger than the length of vehicles.3 NotationsNotation Definition UnitL Number of outgoing lanes in highway N/AB Number of tollbooths N/AD Distance between the tollbooth and the end of the plaza mdlWidth of the lane in highway mdbWidth of the lane in tollbooth area mr Radius for vehicles
37、 turning mThTime of toll service per vehicle human-staed tollbooths st0Time of control (to be defined in Section 4.2) svmSpeed limit in merging area m/sasStarting acceleration of vehicles after toll payment m/s2abDeceleration of braking vehicles m/s24 ModelsIn this section, we first introduce and an
38、alyze the existing model, and then two new modelsinvented by us. And the focus for the new models will only be the establishment of the firstnew model (control time model) and second new model (waiting area model) only deviatesfrom the first one slightly.Team # 70545 Page 5 of 254.1 Existing Model(E
39、M)Under our investigation, the existing solution for the merging after toll can be roughly dividedinto two types.The first solution is in short called “no solution”, which doesnt give the drivers anyinstruction after toll and let them go at their own discretion. Then the second and morecommonly used
40、 solution is to instruct the vehicles to merge at some pre-determined mergingpoint.For the first solution, the eciency of it is mainly based on the drivers own discretions,which can vary a lot among dierent people. And in the light trac situations, this solutionmay be optimal, for the reason that dr
41、ivers can go whatever path they what, and in otherwords their own eciency is maximized. Meanwhile, with the light tracassumption,thereare little probability to crash. However, it will be fairly chaotic in heavy trac, because itis very likely that drivers own optimal path can cross each other. And th
42、en they will mergeat any point so it is also hard to predict what drivers will do. Therefore, in heavy trac, itis neither ecient nor safe.And the second solution provides a relatively orderly merging pattern. Vehicles merge atsome certain merging points and at least the driving direction is predicta
43、ble as long as thedrivers are rational and always follow the instruction. However, some problems also occur atthe merging point. As for most time, the number of tollbooths are larger than the numberof lanes on the highway, there always exist the situation that vehicles from several dierentbooths nee
44、d to merge into one lane. So dierent vehicles from dierent booths can arrive atthe merging point simultaneously, and then only one of them can get through at a time andothers have to wait. This situation can be quite dangerous because drivers always want togo first, and it also increase the deficien
45、cy.4.2 Control Time Model (CTM)Given aforementioned deficiencies of existing merging pattern, we propose a new model,partially based on the current one. Instead of having all the vehicles moving and merging attheir own discretion, control time model will control the departure time of vehicles to ens
46、ureasmoothandsafeemergingprocess.Specifically, for situations where two booths merge into one lane, the second vehicle willonly be allowed to proceed t0seconds after the first vehicle moves forward. The time t0isdefined as the control time. Similarly, for situations where three booths merge into one
47、 lane,the third vehicle will be allowed to proceed t0seconds after the second vehicle moves forward,whilst the second vehicle t0seconds after the first vehicle, as shown in Figure 1 and Figure2.Team # 70545 Page 6 of 25Figure 1: Either A or B is open Figure 2: Either C, D or E is openIn this way, th
48、e regulated merging of the vehicles into another lane would be more ecientthan the situation where vehicles are proceeding without regulation, for drivers should taketime to make decision when multiple booths merge into one lane simultaneously, let alonethe risk for doing so.We model that vehicles s
49、tart with constant acceleration asuntil reach the maximum speedvmin the straight path. They then immediately starts merging into their prescribed road intwo consecutive tangent circle arcs.Figure 3: Two or three tollbooth egress lanes merge into one laneNow we evaluate the “appropriate” control time. When emergency happens, one vehicletake severe brake action with acceleration ab, after response time tres, the posterior vehicletake severe brake action with the same acceleration. Consider the distance of the two vehicles(t = 0 is the time when emergen
