The Keep-Right-Except-To-Pass RuleSUMMARYDriving automobiles on the right is the rule in some countries, the purpose of this paper is to analyze different traffic rules through establishing models.Firstly, we established five models: car-following model, cellular automata model, fluid-dynamics model, speed-safety model and two-lane cellular automata model. We analyzed the models and processed the data by using SPSS and MATLAB, and then we simulated the data by utilizing VISSIM.For question one, we got the indicators of high load degree (6.1≥) and low load degree (7.0≤) by the definition. By comparison, we chose the car-following model and cellular automata model to analyze its impact on traffic flow. Utilizing speed-safety model, we discussed the road safety. We examined the tradeoffs between traffic flow and safety intuitively through diagrams. To a certain degree, the rule is effective in promoting traffic flow.For question two,the original rule is not the most effective one, based on the two-lane cellular automata model, we put forward the more effective rules: the lane-changing rule and the merging rule. It turned out that the rules can promote 8.5% traffic flow and prevent blocking.For question three, according to the data by simulating and the ITS cellular automata model, we received a figure which is the comparison of ITS cellular automaton model and the original model. Then we drew a conclusion that the traffic condition improved a lot under the ITS (Intelligent Transportation System). In this case, the traffic flow promoted 27.3% and the road safety improved significantly.Key words:Traffic flow Cellular automata model ITS VISSIM Car-following modelContents1. Introduction (4)2. The Description of the Problem (4)2.1Build a mathematical model to analyze this rule (4)2.2 Put forward more effective rules (4)2.3 Intelligent Traffic System (4)3. Problem Analysis (5)3.1 Problem One (5)3.2 Problem two....... (5)3.3 Problem three (5)4.Symbols and assumptions (5)4.1Symbols (5)4.2 Assumptions (6)5.Models (7)5.1 Model A: The car-following model (7)5.2 Model B: Microscopic – cellular automata Model (8)5.3 Model C: The Macroscopic – fluid-dynamics Model (10)5.4 Model D: The Speed-safety Model (11)5.5 Model E: The Two-lane cellular automata Model (12)6.Solution and Conclusions of the problem (13)6.1 Problem One (13)6.1.1 The Road load (13)6.1.2 The relationship between vehicle flux and load degree (15)6.1.3 The relationship of traffic safety and load degree (16)6.1.4 Different maximum-speed-limit influence on traffic (17)6.2 Problem Two (18)6.2.1The Lane-Changing rule of different speed (18)6.2.2 Merging Rule (20)6.2.3 The Keep-Left Rule (21)6.3 Problem Three (21)6.3.1 Freeway Tele-Traffic Speed Intelligent Control System (21)6.3.2 The cellular automata model of ITS (23)6.3.3 The cellular automata model of ITS (23)6.3.4 The security system based on ITS (25)7. Strengths and Weaknesses (25)7.1 Strengths (25)7.2 Weaknesses (25)8. References (26)I. IntroductionWith increasing number of vehicles, it is well known that the traffic problem becomes an important topic in nowadays. It is common that driving automobiles on the right is the rule and overtaking is allowed in the vehicular motion process on the multi-lane freeways. In this case, safety and the rate of the vehicles have a relationship with the traffic flux.Whether driving automobiles on the right or on the left do not have a strict boundaries, it is only a custom. We established the Lane-Changing rule of different speed respective and the Merging Rule. In countries where driving automobiles on the left is the norm, the rules can also be used in the keep-left rule.Intelligent Transportation Systems (ITS) are effective ways to solve or alleviate various traffic problems. The main functions of expressway traffic safety early-warning system are to provide corresponding control strategy and prevent the happening of accidents, by the traffic conditions and meteorological conditions, traffic flow characteristics, and so on.II. The Description of the Problem2.1 Build a mathematical model to analyze this rule.Build and analyze a mathematical model to analyze the performance of this rule in light and heavy traffic. You may wish to examine tradeoffs between traffic Flux and safety, the role of under- or over-posted speed limits (that is, speed limits that are too low or too high), and/or other factors that may not be explicitly called out in this problem statement.2.2Put forward more effective rulesIs this rule effective in promoting better traffic flux? If not, suggest and analyze alternatives (to include possibly no rule of this kind at all) that might promote greater traffic flux, safety, and/or other factors that you deem important. In countries where driving automobiles on the left is the norm, argue whether or not your solution can be carried over with a simple change of orientation, or would additional requirements be needed.2.3 Intelligent Traffic SystemLastly, the rule as stated above relies upon human judgment for compliance. If vehicle transportation on the same roadway was fully under the control of an intelligent system –either part of the road network or imbedded in the design of all vehicles using the roadway – to what extent would this change the results of your earlier analysis?III. Problem Analysis3.1 Problem OneIn order to verify whether the rules are effective, we established mathematical models to explain the relationship among the safety, traffic flow, density and speed. Then, it needs to consider how to adopt our models in the analysis of the performance of this rule in light and heavy traffic. At last, we give a relationship of traffic flux and load degree under the rules.3.2 Problem twoThrough the above analysis and access to a large number of data, we can see that the above rule is effective to some extent. But the traffic flux and safety easily affected by road load degree under the rule, therefore, we put forward a more effective rule. We established the Lane-Changing rule of different speed respective and the Merging Rule. The rules can also be used in the keep-left rule.3.3 Problem threeThe increasing demand for transport creates a huge challenge to local and central governments. ITS technologies and concepts have begun to be embedded within the transportation system. We established the ITS cellular automata model and it played an important role in improving the traffic flow, ITS can increase the road safety too.IV. Symbols and assumptions4.1 Symbols Table 1 Symbols and instructionsSymbolsInstructions AR accident rate km veh time ⋅810/σ the speed standard deviation h km /Death traffic accident death rate %V ∆ vehicle operating speed difference h km /Nthe annual average accident frequency times/yearVaverage Speed h km / CR millions of vehicle kilometers casualty rate, time/million vehiclekilometerst i g ,the load degree of section i at time t t i d ,the actual density of section i at time t cri dthe critical density of section i Llength of vehicle dv density of vehicles t i g , the load degree of section i at time tt i occ , the time occupancy ratio of section i at time tcri occ the time occupancy ratio of section i at the maximum capacitythe traffic fluxL the total lengtha the vehicle rate of decelerationmax v the maximum speed limitl Nthe total number of vehicles 4.2 Assumptions●Assume that the vehicles on the freeway are same type ●Assumptions used in modeling examples demonstrate true and reliable data ●Assuming the vehicles are all Normal driving ● Assume that the freeway without forkV. Models5.1 Model A: The car-following modelAs shown in the Fig.1, taking a special case of the car-following model in which vehicle are evenly spaced long the road and move with uniform velocity. With 0d , equal to the car length plus a minimum gap between vehicle, and the distance headway d between vehicles.]1[ Thus)(0d d N L l += With a densitydd L N dv l +==01Fig.1 simple graphThe flux φ is a function of density vdv dv =)(φA choice has to be made to decide the safe velocity for a given distance. One approach could be that a car must be able to come to a complete stop within that distance.ad v 2-=,0<aHowever this approach is inappropriate because driver reaction time isn’t taken into account and drivers generally don’t drive in this manner. A more appropriate approach is to use the road safety “Two -second rule”, 0t d v = Speed limit now needs to be taken into account, otherwise, as the density goes to zero, velocities would go to infinity,⎪⎪⎭⎫ ⎝⎛=0max ,min t d v v This naturally gives rise to the two phases in the fundamental diagram, the free flowphase max 0v t d >and the jammed phase max 0v t d < The resulting flux is then given bydd v d +=0max )(φ , free-flow )()(00d d t dd +=φ, jammed Or ⎪⎩⎪⎨⎧-+≤=-otherwise t dv d d t v dv for dv v dv 00100max max 1)()(φ5.2 Model B: Microscopic – cellular automata ModelCellular automata (CA) models are an idealization of physical systems by using discrete time and space and each unit can only have a finite number of discrete states. CA models are popular because in general they are highly computationally efficient due to their use of integer values for time, space and state. This is the model I have chosen to base my numerical simulation on and it is discussed in more detail as below.In the Nagel-Schreckenberg model (NaSch), the road is divided into discrete cells or lattice sites; each cell is either empty or occupied by exactly one car at each time step. Position, velocity, acceleration and time are treated as discrete integer variables making this a computationally efficient model. The n’t h vehicle can have a velocity max ,,2,1,0v v n =. As a first step n v the road is initialized randomly as being occupied 1 or unoccupied 0. At each occupied site, a car is assigned an integer velocity max v v ≤. Car positions are updated in parallel. Their velocities are set according to the following rules. i. Accelerate: All vehicles are accelerated by one until a maximum of max v .This represents how drivers like to drive at the speed limit where possible.),min(max 1v v v n n +→ii. Decelerate: If there is a car closer than n v , decelerate so as not to crash.),min(1-→n n n d v viii. Randomization: With a given probability p , randomly decrease the velocity by one. This represents imperfect driving and over-braking.)0,1max(-→n n v v With probability piv. Move: With all the velocities set, the position of each car is updated.n n n v x x +→These steps can be best understood with a visual example in Fig.2. To avoid creating and destroying vehicle at either end of the road periodic boundary conditions can be used. For L v x n n >+, where L is the length of the road:L v x x n n n -+→Fig.2 Example for the application of the update rules.We have assumed max v =2 and p =1/3.Therefore, on average one-third of the vehicle qualifying will slow down in the randomization step.An addition to the NaSch model is t he “slow to start” probability p . When vehicle come to a complete stop, there is a probability p that they do not move accelerate at all,in that time step. This reproduces the phenomenon of how drivers are slow to move off from a standing start. In real situations this further add to the problem of phantom traffic jams and the size of the tail back associated with them.]2[5.3 Model C: The Macroscopic – fluid-dynamics ModelAt a macroscopic scale traffic on a road appears to Flux like a fluid in a stream. A macroscopic theory can be developed by adapting the hydrodynamic theory of fluids. Traffic can be thought of as essentially a one-dimensional comprisable fluid. Away from junctions no vehicles enter or leave the system so there is a conservation of vehicles.]3[ Speed v ,density dv and flux φare called three elements of traffic flux ，flux is defined as the product of density and speed ：vdv =φWhen there is a few vehicle, vehicles run with free-flux speed f v ,We defined this state of traffic flux as sparse flux; With the increase of vehicle ，the running speed will decrease, When the vehicle density reaches a critical value(Blocking density j k ),there will be a traffic jam.According to the differential equation of motion ，we established a multiple lanes suitable for overtaking.Suppose flux of in and out of the lane line is r ，we can derive the equation belowdxdr x t dv =∂∂+∂∂φ Overtaking rate n defined as the ratio of overtaking flux and driveway flux.φ/r n =,)1,1(-∈n We can get the speed-density model as below⎪⎪⎪⎩⎪⎪⎪⎨⎧<<-≤≤----≤≤-+=nj n j r j f n j f n j r j f n f n j f r f f e k dv e k e k dv m v e k dv k m m e k e dv k m v e k m m e k dv k e dv k m m v v 21)1(2114141141])1([φφφ Speed is a function of time and distance, thus speed can be described by),(t x v v =Acceleration is the derivative of velocity versus timexv v t v dt dx x v t v dt t x dv dt dv ∂∂+∂∂=∙∂∂+∂∂===),(αWe transformed the influence of overtaking into viscous resistance w τ and we used macro-variable instead of micro-variable, therefore we can getw e t t x v dv v Td dv τ+-=)],()([1Viscous resistance xdvn v v v f w ∂∂+-=])([τ We can derive the equations belowxdv n v v v v dv v T x v v t v f e ∂∂+-+-=∂∂+∂∂])([])([1 We can derive the equations belowt r u k u k xt k k n i n i n i ni n i n i n i ∆+-∆∆+=--+)(111 )}]()([)(1)(){(1111n i n i n i f n i n i e n i n i e n i n i n i n i k k v v v v v v Tv v v v v x t v v -+-+-+-⋅-+∆∆+=---+T he above formula satisfy the condition mvv k dv x t f j∆∆≤∆5.4 Model D: The Speed-safety ModelThe frequency of accidents, mainly related to the speed discreteness, rather than magnitude of the speed. We made regression analysis to accident rate, the speed standard deviation and average velocity.The relational model []4are shown as fluxing:Speed change is directly related with several of the accident rate, average speed andthe calculation of driving speed discreteness, the greater the chance of traffic accidents. Form the model, we can see that the rate of speed change is directly related to the probability of accidents, average speed and the calculation of driving speed discreteness is higher, the likelihood of accidents are greater.σ0553.05839.9e AR =Car in the traffic accident ability may cause transformation happen, the kinetic energy will turn into energy and thermal energy to make the vehicle deformation, and the kinetic energy depends on the quality and speed. Vehicle energy changes resulting from accident square of the quality and speed. Obviously, the greater physical energy is, the more is the energy conversion , the more serious is the consequences of the accident , all above are causing greater casualties .The approximate function is shown in following []5:4)24.114(V Death ∆=The Burg formula:V VN ∆=536.1ln By accessing to information , we drawn a conclusion that when the average running speed decreases 1h km /, traffic accident casualty rate will decrease by 7%.The formula is:366.30405.000268.0-+-=Diff V CR5.5 Model E: The Two-lane cellular automata Model),1min(max v v d i i +<i other i d d >, safe back i d d >,other i d ,: the cell number of vehicle i and the former vehicle on the adjacent lane at time t, back i d ,: the cell number of vehicle i and the latter vehicle on the adjacent lane at time t, safe d :the safety distance.),min(max 1v v d i i +< refers to the lane changing motivation which show that vehicle i can not run as the expected speed on thelane .The vehicle i in the adjacent lane have better driving conditions than the original lane.i other i d d >, refers to the condition of the adjacent lane, safe back i d d >,refers to safety conditions which show that lane-changing will not cause congestion of the latter vehicles.VI. Solution and Conclusions of the problem6.1 Problem One6.1.1 The Road load1) Road load analysis is an important research direction of the road capacity analysis and management, commonly used in road or intersection level of service evaluation. It is calculated based on the ratio of road traffic to their capacity, i.e. C V /. Traffic density is an important indicator of the theory , which is the number of the existence of vehicle at some point and it reflects the road space utilization. When road traffic Flux satisfies its traffic capacity, road has the highest utilization rate, traffic running well, density value at this time is called the critical density. We define roadways actual density and the ratio of the critical density as a new calculation method, and as the important indexes for evaluation of road traffic running state.]6[ 2) The definition of road loadThe road load degree refers to at one point, the ratio of the actual density and the density of largest capacity in this section. Road load degree describes the levels of usage use of the road overall capacity, and reflects the utilization of road resources in space. calculation method ：crit i t i d d g ,,=Obviously,0,≥s t i g .The load degree of the section closer to 0 , indicating that the lower extent of road traffic load , the better traffic running state; the load degree of the section closer to 1 , indicates that traffic load is close to saturation, the traffic Flux is close to the traffic capacity; the greater the load degree of the section, indicating that the heavier extent of road traffic load , the worse traffic running state.The relationship between road density and time occupancy ratio: time occupancy ratio refers to the ratio of the vehicle occupancy time accumulative total value and the determination of the time on the observation section of the road. Time occupancy ratio is not only related to traffic volume, but also related to the length and density of the vehicle. Assume the type and length of vehicle are the same of the observation section of the road, we can get the formula:dv L occ ⨯=In the case of vehicles with a certain length, the density has a linear relation with time occupancy ratio. Therefore, we can use the ratio of the actual time occupancy ratio and the time occupancy ratio at the maximum capacity to calculate the load degree of the section under specified conditions.]7[crit i crit i crit i s ti occ occ d L d L d d g ,,,,=⨯⨯==The relationship of load degree and vehicle speed is shown in Table 2 , we got the Fig.3by software Excel.Table 2 The standard of road evaluationFig.3 The relation ship of load degree and speedThe level o f trafficstateSpeed (km) The loaddegreeClassify the degree very smooth105~110 ≤0.2low loadcondition100~1050.2~0.4 basically smooth 95~100 0.4~0.6 90~950.6~0.7 travel steadily 85~90 0.7~0.9 normal loadcondition80~850.9~1 travel slowly 75~80 1~1.2 70~751.2~1.4 general congestion65~70 1.4~1.6 high loadcondition60~65 1.6~1.8 severe congestion55~60 1.8~250~55≥2Based on the level of traffic state, we defined high load condition when load degree is higher than 1.6 and low load condition when load degree is less than 0.76.1.2 The relationship between vehicle flux and load degreeThe relationship between vehicle flux and load degree gives a good understanding of what is happening to traffic. By comparing Model B and Model C, we found that the latter is more effective. According to the Model B, We get a combination of statistics as shown in Table 3,by analyze the statistics by software MATLAB, we got a diagram which is a plot of traffic flux and density as shown in Fig.4. At low densities when the traffic is in a free Fluxing phase, flux increases linearly with density. At high densities the traffic is in a congested state and flux decreases with density to a limit of no vehicles moving on a full road. Near the point of maximum flux there is a phase transition from free traffic to congested traffic.]8[Table 3The relationship of flux and densityFig.4 The relationship of flux and density6.1.3 The relationship of traffic safety and load degreeDensity /(vehicle per meter)57.915.622.128.339.751.762.998.6182.3Flux/(ve hicle per second)0 360.5 940.531594.322287.372973.362610.82336.841631.221087.37561.06What is known to all is that running speed and the speed directly affects the probability of highway accidents. If running speed is too high, the driver awareness of road conditions, the judgment on the driving environment will reduce, resulting in increasing braking distance[]9. On the contrary,speed is too slow can make overtaking frequency increased so that the probability of an increase in traffic accidents.Through access to information we knew that when the speed is greater than 60 km/h, every 5 km/h, the accident rate is twice higher than the original, at the same time, the severity of the accident will show exponential growth]10[as shown in Table 4.[]11.According to the Model D, we got the intuitive graphical as shown in Fig.5by SPSS.Table 4 The relationship between speed and accident riskThe speed of the vehicle (km) The relative value of accident50~60 1.0060-70 2.0070~80 4.1680~90 10.6090~100 31.81100~110 56.55Fig.5 The relationship of traffic safety and load degreeIn summary, the severity of the accident and the speed is not a linear relationship , the accident severity will produce more important along with the change of velocity change,the higher operating speed value , the greater the amount of change in the speed when the accident occurred. At last, the accident seriously degree is higher.6.1.4 Different maximum-speed-limit influence on trafficWe got a combination of statistics from document literature, by the statistics, we made the Table 5.]12[We got the intuitive graphical as shown in Fig.6 by MATLAB.Table 5 The relationship of flux and max vFig.6 The relationship of flux and max vIn summary, at low load degree when the traffic is in a free fluxing phase, flux increases linearly with density; at high load degree the traffic is in a congested state and flux decreases with density to a limit of no vehicles moving on a full road; near the point of maximum flux there is a phase transition from free traffic to congested traffic. The severity of the accident and the speed is not a linear relationship , the accident severity will produce more important along with the change of velocity change, the higher operating speed value , the greater the amount of change in the speed when the accident occurred.max v /(h km /)10 20 30 40 50 60 70 80 90 100 110 120Flux/(v ehicle/s) 0.081 0.154 0.229 0.286 0.36 0.356 0.362 0.36 0.359 0.367 0.369 0.3626.2 Problem TwoThrough the above analysis, we can see that the above rule is effective to some extent. But the traffic flux and safety easily affected by road load degree under the rule, therefore, we put forward a more effective rule.6.2.1 The Lane-Changing rule of different speedAccording to the Model B (the Na model), the road is divided into discrete cells or lattice sites; each cell is either empty or occupied by exactly one car at each time step. We established the Lane-Changing rule of different speed respectively. The NaSch model as follows:The acceleration process )1,min(max +→i i v v v The moderating process ),min(i i i d v v →Delay position stochastic as probability p )0,1max(-→i i v v Update position i i i v x x +=Based on the Model E (The STCA model), we put forward the lane-changing rule according to the real characteristics of changing lane.]13[ The rule is as follow:1=i T01=+i T),1min(max 1v v d i i +<-2,≥back i d i other i d d >, other back i v v ,≥change f P rand ,()<0=i T 11=-i T),1min(max 11v v d i i +<--safe back i d d >, ),1min(max ,v v d i other i +≥change s P rand ,()<Each lane can be seen as a one-dimensional cellular chain composed of cellular automata, the traffic is the mixed traffic flow, and the vehicle is divided into two types: fast type vehicle and slow type vehicle. Each vehicle takes up a cellular automata, we simulate by using periodic boundary conditions. So we can derive the equations belowL N l l 2/=ρ∑==lN i lilvN vl 01i l l v ρφ=Utilizing the statistics from literature ]14[, we got Fig.7 by the software MATLAB. Fig.7 shows that the average traffic flow under the new rule is higher the original rule, and it can still maintain traffic flow under the high density. From a certain extent, the newrule is more efficient than the original rule.Fig.7 The relationship of flux and densityAnother advantage of this new rule is to avoid congestion, as shown in the Fig.8, (a) is the jam of the slow vehicle, and F is the fast vehicle, S is the slow vehicle, C is the random vehicle. (b) shows that the slow vehicle ’s active lane changing make jam disappear at the next time step, (c) shows that the slow vehicle ’s active lane changing make jam disappear at the next time step.C cF S(a)C SF(b)C FS(c)Fig.8 The method of preventing the formation of block6.2.2 Merging RuleAs a first step to developing the model to reproduce a real road network we chose to have two roads merge into one. This is analogous to an on-ramp merging onto a single lane motorway. As shown in Fig.9 the two roads exist in parallel, independent of each other. Then there is a merging zone in which cars on road B must try and merge onto road A before road B comes to an end. Each road is updated independently, alternating between each, for every time step.The system is very dependent on the rules chosen for merging traffic. Priority should be given to those already on the main road A and cars on road B should filter in when there is space for them to do so. Our initial set of Merging Rule allowed B cars to move over whenever there was space. This led to virtually no flow on A while B cars were free flowing. This was because a car from B might slot in directly in front of an A car causing it to come to a complete stop in the next time step and thus create a traffic jam on road A and thereby making it even easier for even more B cars to move in. We imposed the rule that a car on road B can only move over if there are two empty spaces free, i.e. the empty space that the car is moving into and an empty space behind the one that it is moving into.Fig.9 The schematic diagram of Merging RuleMerging Rule, for a car on road B in the “merging zone”:i. The car’s velocity is still set by the road that it is on.ii. The car will try and move over at the earliest possible time when there are at least two spaces available on road A.iii. If the car succeeds in moving over it will move up on that new road as much as it can, until it catches up with another car or to the limit that its velocity allows.iv. If it cannot move over, its velocity is reduced by one and it moves forward on its own road to the limit where it reaches the end of the road, at which point it must stop. It should be noted that the length of the merging zone can be set to one if it is required to model a T junction with a yield sign.6.2.3The Keep-Left RuleBy looking up a large number of date, we draw a conclusion that driving automobiles on the left is the same as driving automobiles on the right, so the rule can be used in the country where driving automobiles on the left.6.3Problem ThreeThe increasing demand for transport creates a huge challenge to local and central governments. As it is clear that simply building new roads will not be the answer, optimal transport strategies will be needed.]15[6.3.1 Freeway Tele-Traffic Speed Intelligent Control SystemFig.10 The function diagram of FTSICS systemAs Fig.10 shows, freeway Tele-Traffic Speed Intelligent Control System (FTSICS) should gather the information of each road and manage it. According to the state of road traffic information, weather information and road network structure, FTSICS can calculate the limit speed of each road according to the traffic state information, weather information, and road network structure information and send the message to motorist, thus making the overall coordination of network traffic state.]15[。
