A Stochastic Traffic Cellular Automata with Controlled Randomness

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1 A Stochastic Traffic Cellular Automata with Controlled Randomness Lorita Angeline 1, Mei Yeen Choong, Bih Lii Chua, Renee Ka Yin Chin, Kenneth Tze Kin Teo 2 Modelling, Simulation & Computing Laboratory Faculty of Engineering, Universiti Malaysia Sabah, Kota Kinabalu, Malaysia msclab@ums.edu.my, angeline.lorita@gmail.com 1, ktkteo@ieee.org 2 Abstract Traffic flow pattern appears to be random and makes it more challenging to model and analyse. This paper aims to present a traffic model based on NaSchr model, but with a revised rule for randomisation. User can trigger a manual braking on certain cars and its effect on traffic flow can be observed and analysed. The traffic cellular automata model with manual braking mode is applied to a single lane with open boundary conditions. A new variable bad-agent is introduced to represent the tailgater on road that cause the inevitable braketapping on drivers behind them, thus the emergence of traffic waves. The simulation results show that when cars come too close to each other at high speed, certainly it creates noise. This noise is the traffic waves that propagate backward. Keywords Cellular Automata; TCA; NaSchr model; traffic jam; traffic waves I. INTRODUCTION TO TRAFFIC CELLULAR AUTOMATA Due to the dynamic nature of traffic flow, one might consider it to be difficult to define a set of equations to model and estimate traffic congestion. However, physicists and traffic engineers have been employed modeling techniques to analyse traffic patterns for several decades [1]. Traffic modelling can be characterized into three models: microscopic models, mesoscopic models and macroscopic models. Cellular Automata (CA) is a discrete system with grid of simple finite state machines. The system changes its states synchronously, according to a local update rule that specifies the new state of each cell based on the old states of its neighbours [2]. The implication of discrete is that space, time and properties of the automaton can take only a finite number of states. Thus it belongs to the microscopic family models. The aim of this paper is to study the popular NaSchr benchmark model. This paper also presents a traffic model based on NaSchr model and revised its rules. In this new model, user able to mimick the randomisation in traffic nature by triggering a manual brake on certain cars at certain times and analysed its effect on traffic flow. This paper is organised as follows: the next section comprises the study of NaSchr model and its configuration. Section III presents a new stochastic traffic cellular automata (TCA) model with manual braking. Section IV presents an in depth experimental results and analysis. Overall project and future extensions are presented in Section V. CA Terminology Boundary conditions: Since the road is finite, boundary conditions need to be included. There are two possibilities: open and close boundary conditions. Cell (site): a single element of a cellular space, the smallest unit of the space. Cellular space: A lattice (grid) space made up of cells. Cellular automaton: A structure built in a cellular space, an automaton built of cells. The term cellular automata is plural. If dealing with just one system a cellular automaton (singular). Also refer as CA. Closed boundary conditions: A car entering the road, reaches the end and restarts at the beginning. The road actually does not have a beginning or end. Configuration: A snapshot of all cell states, representing a single point in time. Usually is the starting point or a result of running a cellular space. Generation: CA living over a period of time (it is not a real world time). Passage of time in a cellular space is measured in generations (some paper refer to it as iteration/simulation steps/simulation lengths) Local rule: The rule governing the transition between states. It is called local because it only uses the neighbourhood as its input. Neighbourhood: The surrounding cells that influence the next state. The simplest neighbourhood in one dimension for any given cell would be the cell itself and its two adjacent neighbours: one to the left and one to the right. Open boundary conditions: A car entering the road, reaches the end and can leave the road (to another section). CA terminologies are based on [2 4] /17/$ IEEE 68

2 II. NASCHR PROTOTYPE MODEL NaSchr model is a stochastic discrete automaton model to simulate freeway traffic. Reference [5], Kai Nagel and Schresckenberg present this model in The authors applied the Monte-Carlo simulations of the model to show a change from smooth traffic flow to stop-and-go traffic waves as vehicle densities increased. The model is defined on a onedimensional array of L sites (cells) with open or periodic boundary conditions. Each site (cell) may either be taken by one vehicle or it may be vacant. Each vehicle has an integer speed with values between zero and a defined v max. One update of the system consists of the following four consecutive rules, which are performed in parallel for all vehicles. Table 1 shows NaSchr model rules. Rule 3 is necessary in simulating realistic traffic flow; otherwise the dynamics is entirely deterministic. It takes into account natural speed fluctuations due to human behaviour and varying external conditions. Without this uncertainty, every initial configuration of the vehicles and its resultant velocities will yield a stationary pattern and the flow will be appear shifted backward, one site per time step. Reference [5] presents the results from systems with closed boundary conditions. It is also known as traffic on a circle. As the total number N of cars in the circle cannot change throughout the dynamics system, it is feasible to define a constant density as in (1). Laminar (smooth) traffic at low densities and congestion clusters (small jams) at higher densities. Congestion clusters are emerged randomly due to speed-fluctuations of the cars. TABLE I. 1) Acceleration: 2) Deceleration 3) Randomisation (in braking) 4) Car advancement x i v i v max g i NASCHR MODEL RULES NaSchr model rules if (v i < v max && g i > v i + 1) v i = v i + 1 if (g i < v i ) v i = g i - 1 with probability p, if v > 0 v = v - 1 each vehicle is advanced by v sites. = position of the ith vehicle = speed of ith vehicle = maximum speed = gap between the ith and the (i+j)th vehicle (i.e., vehicle immediately ahead) and is given by g i = x i+1 - x i - 1 (1) TABLE II. NASCHR MODEL PARAMETERS VALUE Fig. 1 shows the fundamental diagram for the NaSchr model. Three values of ρ is applied to demonstrate the effect of randomness. Table II shows the parameters value. It can clearly be seen in Fig.1 that a shift in density takes place between 0.1 < ρ < 0.2. Two observations can be made from NaSchr model: Fig. 1 (a) Fig. 1 (b) Fig. 1 (c) Road length Num. of sample V max Randomess, ρ Length (a) ρ = 0.1 (b) ρ = 0.6 (c) ρ = 0.9 Fig. 1. NaSchr model rule 3 (randomness, ρ) 69

3 III. A TCA MODEL WITH CONTROLLED RANDOMNESS A TCA model is developed with open boundary conditions, where cars (samples) entering the road, reach the end and leave the road. This model is based on NaSchr stochastic model, with an additional rule manual braking mode. Manual braking in this simulation is different from the previous NaSchr model, where random braking is applied randomly to any car at any time. It is introduced into the simulation in order to analyse the emergence of traffic jam. The model can be described with one main algorithm and two sub-programs. Table III shows the algorithm for the main program. There are several parameters need to be set by user. Such parameters are road length, chart width, number of cars travel on road and maximum velocity. Table IV shows the road length is set to 400, however such lengthy road will be hard to observe and analyse. Thus, user can set the road to be sectioned into 50 cells per row. The extra cells will be wrapped around to second row and so on. Setting chart width to 50 does this. The while loop is terminated if either of the two conditions are met. Condition 1: All cars leave the road. Condition 2: User halt the program. The main program also contains two sub-programs, velocity update and car advancement. Table IV and Table V shows the sub-programs respectively. Table IV is responsible to update the car s velocity according to certain rules. The program is set to seek maximum velocity with collision avoidance. User also able to manually apply brake to certain cars. This feature is useful to analyse the emergence of traffic jam. This sub-program is terminated if it reached the end of the road length. Table V is responsible for car advancement on the cellular automata platform, and exiting the road. TABLE III. TCA MAIN ALGORITHM User settings: 1 road_length = 400; chart_width = 50; numberofcars = 10; vmax = 3; 2 while True 3 if (CountAddedCar < numberofcars) 4 Add car into simulation; 5 if (CountAddedCar > 0 && CountAddedCar > numberofcars) 6 break while loop; %break while loop at line 2 7 else 8 go to sub-program velocity update ; 9 go to sub-program car movement ; 10 if (StopProgramSignal = 1) % user halt the program 11 break while loop; %break while loop at line 2 12 else 13 Repeat while loop (line 2); 14 end 15 end 16 end 17 end 18 Update variables; 19 Calculate density; 20 Calculate flowrate; TABLE IV. SUB-PROGRAM VELOCITY UPDATES 1 Variables: index referring to cell under consideration bf used to flag collision avoidance 2 distance = 0; 3 bf = 0; 4 for (j = 1:road_length; j < road_length; j++) 5 for (k = 1:vmax; k<vmax; k++) % seek vmax ahead 6 distance = k; 7 if (j+k <= road_length) % safe to land here? 8 index = j+k; 9 else 10 index = j+k-road_length; % wrapping 11 CarOutNowVelo = 1; % this car will exit from road in the next simulation 12 end 13 if (road(index) ==1 && CarOutNowVelo == 0) 14 % if there s another car on index, break for loop (line 5) 15 bf =1; 16 break for loop; 17 end 18 end 19 if (velocities(j) < vmax) % acceleration 20 velocities(j) = velocities(j)+1; 21 end 22 if (velocities(j) > distance -1 && bf == 1) % collision avoidance 23 velocities(j) = distance - 1; 24 end 25 if (ManualBraking = 1) % Manual braking triggered by user 26 velocities(j) = velocities(j) -1; % user can adjust the weight of the manual braking. 27 end 28 end % loop up line 4, until reach end of road_length TABLE V. SUB-PROGRAM CAR ADVANCEMENT Variables: 1 CarOutNow is used to check for car exiting from road 2 CarOutNow = 0; 3 for (j = 1:road_length; j < road_length; j++) 4 if (road(j) == 1) 5 if (j + velocities(j) <= road_length) 6 index = j + velocities(j); 7 else 8 index = j + velocities(j) road_length; % wrapping CarOutNow = 1; % this car will exit from road 9 in the next simulation 10 end 11 if (CarOutNow == 1) % if current j exiting from road road_next(index) = 0; % set current j car to exit 12 from road in next simulation step velocities_next(index) = 0; % set velocity of current j car 13 to 0 in the next simulation step 14 CarOutNow = 0; 15 else 16 road_next(index) = 1; 17 velocities_next(index) = velocities(j); 18 end 19 end 20 end % loop up line 3, until reach end of road_length 70

4 IV. EXPERIMENTAL RESULTS AND DISCUSSIONS A new variable is introduced into the NaSchr prototype model. This variable also known as bad-agent, is intended to perform manual braking. However, one important hypothesis to be emphasized here is that, traffic waves are natural byproduct of tailgatting. Tailgatting caused the inevitable braketapping in order to avoid collision. In this traffic model, the brake-tapping is used to re-create the hypotesised traffic waves, but in no way putting the blame on drivers that hit the brake. Although, brake tapping has been discussed as the cause of traffic waves [6]. The first part of this experiment will analyse the effect of one bad-agent on road. The second part will analyse the effect of two bad-agent on road. A. One Bad-Agent One car is marked, to represent the bad-agent. The parameters of the model is set accordingly (road_length = 400, num.of samples = 10, v max = 3, bad-agent = car-3). Fig. 2 4 show the results when braking weight = 1 (refer to Table VI). Notice how the space between cars is reduced, although none of the vehicle is forced to slow down. Although travelling at maximum speed, the cars still able to avoid collision. In accordance with the NaSchr rule, vehicle is free to accelerate to maximum speed, as long as the gap between the ith and the (i+j)th vehicle is greater than speed + 1 (gi > vi + 1). Fig. 5 9 show the results when braking weight = 2. In the initial simulation the slightly heavier brake caused a bigger gap (8 cells) between car-2 and car-3 (previously, brake value-1 only caused 8 cells of gap after the 3 rd braking). Fig. 5 shows the simulation steps at 47, where the speed would be 1 after braking, however the yellow cell indicates the speed quickly resume to speed 2. Fig. 6 9 show the simulation after the 6 th manual braking is triggered on car-3. Notice how the yellow cell (reduced speed) travels backwards. A clear pattern of such traffic waves travel backwards can be seen in the gap between cars, however all cars managed to resume to it maximum speed once the waves dissipated (refer to Table VII). Fig. 5. Steps = 47, bad-agent = 1, braking weight = 2 Fig. 6. Steps = 86, bad-agent = 1, braking weight = 2 Fig. 7. Steps = 88, bad-agent = 1, braking weight = 2 Fig. 8. Steps = 91, bad-agent = 1, braking weight = 2 Fig. 9. Steps = 92, bad-agent = 1, braking weight = 2 TABLE VI. RESULTS, BAD-AGENT = 1, BRAKING WEIGHT = , 5, 5, 5, 5, 5, 5, 5, , 8, 3, 4, 5, 5, 5, 5, , 14, 3, 3, 3, 3, 5, 5, 5 Fig. 2. Steps = 46, bad-agent = 1, braking weight = , 20, 2, 3, 3, 3, 3, 3, 3 TABLE VII. RESULTS, BAD-AGENT = 1, BRAKING WEIGHT = , 8, 2, 5, 5, 5, 5, 5, , 13, 1, 3, 3, 5, 5, 5, 5 Fig. 3. Steps = 91, bad-agent = 1, braking weight = , 23, 2, 1, 3, 3, 3, 3, , 23, 3, 3, 2, 1, 3, 3, , 23, 3, 3, 3, 2, 1, 3, , 23, 3, 3, 3, 3, 3, 2, 1 Fig. 4. Steps = 134, bad-agent = 1, braking weight = , 23, 3, 3, 3, 3, 3, 3, 2 71

5 B. Two Bad-Agent The previous section shows there is no significant effect when there is only one car misbehaved on road. Although the bad-agent reduced its speed multiple times, it does not affect the following cars speed. This section introduces two misbehaved cars with manual braking at different weightage. The parameters of the model is set accordingly (road_length = 400, num.of samples = 10, v max = 3, bad-agent = car-1 and car- 4). Fig show the results when braking weight = 1 (Table VIII summarised the overall results). The simulation started with 10 cars moving at its maximum speed constantly. No brake has been triggered at this moment. The first brake is triggered at steps = 67, causing the target vehicle to reduce its speed by 1. The initial constant gap, 5-car-distance between each car is disrupted, with a big separation between car-3 and car-4 (10 cells). Fig. 10 shows the simulation steps at 84, where braking is triggered for the 17 th time. The unnecessary braking on road caused vehicles to cluster in packs. If enough of these slowdowns occur in a short time, a traffic jam forms. However the gap between car-3 and car-4 remain constant. Fig. 11 and Fig. 12 show the simulation steps from 96 to 139, where brake is triggered for the 18 th time. Once the braking is stop, the gap between car-3 and car-4 is reduced. Also notice all cars remain at a constant gap of 3 cells from each other. Fig show the results when braking weight = 2 (Table IX summarised the overall results). The first braking caused a gap of 8 and extended to 12 after the third braking. Notice the pattern of separation between car-3 and car-4. As the iteration goes on, the cars able to pick up to its maximum speed and reduce the gap back to 9. Fig show the simulation after the 6 th manual brake is triggered on both car-1 and car-4. The reduced speed in car-3 (steps = 96) disappear in Fig.16 (steps = 98) when the first three cars resume to their maximum speed and safe to maintain at that speed. Notice how the yellow cell (reduced speed in car-4) travels backwards. Although the manual braking supposed to reduce its speed to 1, but the cars quickly resume to speed 2. Fig. 13. Steps = 34, bad-agent = 2, braking weight = 2 Fig. 14. Steps = 52, bad-agent = 2, braking weight = 2 Fig. 15. Steps = 96, bad-agent = 2, braking weight = 2 Fig. 16. Steps = 98, bad-agent = 2, braking weight = 2 Fig. 17. Steps = 100, bad-agent = 2, braking weight = 2 TABLE VIII. RESULTS, BAD-AGENT = 2, BRAKING WEIGHT = , 5, 5, 5, 5, 5, 5, 5, , 3, 10, 2, 3, 5, 5, 5, , 2, 11, 2, 2, 3, 4, 5, , 2, 11, 2, 2, 2, 2, 2, , 3, 9, 3, 3, 3, 3, 3, , 3, 9, 3, 3, 3, 3, 3, 3 Fig. 10. Steps = 84, bad-agent = 2, braking weight = 1 TABLE IX. RESULTS, BAD-AGENT = 2, BRAKING WEIGHT = , 5, 5, 5, 5, 5, 5, 5, , 5, 8, 2, 5, 5, 5, 5, 5 Fig. 11. Steps = 96, bad-agent = 2, braking weight = , 2, 11, 2, 2, 5, 5, 5, , 1, 12, 2, 1, 3, 5, 5, , 3, 11, 1, 3, 3, 3, 3, , 2, 10, 3, 2, 1, 3, 3, , 3, 9, 3, 3, 3, 2, 1, 3 Fig. 12. Steps = 139, bad-agent = 2, braking weight = , 3, 9, 3, 3, 3, 3, 3, 2 72

6 The results above show how brake-tapping create traffic waves. But drivers that tapping their brake is not the root of the problem here. Consider the scenario as in Fig Suppose car-1 is travelling faster than the other cars. Car-2 would speed up to its maximum speed, and so do car-3 and all the other cars behind it. Now if car-1 slow down slightly, after a short interval, car-2 would have decelerated too. But because of the time delay (mechanical delay and driver s reaction time), car-2 would get riskily too close to car-1 [7]. Car-3 would have to decelerate too, and get dangerously too close to car-2. And because of the amplified time delay, car-4 would have to brake even harder to avoid collision. As the time delay is piling up, some cars would have to brake to a complete stop (hard braking). In the above case, traffic waves seem like triggered by car- 1 that slow down slightly. In real traffic nature, car-1 would have to slow down for numerous reasons; it could be broken down, or comes to an intersection, or it just a tailgater that need to jump ahead any spaces. Whatever the reasons, these forced all the cars behind it to decelerate as it entered the cluster of congestions. A driver does not have choice but to brake in order to avoid collision, but a driver have choice not to tailgating. As mentioned earlier in this section, the brake-tapping is applied in this trafic model with the purpose to re-create the by-product of tailgatting. It is crucial to identify correctly the cause of the traffic waves so that an effective solution can be proposed to mitigate the emergence of traffic waves. It also matter how hard a driver brake. Comparing Table VIII and Table IX, it takes 18 brake-tapping with the weightage of 1 to cause separation gap of three, and 6 brake tapping of weightage of 2 to cause similar separation gap. In other words, hard braking cause traffic waves to emerge faster [8, 9]. In light traffic and with adequate spacing between cars, the traffic waves would evaporate instantly. However in heavy traffic, it would be intensified and create a long tailback [10]. One hypothesis can be draw from here is that, when cars travelling at high speed are too close with each other, a tiny change of speed can propagates into a stop-and-go waves. Thus, if the cars have adequate space between each other then it would reduce the chances of hard braking. That certainly means stop tailgating by hurrying to the empty space opens up ahead. V. CONCLUSIONS This paper begin with an overview of cellular automata platform, its terminology and configuration. NaSchr model, a stochastic discrete automaton model also discussed in detail. Its configuration and simulation results shows that it is practical to emulate the spontaneous formation of traffic jam. However, it is necessary to analyse the emergence of traffic jam. Thus, the randomisation rule in NaSchr model is amended with a manual braking mode. The NaSchr based TCA model is simulated with open boundary conditions and user may adjust the weight of the braking value. The reason behind this analysis is to investigate the effect of hard braking on traffic flow. A new variable bad-agent is introduced, to represent the noise that might occurs on road (broken down car, merge zone and tailgatting). User can manually trigger brakes on certain cars and observe its effect on traffic flow. The results obtained show that: 1. Sudden brake may disrupt traffic flow 2. The weight of the braking value does matter 3. Traffic waves propagates backward Future research extensions include identifying appropriate spacing between cars that travelling at high speed. With proper space between cars, then the chances of hard brake would be reduced. It is also useful to analyse the influence of good driving behaviour (travelling at average speed and bringing space into the congestions) and how it can mitigate the congestions. ACKNOWLEDGMENT The authors would like to acknowledge the Ministry of Higher Education (KPT) for supporting this research under Exploratory Research Grant Scheme (ERGS), grant no. ERG0042-ICT-1/2013 and MyBrain15 programme.. REFERENCES [1] M.Y. Choong, R.K.Y. Chin, K.B. Yeo and K.T.K. Teo, Trajectory pattern mining via clustering based on similarity function for transportation surveillance, International Journal of Simulation Systems, Science & Technology, vol. 17, no. 34, pp [2] A. 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