Modeling Traffic Flow for Two and Three Lanes through Cellular Automata

This paper describes two new models for vehicular traffic flow multilane. One applied to vehicular traffic flow for two lanes and one to three lanes. The models are compared with model Rickert, Nagel, Schreckenberg and Latour (RNSL) for the case of two lanes, and the model Daoudia and Moussa (D-M) for the case of three lanes. The models were compared using the parameters of the fundamental diagram and considering the effect of slow vehicles.


Introduction
At present different models of vehicular traffic flow in terms of the theory of Cellular Automata (CA) have been made [4], [5], [6]. This approach began in 1992 with Nagel and Schreckenberg [2]. They proposed a model for vehicular traffic flow of a single lane probabilistic (NaSch model), which is carried out under four rules. The model NaSch reproduces phenomena encountered in the actual flow of traffic jam as ghosts, however, the model is minimal in the sense that any simplification of some of the rules does not reproduce realistic results. Moreover, the traffic flow is generally composed of various types of vehicles with different desired speeds and moving in more than one lane. If the model NaSch considered various types of vehicles, is the result of creation of platoons, that means slow vehicles are followed by rapid vehicles reducing the average speed of the free flow system slow vehicles. In 1995 Rickert, Nagel, Schreckenberg and Latour published a model of vehicular traffic flow (RNSL model) in two adjacent lanes and independent, and this they added rules allowing the simulation of lane change [3]. This paper presents the proposal of two models of traffic flow: two and three lanes through cellular automata. There are two types of vehicles each one with different speeds. Analyzes the effect of slow vehicles in a heterogeneous system and compared to the model RNSL for the case of two lanes, and with DM model for the case of three lanes parameters via basic diagram.

NaSch Model
NaSch model is defined on a probabilistic CA. The model is defined on a 1dimensional array of L cells, cell or sites with periodic boundary conditions. Each cell represents a division of the road of 7.5 m. Each site can be empty or occupied by a car. Each car has an associated speed v i whole between 0 and v max = 5(≈ 135km/h), the maximum speed that can reach cars. There is a variable that determines the progress of all cars known as gap, which represents the total number of empty sites in front of a car. That is, if x i represents the i-th position of the vehicle, then gap(i) = x i+1 − x i − 1. In each discrete time step, the system is composed of N cars updated in parallel according to the following rules: Step 1: acceleration: Step 2: braking: Step 3: randomness: Step 4: update: Step 1 reflects the rush of drivers to accelerate until full speed v max ; step 2 is to prevent accidents, the driver must stop if the speed exceeds the distance that separates it from the car ahead; in step 3 are considered the different parameters on the behavior of drivers to accelerate or decelerate from which are created traffic jams; to step 4 gives the system configuration at time t + 1 from time t.

RNSL Model
The NaSch model is not able to modeling the full complexity of the flow of traffic because the traffic flow is composed of various types of cars with different desired speeds, besides moving over a lane. The RNSL model generalizes the model NaSch to 2 lanes, for this, CA considered a 1-dimensional consisting of two adjacent arrays 1-dimensional size L over which the NaSch model has been implemented in each arrangement and rules added to simulate the lane change.
The update of the system is carried out in two sub-steps: 1. Analyze the lane change under the new set of rules. If a vehicle meets the conditions of lane change, it is moved to the next lane transversely without moving forward. This sub-step is not possible in reality, but with the following sub-step simulation get the desired lane change.
2. Update the system according to the rules NaSch model independently in each arrangement. This sub-step setup makes use of the sub-step 1.
The lane change model is as follows: If k is the position of the k-th site on lane j. Then k −j ≤ k denote the site on the lane j that containing the nearest car to site from backward k; and k +j > k −j the site on lane j that containing the nearest car to site k from forward. If x i j represents the i-th position of the vehicle in lane j, then the following variables are defined: , the total number of empty places in front of a vehicle.
the number of empty places in front of the rail k, to where you want to make the change. The RNSL model is defined by the following set of rules that are applied in parallel to each of the vehicles of the system at each time step t.    The lane change model is as follows: The rules that define the lane change speed of a car v i are: This criterion is applied to any rail j that containing the considered car.
2. For safety criterion are two cases: (a) If the i-th vehicle is in the lane j = 1 or j = 3 and are met: iii. rand() < p change , then the car switch to rail (2) at lane j, placing in position i of the rail j.
3. The system is updated by applying the rules NaSch model independently to each lane.
The following example shows a deficiency that presents the D-M model. Consider the three-lane model as shown in Figure 2. Assume that the speed of the vehicle a located in the position 9 of the rail (1) is equal to v max = 5 (≈ 135km/h) and the speeds of the cars b and c located at position 14 of the rail (1) and at position 9 of the rail (2) respectively are 2(≈ 54km/h). Since as 5 = v hope (8) > gap(9) 1 = 4, incentive criterion is met for the lane change for the car a. Must 11 = gap o (9) 12 > gap(9) 1 , then meets the first part of the safety criterion, however, as 2 = gap o,back (9) 12 < v max , not met then the second part of the safety criterion and therefore the car a remain on the rail (1) in the next time step; however, reality, a car speed equal to 135km/h and in these conditions can change the channel (2) without any problems. The proposed model is looking to correct this deficiency presents the D-M to allow the lane change, to thereby have a more robust 2 models nearest to reality.

Proposed model for two lanes
The proposed model which we refer to as 2C model consists of two sets of rules: 1) lane-changing rules and 2) displacement rules. 1) Rules lane change: (b) Safety criteria:

Proposed model to three lanes
The proposed model which we refer to as 3C consists of two sets of rules: 1) lane-changing rules and 2) displacement rules. Below is mention each of the rules.

Rules lane change:
(a) Incentive criteria: v hope j (i) > gap(i) j with v hope j (i) = min(v i + 1, v max ). This criterion is applied to any rail j that containing considered car.
(b) For safety criterion are two cases: i. If the i-th vehicle is in the lane j = 1 or j = 3 and are met: speed of the car at the site i −2 , C. rand() < p change , then the car changes lane j to lane (2), placing in position x i j + v hope j (i) on the rail (2). ii. If the i-th car is located in the rail (2) and are met: (c) Rules of displacement: The system is updated by applying the rules NaSch model independently to each lane without modifying those cars that made the lane change.
7 Simulation of the proposed models were performed on a lattice of size 300 sites ≈ 2.25km with periodic boundary conditions. The system is heterogeneous. In the initial state, cars are randomly distributed in the lattice with zero initial velocity, with the proportion of 10% slow cars with maximum speed v slow = 2 and 90% of maximum speed fast cars v f ast = 4. The lane change probability is 0.85, and the probability for each track brake is 0.15. The simulation results were recorded after the first 1000 time steps. We denote ρ to the total system density, and φ the number of vehicles passing through a position of the lattice per unit time T . That is: where n i,i+1 (t) = 1 if the movement of a car is detected among the sites i and i + 1 at iteration t → t + 1. A φ is known as system flow.

Conclusions
There have been two models of traffic flow multi-lane, one associated with the flow of traffic to two lanes and one of three lanes, to which we refer to as model 2C and 3C respectively. The 2C model was compared with the model RNSL, and 3C model was compared with the model D-M, via both fundamental diagram. It is observed that the traffic flow of the proposed models is increased compared with the model D-M and RNSL, this is because the proposed models allow lane changes that may occur in the real flow and that with the RNSL and D-M models is not possible causing a greater amount of platoons and therefore decreasing traffic flow.