The effect of mixture lengths of vehicles on the asymmetric exclusion model is studied using numerical simulations for both open and periodic boundaries in deterministic parallel dynamics. The vehicles are filed according to their length, the small cars type 1 occupy one cell whereas the big ones type 2 takes two. In the case of open boundaries two cases are presented. The first case corresponds to a chain with two entries where densities are calculated as a function of the injecting rates α1 and α2 of vehicles type 1 and type 2 respectively, and the phase diagram (α1,α2) is presented for a fixed value of the extracting rate β. In this situation the first order transition from low to high density phases occurs at α1 + α2 = β and disappears for α2 > β. The second case corresponds to a chain with one entry, where α is the injecting rate of vehicles independent of their nature. Type2 are injected with the conditional probability αα2, where 0 ≤ α2 = nα ≤ α and n is the concentration of type2. Densities are calculated as a function of the injecting rates α, and the phase diagrams (α,β) are established for different values of n. In this situation the gap which is a characteristic of the first order transition vanishes with increasing α for n = 0. However, the first order transition between high and low densities exhibit an end point above which the global density undergoes a continuous passage. The end point coordinate depends strongly on the value of n. In the periodic boundaries case, the presence of vehicles type2 in the chain leads to a modification in the fundamental diagram (current, density). Indeed, the maximal current value decreases with increasing the concentration of vehicles type 2, and occurs at higher values of the global density.

In this paper, we have investigated two observed situations in a multi-lane road. The first one concerns a fast merging vehicle. The second situation is related to the case of a fast vehicle leaving the fastest lane back into the slowest lane and targeting a specific way out. We are interested in the relaxation time τ , i.e., which is the time that the merging (diverging) vehicle spends before reaching the desired lane. Using analytical treatment and numerical simulations for the NaSch model, we have found two states, namely, the free state in which the merging (diverging) vehicle reaches the desired lane, and the trapped state in which τ diverges. We have established phase diagrams for several values of the braking probability. In the second situation, we have shown that diverging from the fast lane targeting a specific way out is not a simple task. Even if the diverging vehicle is in the free phase, two different states can be distinguished. One is the critical state, in which the diverging car can probably reach the desired way out. The other is the safe state, in which the diverging car can surely reach the desired way out. In order to be in the safe state, we have found that the driver of the diverging car must know the critical distance (below which the way out will be out of his reach) in each lane. Furthermore, this critical distance depends on the density of cars, and it follows an exponential law.

The correlation between the velocity of two successive vehicles as a function of time headway is studied in the one-dimensional cellular automata (CA) NaSch model within parallel dynamic update. It is found that a strong correlation occurs in short time headway. The behavior of the correlation velocity as a function of the car density in different traffic states is also investigated. Moreover, our study is also extended to a more complicated situation where the two vehicles are separated by a number n of other vehicles. It is shown that the velocity correlation coefficient depends strongly on the number n of vehicles in between and on their positions.

We propose a traffic flow model which takes into account the disorder in the length and the maximal speed of cars. Using cellular automaton model in parallel dynamics we have studied the behavior of traffic flow, especially the condition of formation (active phase) and the dissociation (absorbing phase) of platoons. It is found that the transition from active to absorbing phase depend on the length of the slow vehicles L2. Indeed, the transition is discontinuous for L2 = 1 and continuous when L2 > 1. In the later case, the critical exponent is calculated near the transition density, and the space-time diagram shows the presence of the two phases. Furthermore, we have found that the fundamental diagram exhibit a plateau which depend on the size of the system.

The satisfaction rate of desired velocity in the case of a mixture of fast and slow vehicles is studied by using a cellular automaton method. It is found that at low density the satisfaction rate depends on the maximal velocity. However, the behavior of the satisfaction rate as a function of the coefficient of variance is independent of the maximal velocity. This is in good agreement with empirical results obtained by Lipshtat [Phys. Rev. E 79 066110 (2009)]. Furthermore, our numerical result demonstrates that at low density the satisfaction rate takes higher values, whereas the coefficient of variance is close to zero. The coefficient of variance increases with increasing density, while the satisfaction rate decreases to zero. Moreover, we have also shown that, at low density the coefficient variance depends strongly on the probability of overtaking.