Abstract
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.