First-passage processes in a deterministic one-dimensional cellular automaton model of traffic flow

We present analytical results for first-passage processes in a deterministic one-dimensional cellular automaton (CA) model of traffic flow. Starting at time $t=0$ from a random initial state with car density p, at every time step $t\ge 1$ each car moves one step to the right if the cell on its right is empty, and is stopped if it is occupied by another car. The model, which coincides with CA rule 184 in Wolfram's numbering scheme, exhibits a continuous dynamical phase transition at $p=1/2$, between a low-density free-flowing phase and a high-density congested phase. Using the framework of first-passage processes, we derive a closed-form expression for the distribution $P(T_{FS}=t)$ of first-stopping (FS) times, which is the probability that a randomly selected car will be stopped for the first time at time $t$. We also obtain a closed-form expression for the stopping probability $P_S(t)$, which is the probability that a randomly selected car will be stopped at time $t$. In the low-density phase of $0<p<1/2$, the probability $P_S(t)$ yields a closed-form expression for the distribution $P(T_{LS}=t)$ of last-stopping (LS) times, which is the probability that a randomly selected car will be stopped for the last time at time $t$, beyond which it will move freely indefinitely. In this regime, we analyze the relation between the LS time and the number of stopping events $N_S$ which take place up to that time. We present closed-form expressions for the joint distribution $P(T_{LS}=t,N_S=n)$, for the two conditional distributions that emanate from it and for the marginal distribution $P(N_S=n)$. These results provide insight on the time scales of congestion and relaxation in deterministic traffic flow from the point of view of individual cars. In a broader context, they provide insight on complex relaxation processes that involve many interacting particles, such as deterministic surface growth.