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≥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