Modeling gridlock at roundabout

We propose a cellular automaton model to study the traffic patterns at a roundabout. We obtain the complete phase diagram which consists of four distinct phases: free flow, congestion, bottleneck, and gridlock. For the six possible transitions among these four phases, we observe only five transitions. Transition between congestion and gridlock is forbidden. Transition between free-flowing and gridlock is abrupt. Transition between bottleneck and gridlock is smooth. Instead of directly related to congestion, gridlock can be taken as an extreme limit of bottleneck. We show that both bottleneck and gridlock are caused by the traffic interweave, which can be well characterized by a new parameter χ. Theoretical implications are discussed. The cellular automata can serve as a basic model for practical applications.