Phase transitions and optimal transport in stochastic roundabout traffic

We study traffic in a roundabout model, where the dynamics along the interior lane of the roundabout are described by the totally asymmetric simple exclusion process (TASEP). Vehicles can enter the interior lane or exit from it via S intersecting streets with given rates, and locally modified dynamics at the junctions take into account that collisions of entering vehicles with vehicles approaching the entrance point from the interior lane should be avoided. A route matrix specifies the probabilities for vehicles to arrive from and to exit to certain intersecting streets. By subdividing the interior lane into segments between consecutive intersecting streets with effective entrance and exit rates, a classification of the stationary roundabout traffic in terms of TASEP multiphases is given, where each segment can be in either the low-density, high-density, or maximum current TASEP phase. A general methodology is developed, which allows one to calculate the multiphases and optimal throughput conditions based on a mean-field treatment. Explicit analytical results from this treatment are derived for equivalent interesting streets. The results are shown to be in good agreement with kinetic Monte Carlo simulations.