Modeling Traffic Fluctuations and Congestion on a Road Network

Statistical mechanics points out as fluctuations have a relevant role for systems near critical points. We study the effect of traffic fluctuations and the transition to congested states for a stochastic dynamical model of traffic on a road network. The model simulates a finite population that moves from one road to another according to random transition probabilities. In such a way, we mimic the traffic fluctuations due to the granular feature of traffic and the dynamics at the crossing points. Then the amplitude of traffic flow fluctuations is proportional to the average flow as suggested by empirical observations. Assuming a parabolic shaped flow-density relation, there exists an unstable critical point for the road dynamics and the system can perform a phase transition to a congested state, where some roads reach their maximal capacity. We apply a statistical physics approach to study the onset congestion and we characterize analytically the relation between the fluctuations amplitude and the appearance of congested nodes. We verify the results by means of numerical simulations on a Manhattan-like road network. Moreover we point out the existence of oscillating regimes, where traffic oscillations back propagate on the road network, whose onset depend sensitively from the traffic fluctuations and that have a strong influence on the hysteresis cycles of the systems when the traffic load is modulated. The comparison between the numerical simulations and the empirical traffic data recorded by an inductive-loop traffic detector system (MTS system) on the county roads of the Emilia Romagna region in Italy is shortly discussed.