Stochastic replicator equation for understanding traffic congestion

We study traffic congestion in multilines at moderate densities. Here, we consider that the congestion arises when drivers wait for passengers no matter the traffic condition. This situation is modeled by using a stochastic replicator equation. Its numerical solution shows that in the transient state, and although the traffic lines are empty, the variance of the system increases until it reaches a maximum point, which results in the maximum traffic congestion. After this critical point, the variance decreases until it recovers the Nash equilibrium and the traffic lines become free again. In addition, the variance is calculated analytically by solving the corresponding Fokker-Planck equation using the homotopy-Padé approximation.