Steady-state solution of traffic flow on a simple road network

The steady-state traffic flow on a simply circled road network is analytically studied using the Lighthill-Witham-Richards (LWR) model. The network is typically composed of a diverging and a merging junction together with three connected road sections. At the diverging junction, traffic flow is assigned to satisfy the user-equilibrium condition. At the merging junction, queuing or shock structures due to the bottleneck effect is taken into account. We indicate that the solution depends on the total number of vehicles on the road network, and that the bottleneck effect gives rise to not only capacity drop but inefficient utilization of the two road sections upstream the merging junction. To further validate the derived steady-state solution, a first-order Godunov scheme of the LWR model is adopted for simulation of traffic flow in each road section and the demand-supply concept is applied to provide boundary values at the junctions for the scheme. By varying the total number of vehicles from zero to the maximum, the simulation shows that a randomly distributed state of traffic flow is bound to evolve into a steady state, which is completely in agreement with the analytical solution.