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.

使用 Lighthill-Witham-Richards (LWR) 模型对简单圆形道路网络上的稳态交通流进行分析研究。该网络通常由一个分流和合流路口以及三个相连的路段组成。在分叉路口，交通流被分配以满足用户平衡条件。在汇合点，考虑了由于瓶颈效应引起的排队或冲击结构。我们指出，该解决方案取决于道路网络上的车辆总数，瓶颈效应不仅会导致容量下降，还会导致合并路口上游两个路段的低效利用。为了进一步验证导出的稳态解，LWR模型的一阶Godunov方案用于模拟每个路段的交通流，并应用供需概念为该方案提供路口处的边界值。通过将车辆总数从零变到最大，仿真表明交通流的随机分布状态必然会演变为稳定状态，这与解析解完全一致。