A bathtub model of transit congestion
将乘客上下车造成的摩擦效应纳入公交线路的各向同性模型，推导出网络下车函数，分析超拥堵现象、多重均衡及社会最优，并用复制动态研究均衡稳定性。
Studies of transit dwell times suggest that the delay caused by passengers boarding and alighting rises with the number of passengers on each vehicle. This paper incorporates such a “friction effect” into an isotropic model of a transit route with elastic demand. We derive a strongly unimodal “Network Alighting Function” giving the steady-state rate of passenger flows in terms of the accumulation of passengers on vehicles. Like the Network Exit Function developed for isotropic models of vehicle traffic, the system may exhibit hypercongestion. Since ridership depends on travel times, wait times and the level of crowding, the physical model is used to solve for (possibly multiple) equilibria as well as the social optimum. Using replicator dynamics to describe the evolution of demand, we also investigate the asymptotic local stability of different kinds of equilibria.