Modelling passenger assignments in public transport networks is a fundamental task for city planners, especially when deliberating network infrastructure decisions. A key aspect of a realistic model for passenger assignments is to integrate selfish routing behaviour of passengers on the one hand, and the limited vehicle capacities on the other hand. We formulate a side-constrained user equilibrium model in a schedule-based time-expanded transit network, where passengers are modelled via a continuum of non-atomic agents that want to travel with a fixed start time from a user-specific origin to a destination. An agent's route may comprise several rides along given lines, each using vehicles with hard loading capacities. We give a characterization of (side-constrained) user equilibria via a quasi-variational inequality and prove their existence by generalizing a well-known existence result of Bernstein and Smith (Transp. Sci., 1994). We further derive a polynomial time algorithm for single-commodity instances and an exact finite time algorithm for the multi-commodity case. Based on our quasi-variational characterization, we finally devise a fast heuristic computing user equilibria, which is tested on real-world instances based on data gained from the Hamburg S-Bahn system and the Swiss long-distance train network. It turns out that w.r.t. the total travel time, the computed user-equilibria are quite efficient compared to a system optimum, which neglects equilibrium constraints and only minimizes total travel time.

翻译：公共交通网络中的客流分配建模是城市规划者的一项基础性任务，尤其是在审议网络基础设施决策时。一个现实客流分配模型的关键方面在于，一方面要整合乘客的自私路径选择行为，另一方面要考虑有限的车辆容量。我们在一个基于时刻表的时间扩展公交网络中，建立了一个带边约束的用户均衡模型。在该模型中，乘客被建模为连续的非原子智能体，他们希望在固定的出发时间从各自特定的起点前往目的地。一个智能体的路径可能包含沿给定线路的若干次乘车，每次乘车均使用具有严格载客容量限制的车辆。我们通过一个拟变分不等式刻画了（带边约束的）用户均衡，并通过推广Bernstein和Smith（Transp. Sci., 1994）的一个著名存在性结果证明了其存在性。我们进一步推导了单商品实例的多项式时间算法，以及多商品情况下的精确有限时间算法。基于我们的拟变分不等式刻画，我们最终设计了一种快速启发式算法来计算用户均衡，并在基于汉堡S-Bahn系统和瑞士长途铁路网络数据的真实世界实例上进行了测试。结果表明，就总旅行时间而言，计算得到的用户均衡与忽略均衡约束、仅最小化总旅行时间的系统最优解相比，效率相当高。