This paper proposes a methodology to estimate the passenger macroscopic fundamental diagram for bi-modal urban corridors while accounting for the stochastic nature of bus operations. The proposed framework extends the existing variational theory (VT) approaches by: (i) introducing a probabilistic VT graph, where the costs are computed using an efficient stochastic shortest path algorithm; (ii) capturing the effects of stochastic moving bus bottlenecks and the correlation of bus arrival times; (iii) incorporating a macroscopic passenger model that reflects the passenger dynamics for the different modes; and (iv) accounting for the effects that the traffic conditions might have on bus operations.

Using a Monte-Carlo simulation and empirical data from a bi-modal corridor in Zurich, Switzerland, we not only successfully validate the results yielded by our stochastic VT approach, but also show its applicability on a real corridor. A comparison with a deterministic VT approach reveals the value of the proposed framework, especially for corridors with a high bus frequency and considerable stochasticity. The results demonstrate that incorporating stochasticity and the traffic conditions is essential if buses run with relatively short and variable headways. Moreover, we introduce an innovative application example for the evaluation of different bus lane layouts, aiming to maximize the passenger throughput along a bi-modal urban corridor. The application example shows that the proposed framework can be used as an efficient modeling tool for practitioners. In particular, it can be used to identify a proper lane allocation strategy by computing the critical density of cars when a mixed lane should be switched to a dedicated bus lane or vice versa. It is important to note that such application would not have been possible without our proposed VT extensions, which account for both passenger dynamics and the impact of traffic conditions. Finally, given that the proposed methodology is generic, it can be easily extended to various traffic problems involving stochasticity.

本文提出了一种方法，在考虑公交运营的随机性的同时，估算了双峰型城市走廊的乘客宏观基本图。所提出的框架通过以下方式扩展了现有的变分理论（VT）方法：（i）引入概率VT图，其中使用高效的随机最短路径算法计算成本；（ii）捕捉随机移动的公交瓶颈的影响以及公交到达时间的相关性；（iii）纳入一个宏观的乘客模型，以反映不同模式的乘客动态；（iv）考虑交通状况可能对公交车运营的影响。

使用蒙特卡洛模拟和来自瑞士苏黎世的双峰走廊的经验数据，我们不仅成功地验证了随机VT方法产生的结果，而且还证明了其在真实走廊上的适用性。与确定性VT方法的比较揭示了所提出框架的价值，特别是对于总线频率高且随机性较高的走廊。结果表明，如果公交车行驶时距相对较短且变化不定，则必须考虑到随机性和交通状况。此外，我们介绍了一个创新的应用示例，用于评估不同的公交专用道布局，旨在最大程度地提高双峰城市走廊沿线的旅客吞吐量。应用示例表明，所提出的框架可以用作从业人员的有效建模工具。特别是，当混合车道应切换到专用公交车道时，可以通过计算汽车的临界密度来确定合适的车道分配策略，反之亦然。重要的是要注意，如果没有我们提议的VT扩展，那么这种应用是不可能的，因为VT扩展既考虑了乘客动态又影响了交通状况。最后，由于所提出的方法是通用的，因此可以轻松地扩展到涉及随机性的各种交通问题。重要的是要注意，如果没有我们提议的VT扩展，就不可能实现这种应用，因为VT扩展既考虑了乘客动态又影响了交通状况。最后，鉴于所提出的方法是通用的，因此可以轻松地扩展到涉及随机性的各种交通问题。重要的是要注意，如果没有我们提议的VT扩展，那么这种应用是不可能的，因为VT扩展既考虑了乘客动态又影响了交通状况。最后，由于所提出的方法是通用的，因此可以轻松地扩展到涉及随机性的各种交通问题。