## 1 Introduction  
### Background  
High-speed railways provide swift and comfortable travel experiences that are highly valued by passengers. Nonetheless, when passengers arrive at their destination and disembark, it is the most stressful time for station managers. Based on the feedback from these station managers with years of operational experience, we believe that the limited platforms and narrow exits necessitate that passengers efficiently navigate the station within a specified timeframe, resulting in a buildup of congestion on the platform. This congestion heightens the risk of passengers falling from the platform; the denser the crowd, the greater the likelihood of such incidents [1; 2]. Therefore, the aim of our research question is to address the issue by optimizing the current exit width to reduce congestion and the risk of falling off the platform. The optimized exit width range can ensure that the disembarking passengers complete their exit process within the designated timeframe, which means safe, non-congested evacuation. As illustrated in Figure 1, which depicts a typical high-speed rail platform layout, the exit process unfolds as follows: Upon the train's arrival, passengers get off the train and proceed from points A1, A2, B1, B2, C1, and C2 in the direction of the red arrows, converging at the designated purple points A, B, and C. Subsequently, passengers move from points A, B, and C along the red arrows to eventually go through exits D or E, thus completing their disembarkation. In this process, the unidirectional nature of the exits requires passengers from points A and C to first reach point B before they can continue towards exits D or E.  
Due to the monitoring areas in Figure 1 being adjacent to the tracks, the likelihood of passengers falling onto the tracks is significantly increased upon entry. Thus, the monitored areas are the hazardous zones. Generally, out of instinct to avoid danger, passengers are aware that the monitored zones are hazardous and tend to avoid them deliberately. However, during peak travel times, the narrow exits can lead to congestion at point B, potentially forcing passengers into the monitored areas and thus increasing the risk of falls. Consequently, surveillance cameras are essential in these areas to ascertain when passengers cross into them. Relying solely on video surveillance as a preventative measure against falls is very costly, and reducing this cost is a primary concern for station managers. Figure 1: A diagram of the passenger exit process (drawn based on actual data of the platform configuration by Microsoft Visio 2010). In addition, it can be found that simply widening the exits will shrink the bottleneck areas. These narrow bottlenecks can also cause congestion that forces passengers to fall down onto the tracks from bottleneck areas. Therefore, blindly expanding exits is not feasible (infinitely increasing the exit width is not allowed). On the basis of the above introductions, we believe that a feasible key point to improving passenger safety on the platforms in a real application is to redesign the defective exit configurations, especially the width. In other words, we should determine a fully demonstrated width for exits on a typical platform. The process of passengers exiting the platform can be viewed as a problem of passenger flow assignment on paths based on individual microscopic choices, while the goal of completing the non-emergency evacuation within the designated timeframe to alleviate congestion requires estimating the travel time under combinations of paths and periods. Thus, determining how to identify the characteristics of passenger flow assignment on the platforms and construct a mathematical function with density, speed, and flow to accurately calculate the total travel time are the major challenges in this paper. This paper proceeds as follows: Section 1 lists the research background and research aim, highlighting our contributions through a literature comparison. Section 2 constructs the mathematical model formulation.