行人在十字路口的行為

Abstract

本篇論文主要探討只有行人的十字路口，沒有任何車輛，並採用「離心力模型」模擬行人的流動，此外，我假設道路都是單向道，也就是說，同一條路上並不會有反方向移動行人的交互作用。但這個模型有幾點缺失，譬如說，兩個陌生人理應不會靠得太近，就像是他們之間有個排斥力一樣，但這個模型顯示如果兩個人極度靠近，他們之間的作用力竟然變成吸引力了；除此之外，如果兩個行人在同一條路線上走，當他們遇上的時候並不會避開，而是站在原地大眼瞪小眼；另外，我也增加「能見度」的概念，也就是說，當某人沒看到另一位行人時，則某人就不會受到那位行人的作用力影響。修改模型並收集不同號誌燈秒數的數據結果之後，我發現在沒有任何交通號誌的情形下，行人流動是最有效率的，這也就是為什麼室內場館、夜市徒步區等等都不會有號誌燈控制行人的原因，而且我還發現垂直的兩條道路的行人通量是呈現負相關的，也就是說，若現在水平方向道路行人通量較大，鉛直方向道路行人通量就會比較小，通量是一個隨時間震盪的函數。 最後，由於使用「離心力模型」跑模擬的時間相當長，我發展另一套簡化模型來描述無號誌燈的十字路口，採用離散的空間以及時間，每位行人以特定的機率法則決定前進與否，除此之外，我將二維的平面道路化簡為一維的線，所以十字路口看起來就像是兩條線交叉，這個新模型大幅減少電腦模擬的時間。收集多種道路條件的模擬數據，並與「離心力模型」的數據作比較，發現誤差相當小、結果相當貼合，以後就可以用這個新方法更快速地進行其他研究了。

In this study, we will focus on the pedestrian traffic on roads without any vehicles, and the base model we will adopt is the so-called centrifugal force model [1, 2]. To simplify the model, we will only consider unidirectional motion on each road, i.e., the pedestrians won’t walk into one another in opposing directions. Though this base model has its own merits, certain undesirable features are present. For example, the generally repulsive force between two people walking nearby can become attractive if their distance is too small. Another unphysical feature is that people do not avoid each other sidewise if one catches up with the person right in front of him along a straight path. When this happens, the two people will just stand still in this model. In the modified model I have studied, we introduce the concept of visibility. That is, one pedestrian will not be influenced by those who are out of his sight, as is the case when one is approaching a corner of a cross street. Having run through the model many times using various parameter values, we found that, under typical and realistic circumstances, using a traffic light to monitor the pedestrian flow is never a good idea: It almost always makes the flow less efficient. Despite this finding, it remains to be clarified if this is the major reason why one does not see traffic lights in the night markets, or even in a sport stadium or concert hall. Another finding, which seems intuitively obvious, is that the correlation coefficient of the pedestrian flux between two cross roads is negative. Put in plain English, it means that the flux of pedestrians in the perpendicular direction becomes small when the flux in a road is large. In other words, the system will automatically adjust the intersecting fluxes so that each road alternates in somewhat dominating the traffic in a specific direction. Typically, the flux in each direction oscillates in time. Finally, because simulating the general centrifugal force model is time-consuming, and also because the model by itself does not provide any direct physical insight as to why the simulation results are the way they are, I have developed a simplified model to simulate the situation without traffic lights, hoping that it will shed more light to the problem. In this model, the two-dimensional roads are simplified to straight lines, and borrowing ideas from cellular automata, I also discretized space and time, and every pedestrian follows a certain probability rule to move forward. It is interesting to note that this highly simplified model does not just reduce the simulation time tremendously, it can also quantitatively reproduce salient features present in the general centrifugal force model. But exactly how one may mathematically show that the original model is well-approximated by the simplified model still remains to be investigated.