Math Explains Why Your Bus Is Late数学解决公交车晚点问题

维卡什·V. 加亚

Bunching up at stops makes everyone run off schedule.公交串車耽误了每个人的时间。

Have you ever waited for your bus at a bus stop for a very long time—only to be greeted by two or more buses arriving together？

This phenomenon， known as “bus bunching，” is a problem that bus transit systems around the world have been trying to solve for decades. During this time， researchers have used mathematical models to study the behavior of bus transit systems to better understand why this happens. The mathematics identify what causes this problem—and also suggests that bus-tracking technology can be combined with simple control algorithms to improve the situation.

Bunching is annoying for riders， since it increases both the average time spent waiting for the bus and the variability in this waiting time.

Bunching also makes the bus system less reliable， because it causes buses to get off schedule. The long waits induced by bunching can also cause people to shift away from buses toward other， less sustainable modes of transportation.

Bus bunching occurs because bus routes are inherently unstable. When the buses are on schedule， everything seems to work fine. They travel from stop to stop， waiting at each for passengers to exit or climb aboard. However， once a bus gets behind schedule， it’s nearly impossible for it to get back on track. It will continue to get further and further behind schedule until the next bus on the route catches up.

The same thing happens to buses that are early： They continue to get earlier and earlier as they travel through their route， until they catch up to the bus just ahead.

Equations that describe how buses move along a route identify why this happens. The time buses spend serving passengers at a stop is related to the amount of time between consecutive bus arrivals， commonly known as bus headway. When a bus runs late， its headway increases and more passengers arrive that need to be served at its next stop. But the more passengers waiting at a stop， the longer a bus needs to spend there. So late buses need to spend more time at each subsequent stop， causing them to run even later. The opposite happens for a bus that’s early. This cycle continues until multiple buses eventually catch up to each other and bunch.

So what can be done to stop this？ Transit agencies have worked with researchers such as ourselves to propose many different ideas to eliminate bus bunching.

One strategy is to instruct late buses to skip stops where passengers don’t need to get off or to limit the number of people allowed to board late buses at each stop. Both of these allows the late bus to spend less time at each stop， which allows it the opportunity to catch up. Of course， doing so can leave potential users stranded.

Another common strategy is to build more time than needed into a bus’s schedule. This additional time—called slack—helps accommodate the variability in bus travel time. Buses that are early are instructed to hold at selected stops until the scheduled time to depart. However， this strategy does not help late buses recover. It’s also susceptible to any disturbances that cause buses to get off schedule. Delaying or holding buses in this way also reduces the speed at which passengers can travel along the route.

New technology may be able to help. Transit agencies can now track the location of buses in real time and offer tailored feedback to drivers. These novel strategies treat consecutive buses as if they were all connected by springs. Buses that are too close together along the route are given instructions to help “push” them apart， while buses that are too far apart are given information to help “pull” them back together. Drivers might be told to spend this much extra time to spend at a stop or to travel that much slower or faster along a route.

Researchers have developed algorithms that agencies can use to provide such instructions to individual buses and avoid bus bunching. These instructions could be sent from dispatchers at the transit agency who monitor the system and provide simple guidance to drivers or through on-board computers that calculate exactly what drivers should do to prevent bunching automatically. Computer simulations and field tests suggest that these dynamic strategies may one day make bunching a thing of the past.

你有没有遇到过在公交车站等车很久却等不来，到最后一来就来了两辆甚至很多辆？

这一现象被称作“公交串车”，数十年来世界各地的公交系统都在尝试解决这个问题。在这几十年里，研究人员运用了数学模型分析公交系统的运行方式，从而更好地理解为什么会出现这样的情况。数学运算找出了问题的症结所在——而且让人们想到，可以用公交追踪技术结合简单的控制算法来改善这种情况。

串车让乘客们恼火不已，因为它既增加了等公交花费的平均时间，也增加了等待期间的不确定性。

串车也让公交系统变得更不可靠，因为它会导致公交偏离预定的时间表。由串车引起的漫长等待还会让人们放弃公交，转向其他不那么可持续的交通方式。

之所以会出现公交串车，是因为公交车的路线原本就不稳定。公交车按计划行进时，一切看起来都运转良好。它们往来于各个站点，逐站停靠，等待乘客上下车。然而，一辆车一旦误了时间，就几乎不可能回归正轨了。它只会越来越落后于时间表，直到被路线上的下一趟车赶上。

那些提前到的公交车也会遇到同样的事情：它们会在路线上越跑越快，直到赶上前面的那趟车。

描述公交车如何在路线中行进的方程式解释了为什么会发生这样的情况。公交车在站点服务乘客的时间和前后两趟公交到站的间隔时间有关，后者常常被称作发车间隔。一趟车慢了，它的发车间隔就会增加，下一次停靠时就会有更多乘客需要服务。但是某一站点等车的乘客越多，公交车在该站点停靠的时间也就越长。因此，晚点的公交需要在后续的每一个站点花费更多的时间，这就导致了它们越来越晚。提早到站的公交则会遇到相反的情况。这种循环持续不断，直到最终多辆公交串到了同一站点。

所以，我们能做些什么来阻止这样的情况呢？公交公司和我们这样的研究者展开了合作，提出了许多不同的想法，来消除公交串车现象。

策略之一就是，指引晚点的公交跳过那些没有乘客要下车的站点，或者限制每一站允许登上晚点公交的乘客人数。这两招都能让晚点的公交减少在每一站停留的时间，这样它就有机会迎头赶上了。当然，这样做可能会让想乘车者被迫滞留。

另一个常用的策略就是，在公交时间表中设定比实际所需更多的时间。这部分额外的时间——又称松弛时间——能帮助司机根据公交行进时遇到的变数进行灵活调整。提早到的公交车将收到指示，在指定站点停留，等到原计划时间再出发。然而，这一策略并不能帮助晚点的公交赶上来。它也很容易受到种种致使公交偏离时间表的因素干扰。用这种方式让公交车拖延或停留，还会减慢乘客在路线中行进的速度。

新的技术或许帮得上忙。现在，公交公司可以实时追踪公交车的定位，并为司机提供定制的反馈。根据这些新奇的策略，前前后后的公交车仿佛都被彈簧连接在一起一般。路线上相隔太近的公交车会收到指示，帮助它们“推”离彼此;而相隔太远的公交车则会得到相应的信息，帮助它们“拉”近距离。司机可能会被告知在某一站点多停留这么多时间，或者在行进时增加或减少那么多速度。

研究者们研发出了各种算法，公司可以用它们来给各辆公交车下达类似的指示，从而避免公交串车。指示既可以由公交公司的调度员传达，他们监视着整个系统，并为司机提供简单的指引;也可以通过车载电脑来呈现，这些电脑可以自动计算出司机具体应该怎么做来避免串车。电脑模拟和实地试验表明，有朝一日这些动态策略也许会让串车成为历史。

（译者为“《英语世界》杯”翻译大赛获奖选手）