Math Explains Why Your Bus Is Late數學解決公交車晚點問題

維卡什·V. 加亞

Bunching up at stops makes everyone run off schedule.公交串車耽誤了每個人的時間。

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.

你有沒有遇到過在公交車站等車很久卻等不來，到最后一來就來了兩輛甚至很多輛？

這一現象被稱作“公交串車”，數十年來世界各地的公交系統都在嘗試解決這個問題。在這幾十年里，研究人員運用了數學模型分析公交系統的運行方式，從而更好地理解為什么會出現這樣的情況。數學運算找出了問題的癥結所在——而且讓人們想到，可以用公交追蹤技術結合簡單的控制算法來改善這種情況。

串車讓乘客們惱火不已，因為它既增加了等公交花費的平均時間，也增加了等待期間的不確定性。

串車也讓公交系統變得更不可靠，因為它會導致公交偏離預定的時間表。由串車引起的漫長等待還會讓人們放棄公交，轉向其他不那么可持續的交通方式。

之所以會出現公交串車，是因為公交車的路線原本就不穩定。公交車按計劃行進時，一切看起來都運轉良好。它們往來于各個站點，逐站停靠，等待乘客上下車。然而，一輛車一旦誤了時間，就幾乎不可能回歸正軌了。它只會越來越落后于時間表，直到被路線上的下一趟車趕上。

那些提前到的公交車也會遇到同樣的事情：它們會在路線上越跑越快，直到趕上前面的那趟車。

描述公交車如何在路線中行進的方程式解釋了為什么會發生這樣的情況。公交車在站點服務乘客的時間和前后兩趟公交到站的間隔時間有關，后者常常被稱作發車間隔。一趟車慢了，它的發車間隔就會增加，下一次停靠時就會有更多乘客需要服務。但是某一站點等車的乘客越多，公交車在該站點停靠的時間也就越長。因此，晚點的公交需要在后續的每一個站點花費更多的時間，這就導致了它們越來越晚。提早到站的公交則會遇到相反的情況。這種循環持續不斷，直到最終多輛公交串到了同一站點。

所以，我們能做些什么來阻止這樣的情況呢？公交公司和我們這樣的研究者展開了合作，提出了許多不同的想法，來消除公交串車現象。

策略之一就是，指引晚點的公交跳過那些沒有乘客要下車的站點，或者限制每一站允許登上晚點公交的乘客人數。這兩招都能讓晚點的公交減少在每一站停留的時間，這樣它就有機會迎頭趕上了。當然，這樣做可能會讓想乘車者被迫滯留。

另一個常用的策略就是，在公交時間表中設定比實際所需更多的時間。這部分額外的時間——又稱松弛時間——能幫助司機根據公交行進時遇到的變數進行靈活調整。提早到的公交車將收到指示，在指定站點停留，等到原計劃時間再出發。然而，這一策略并不能幫助晚點的公交趕上來。它也很容易受到種種致使公交偏離時間表的因素干擾。用這種方式讓公交車拖延或停留，還會減慢乘客在路線中行進的速度。

新的技術或許幫得上忙。現在，公交公司可以實時追蹤公交車的定位，并為司機提供定制的反饋。根據這些新奇的策略，前前后后的公交車仿佛都被彈簧連接在一起一般。路線上相隔太近的公交車會收到指示，幫助它們“推”離彼此;而相隔太遠的公交車則會得到相應的信息，幫助它們“拉”近距離。司機可能會被告知在某一站點多停留這么多時間，或者在行進時增加或減少那么多速度。

研究者們研發出了各種算法，公司可以用它們來給各輛公交車下達類似的指示，從而避免公交串車。指示既可以由公交公司的調度員傳達，他們監視著整個系統，并為司機提供簡單的指引;也可以通過車載電腦來呈現，這些電腦可以自動計算出司機具體應該怎么做來避免串車。電腦模擬和實地試驗表明，有朝一日這些動態策略也許會讓串車成為歷史。

（譯者為“《英語世界》杯”翻譯大賽獲獎選手）