Maths explains why three buses always come along at once. So could maths fix the problem?

Typical. Bloody typical. Two London buses serving the same route, bunched up in the Clapton area. Image: Felix O, via Flickr.

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. It 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.

Vikash V. Gayah, Associate Professor of Civil Engineering, Pennsylvania State University and S. Ilgin Guler, Assistant Professor of Civil Engineering, Pennsylvania State University.

This article was originally published on The Conversation. Read the original article.

 
 
 
 

What's actually in the UK government’s bailout package for Transport for London?

Wood Green Underground station, north London. Image: Getty.

On 14 May, hours before London’s transport authority ran out of money, the British government agreed to a financial rescue package. Many details of that bailout – its size, the fact it was roughly two-thirds cash and one-third loan, many conditions attached – have been known about for weeks. 

But the information was filtered through spokespeople, because the exact terms of the deal had not been published. This was clearly a source of frustration for London’s mayor Sadiq Khan, who stood to take the political heat for some of the ensuing cuts (to free travel for the old or young, say), but had no way of backing up his contention that the British government made him do it.

That changed Tuesday when Transport for London published this month's board papers, which include a copy of the letter in which transport secretary Grant Shapps sets out the exact terms of the bailout deal. You can read the whole thing here, if you’re so minded, but here are the three big things revealed in the new disclosure.

Firstly, there’s some flexibility in the size of the deal. The bailout was reported to be worth £1.6 billion, significantly less than the £1.9 billion that TfL wanted. In his letter, Shapps spells it out: “To the extent that the actual funding shortfall is greater or lesser than £1.6bn then the amount of Extraordinary Grant and TfL borrowing will increase pro rata, up to a maximum of £1.9bn in aggregate or reduce pro rata accordingly”. 

To put that in English, London’s transport network will not be grinding to a halt because the government didn’t believe TfL about how much money it would need. Up to a point, the money will be available without further negotiations.

The second big takeaway from these board papers is that negotiations will be going on anyway. This bail out is meant to keep TfL rolling until 17 October; but because the agency gets around three-quarters of its revenues from fares, and because the pandemic means fares are likely to be depressed for the foreseeable future, it’s not clear what is meant to happen after that. Social distancing, the board papers note, means that the network will only be able to handle 13 to 20% of normal passenger numbers, even when every service is running.


Shapps’ letter doesn’t answer this question, but it does at least give a sense of when an answer may be forthcoming. It promises “an immediate and broad ranging government-led review of TfL’s future financial position and future financial structure”, which will publish detailed recommendations by the end of August. That will take in fares, operating efficiencies, capital expenditure, “the current fiscal devolution arrangements” – basically, everything. 

The third thing we leaned from that letter is that, to the first approximation, every change to London’s transport policy that is now being rushed through was an explicit condition of this deal. Segregated cycle lanes, pavement extensions and road closures? All in there. So are the suspension of free travel for people under 18, or free peak-hours travel for those over 60. So are increases in the level of the congestion charge.

Many of these changes may be unpopular, but we now know they are not being embraced by London’s mayor entirely on their own merit: They’re being pushed by the Department of Transport as a condition of receiving the bailout. No wonder Khan was miffed that the latter hadn’t been published.

Jonn Elledge was founding editor of CityMetric. He is on Twitter as @jonnelledge and on Facebook as JonnElledgeWrites.