# 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.

# In South Africa's cities, evictions are happening despite a national ban

An aerial view shows a destroyed house in Lawley, south of Johannesburg, on April 20, 2020. The city has been demolishing informal structures on vacant land despite a moratorium on evictions. (Marco Longari/AFP via Getty Images)

On the morning of 15 July, a South African High Court judge ruled that the city of Cape Town’s Anti-Land Invasion Unit had illegally evicted a man when it destroyed the shack where he was living.

That afternoon, the Anti-Land Invasion Unit was out again, removing shacks in another informal settlement.

Evictions were banned in South Africa for nine weeks, after the national government placed the country under a strict Covid-19 lockdown in late March. At present, eviction orders are automatically suspended until the country moves to a lower “alert level” and can only be carried out with a special order from a judge.

Yet major cities including Cape Town, Johannesburg and eThekwini (created through the merger of Durban with several surrounding communities), have continued to use municipal law enforcement agencies and private security companies to remove people from informal housing. In many cases those operations have been conducted without a court order – something required under regular South African law.

Around 900 people were evicted from three informal settlements in eThekwini during the eviction ban, according to the Church Land Programme, a local NGO. Its director, Graham Philpott, says it’s also aware of evictions in other informal settlements.

While evictions aren’t a “new experience” in these communities, the NGO released a report on lockdown evictions because they were “so explicitly illegal”. “There was a moratorium in place,” Philpott says, “and the local municipality acted quite flagrantly against it. There’s no confusion, there’s no doubt whatsoever, it is illegal. But it is part of a trend where the eThekwini municipality has acted illegally in evicting the poor from informal settlements.”

Evictions also took place in Cape Town and Johannesburg during so-called “hard lockdown” according to local activists. In eThekwini and other municipalities, the evictions have continued despite restrictions. In Cape Town, authorities pulled a naked man, Bulelani Qholani, from his shack. That incident, which was captured on video, drew condemnation from the national government and four members of the Anti-Land Invasion unit were suspended.

The cities say they’re fighting “land invasions” – illegal occupations without permission from the land owner.

“Land invasions derail housing and service projects, lead to the pollution of waterways, severely prejudice deserving housing beneficiaries and cause property owners to lose their investments over night,” Cape Town’s executive mayor, Dan Plato said in a statement. (Plato has also claimed that Qholani did not live in the shack he was pulled from and that he disrobed when municipal authorities arrived.)

South African municipalities often claim that the shacks they destroy are unoccupied.

If they were occupied, says Msawakhe Mayisela, a spokesman for the eThekwini municipality, the city would get a court order before conducting an eviction. “Everything we’re doing is within the ambit of the law,” Mayisela says. But “rogue elements” are taking advantage of Covid-19, he added.

“We fully understand that people are desperately in need of land, but the number of people that are flocking to the cities is too much, the city won’t be able to provide housing or accommodation for everyone overnight,” he says.

While eThekwini claims to be a caring city, local activists say the evictions show otherwise.

In one case, 29 women were evicted from shacks during the hard lockdown. With nowhere to go, they slept in an open field and were arrested by the South African Police Service for violating the lockdown, Philpott says.

“These evictions are dehumanizing people whose dignity is already compromised in many ways,” says S’bu Zikode, the president of Abahlali baseMjondolo, a community organization whose Zulu name translates to “the people of the shacks”.

“It has reminded us that we are the people that do not count in our society.”

Municipal law enforcement and private security contractors hired by cities regularly fire rubber bullets, or even live ammunition, at residents during evictions. Some 18 Abahlali baseMjondolo activists have been killed since the organization was founded in 2005, Zikode says, most by the eThekwini Land Invasion Unit and Metro Police.

(Mayisela says that if city employees have broken the law, Abahlali baseMjondolo can file a complaint with the police. “There is no conclusive evidence to the effect that our members have killed them,”  he says.)

Other Abahlali baseMjondolo activists have been killed by what Zikode calls “izinkabi,” hitmen hired by politicians. Two eThekwini city councillors were sentenced to life in prison 2016 after they organized the killing of Thuli Ndlovu, an Abahlali baseMjondolo organizer. A member of the Land Invasion Unit who is currently facing a charge of attempted murder after severely injuring a person during an eviction remains on the job, Zikode says.

South Africa’s 1996 constitution is intended to protect the public from arbitrary state violence and guarantees a right to housing, as well as due process in evictions. But for Zikode, the South African constitution is a “beautiful document on a shelf”.

“For the working class and the poor, it’s still difficult to have access to court. You’ve got to have money to get to court,” he says.

The actions by municipal law enforcement are breaking down social trust, says Buhle Booi, a member of the Khayelitsha Community Action Network, a community group in the largest township in Cape Town.

“There’s a lack of police resources and those very few police resources that they have, they use to destroy people’s homes, to destroy people’s peace, rather than fighting crime, real criminal elements that we see in our society,” Booi says.

For him, it’s a continuation of the practices of the colonial and apartheid governments, pushing poor people, most of whom are Black, to the periphery of cities.

Around one-fifth of South Africa’s urban population live in shacks or informal dwellings, according to a 2018 report by SERI. Many more live in substandard housing. City governments maintain that the shacks destroyed during anti-land invasion operations are unfinished and unoccupied. But Edward Molopi, a research and advocacy officer at SERI, says that this claim is an attempt to escape their legal obligations to get a court order and to find alternative accommodation for affected people.

The roots of the current eviction crisis go back to apartheid, which barred non-white people from living in cities. Between the 1940s and 1970s, tens of thousands of people were forcibly relocated from neighbourhoods like Johannesburg’s Sophiatown and Cape Town’s District Six to remote townships.

In the 26 years following the end of apartheid, deepening economic inequality and rampant unemployment have limited access to formal housing for millions of South Africans. Government housing programs have mostly focused on building small stand-alone homes, often on the peripheries of cities far from jobs and amenities.

While these well-intentioned projects have built millions of homes, they’ve failed to keep up with demand, says Marie Huchzermeyer, a professor at the Centre for Urbanism & Built Environment Studies at the University of the Witwatersrand in Johannesburg. Government-funded housing projects “will never on it’s own be enough,” she says. “It has to be accompanied by land release.”

Government policies call for the “upgrading” of informal settlements and the formalization of residents’ occupation. But “there are still very, very, very few projects” of that nature in South Africa, Huchzermeyer says. “Even if it’s an informal settlement that’s been around for 20 years, there still seems to be a political wish to punish people for having done that.” The government wants people to go through the formal process of being given a house, she says – and for them to be thankful to the government for providing it.

At the municipal level, change will require “real leadership around informal settlement upgrading and around ensuring that land is available for people to occupy,” she says.

Despite the end of enforced racial segregation, spacial apartheid remains a factor in South Africa. There are few mixed-income neighbourhoods. Those who can afford to often live behind walls in sprawling low-density suburbs, while the poor live in overcrowded slums and apartment buildings.

The creation of the apartheid city “didn't happen by chance,” says Amira Osman, a professor of architecture at the Tshwane University of Technology. “It was a deliberate, structured approach to the design of the city. We need a deliberate, structured approach that will undo that.”

Since last fall, Johannesburg’s Inclusionary Housing Policy has required developments of 20 or more units to set aside 30% of those units for low-income housing.

The policy, which faced significant opposition from private developers, won’t lead to dramatic change, says Sarah Charlton, a professor at the Centre for Urbanism and Built Environment Studies, but it is “an important and significant step.”

Zikode isn’t optimistic that change will come for shack dwellers, however.

“People in the high positions of authority pretend that everything is normal,” he says. “They pretend that everyone is treated justly, they pretend that everyone has homes with running water, that everyone has a piece of land – and hide the truth and the lies of our democracy.”

Jacob Serebrin is a freelance journalist currently based in Johannesburg. Follow him on Twitter.