# Driverless cars plus mathematics could equal the end of traffic jams

Here we go again. Image: Getty.

Being stuck in miles of halted traffic is not a relaxing way to start or finish a summer holiday. And as we crawl along the road, our views blocked by by slow-moving roofboxes and caravans, many of us will fantasise about a future free of traffic jams.

As a mathematician and motorist, I view traffic as a complex system, consisting of many interacting agents including cars, lorries, cyclists and pedestrians. Sometimes these agents interact in a free-flowing way; at other, infuriating, times they simply grind to a halt. All scenarios can be examined – and hopefully improved – using mathematical modelling, a way of describing the world in the language of maths.

Mathematical models tell us for instance that if drivers kept within the variable speed limits sometimes displayed on a motorway, traffic would flow consistently at, say, 50mph. Instead we tend to drive more aggressively, accelerating as soon as the opportunity arises – and being forced to brake moments later. The result is greater fuel consumption and a longer overall journey time.

Cooperative driving seems to go against human nature when we get behind the wheel. But could this change if our roads were taken over by driverless cars?

Incorporating driverless cars into mathematical traffic models will prove key to improving traffic flow and assessing the various conditions in which traffic reaches a traffic jam threshold, or “jamming density”. The chances of reaching this point are affected by changes such as road layout, traffic volume and traffic light systems. And crucially, they are affected by whoever is in control of the vehicles.

In mathematical analysis, dense traffic can be treated as a flow and modelled using differential equations which describe the movement of fluids. Queuing models consider individual vehicles on a network of roads and the expected time they spend both in motion and waiting at junctions.

Another type of model consists of a grid in which cars' positions are updated, according to certain rules, from one grid cell to the next. These rules can be based on their current velocity, acceleration and deceleration due to other vehicles and random events. This random deceleration is included to account for situations caused by something other than other vehicles – a pedestrian crossing the road for example, or a driver distracted by a passenger.

Adaptations to such models can take into account factors such as traffic light synchronisation or road closures, and they will need to be adapted further to take into account the movement of driverless cars.

In theory, autonomous cars will typically drive within the speed limits; have faster reaction times allowing them to drive closer together; and will behave less randomly than humans, who tend to overreact in certain situations. On a tactical level, choosing the optimum route, accounting for obstacles and traffic density, driverless cars will behave in a more rational way, as they can communicate with other cars and quickly change route or driving behaviour.

So driverless cars may well make the mathematician’s job easier. Randomness is often introduced into models in order to incorporate unpredictable human behaviour. A system of driverless cars should be simpler to model than the equivalent human-driven traffic because there is less uncertainty. We could predict exactly how individual vehicles respond to events.

In a world with only driverless cars on the roads, computers would have full control of traffic. But for the time being, to avoid traffic jams we need to understand how autonomous and human-driven vehicles will interact together.

Of course, even with the best modelling, cooperative behaviour from driverless cars is not guaranteed. Different manufacturers might compete to come up with the best traffic-controlling software to ensure their cars get from A to B faster than their rivals. And, like the behaviour of individual human drivers, this could negatively affect everyone’s journey time.

But even supposing we managed to implement rules that optimised traffic flow for everyone, we could still get to the point where there are simply too many cars on the road, and jamming density is reached.

Yet there is still potential for self-driving cars to help in this scenario. Some car makers expect that eventually we will stop viewing cars as possessions and instead simply treat them as a transport service. Again, by applying mathematical techniques and modelling, we could optimise how this shared autonomous vehicle service could operate most efficiently, reducing the overall number of cars on the road.

So while driverless cars alone might not rid us of traffic jams completely by themselves, an injection of mathematics into future policy could help navigate a smoother journey ahead.

Lorna Wilson is commercial research associate at the University of Bath.

# Was the decline in Liverpool’s historic population really that unusual?

A view of Liverpool from Birkenhead. Image: Getty.

It is often reported that Liverpool’s population halved after the 1930s. But is this true? Or is it a myth?

Often, it’s simply assumed that it’s true. The end. Indeed, proud Londoner Lord Adonis – a leading proponent of the Liverpool-bypassing High Speed 2 railway, current chair of the National Infrastructure Commission, and generally a very influential person – stood on the stairs in Liverpool Town Hall in 2011 and said:

“The population of Liverpool has nearly halved in the last 50 years.”

This raises two questions. Firstly, did the population of the City of Liverpool really nearly halve in the 50 year period to 2011? That’s easy to check using this University of Portsmouth website – so I did just that (even though I knew he was wrong anyway). In 2011, the population of the City of Liverpool was 466,415. Fifty years earlier, in 1961, it was 737,637, which equates to a 37 per cent drop. Oops!

In fact, the City of Liverpool’s peak population was recorded in the 1931 Census as 846,302. Its lowest subsequent figure was recorded in the 2001 Census as 439,428 – which represents a 48 per cent decline from the peak population, over a 70 year period.

Compare this to the population figures for the similarly sized City of Manchester. Its peak population also recorded in the 1931 Census as 748,729, and its lowest subsequent figure was also recorded in the 2001 Census, as 392,830. This also represents a 48 per cent decline from the peak population, over the same 70 year period.

So, as can be seen here, Liverpool is not a special case at all. Which makes me wonder why it is often singled out or portrayed as exceptional in this regard, in the media and, indeed, by some badly briefed politicians. Even London has a similar story to tell, and it is told rather well in this recent article by a Londoner, for the Museum of London. (Editor’s note: It’s one of mine.)

This leads me onto the second question: where have all those people gone: London? The Moon? Mars?

Well, it turns out that the answer is bit boring and obvious actually: after World War 2, lots of people moved to the suburbs. You know: cars, commuter trains, slum clearance, the Blitz, all that stuff. In other words, Liverpool is just like many other places: after the war, this country experienced a depopulation bonanza.

So what form did this movement to the suburbs take, as far as Liverpool was concerned? Well, people moved and were moved to the suburbs of Greater Liverpool, in what are now the outer boroughs of the city region: Halton, Knowsley, St Helens, Sefton, Wirral. Others moved further, to Cheshire West & Chester, West Lancashire, Warrington, even nearby North Wales, as previously discussed here.

In common with many cities, indeed, Liverpool City Council actually built and owned large several ‘New Town’ council estates, to which they moved tens of thousands of people to from Liverpool’s inner districts: Winsford in Cheshire West (where comedian John Bishop grew up), Runcorn in Halton (where comedian John Bishop also grew up), Skelmersdale in West Lancashire, Kirkby in Knowsley. There is nothing unique or sinister here about Liverpool (apart from comedian John Bishop). This was common practice across the country – Indeed, it was central government policy – and resulted in about 160,000 people being ‘removed’ from the Liverpool local authority area.

Many other people also moved to the nearby suburbs of Greater Liverpool to private housing – another trend reflected across the country. It’s worth acknowledging, however, that cities across the world are subject to a level of ‘churn’ in population, whereby many people move out and many people move in, over time, too.

So how did those prominent images of derelict streets in the inner-city part of the City of Liverpool local authority area come about? For that, you have to blame the last Labour government’s over-zealous ‘Housing Market Renewal Initiative’ (HMRI) disaster – and the over enthusiastic participation of the then-Lib Dem controlled city council. On the promise of ‘free’ money from central government, the latter removed hundreds of people from their homes with a view to demolishing the Victorian terraces, and building new replacements. Many of these houses, in truth, were already fully modernised, owner-occupied houses within viable and longstanding communities, as can be seen here in Voelas Street, one of the famous Welsh Streets of Liverpool:

Voelas Street before HMRI implementation. Image: WelshStreets.co.uk.

The same picture after HMRI implementation Image: WelshStreets.co.uk.

Nonetheless: the council bought the houses and ‘tinned them up’ ready for demolition. Then the coalition Conservative/Lib Dem government, elected in 2010, pulled the plug on the scheme.

Fast forward to 2017 and many of the condemned houses have been renovated, in a process which is still ongoing. These are over-subscribed when they come to market, suggesting that the idea was never appropriate for Liverpool on that scale.

At any rate, it turns out that the Liverpool metropolitan population is pretty much the same as it was at its peak in 1931 (depending where the local borough boundaries are arbitrarily drawn). It just begs the question: why are well educated and supposedly clever people misrepresenting the Liverpool metropolis, in particular, in this way so often? Surely they aren’t stupid are they?

And why are some people so determined to always isolate the City of Liverpool from its hinterland, while London is always described in terms of its whole urban area? It just confuses and undermines what would otherwise often be worthwhile comparisons and discussions. Or, to put it another way: “never, ever, compare apples with larger urban zones”.

In a recent Channel 4 documentary, for example, the well-known and respected journalist Michael Burke directly compared the forecast population growths, by 2039, of the City of Liverpool single local authority area against that of the combined 33 local authority areas of Greater London: 42,722 versus 2.187,708. I mean, what bizarre point is such an inappropriate comparison even trying to make? It is like comparing the projected growth of a normal sized-person’s head with the projected growth of the whole of an obese person, over a protracted period.

Having said all that, there is an important sensible conversation to be had as to why the populations of the Greater Liverpool metropolis and others haven’t grown as fast as maybe should have been the case, whilst, in recent times, the Greater London population has been burgeoning. But constantly pitching it as some sort of rare local apocalypse helps no one.

Dave Mail has declared himself CityMetric’s Liverpool City Region correspondent. He will be updating us on the brave new world of Liverpool City Region, mostly monthly, in ‘E-mail from Liverpool City Region’ and he is on twitter @davemail2017.