¿ìè¶ÌÊÓƵ

You’re cruising down a crowded motorway when, suddenly, traffic slows
nearly to a standstill. A few moments later, again for no apparent reason,
everyone speeds up again. You’ve passed no slip road pouring cars onto the
highway, no crumpled wreck on the roadside, no clue to the cause of the
problem. What happened? Puzzling as this common experience is for most motorists,
it has a simple explanation in the little-known world of traffic flow theory.

Traffic engineers have known for years that traffic behaves much like
a moving fluid, transmitting shock waves of congestion far upstream – and
sometimes downstream – from bottlenecks. Now they are beginning to understand
better how these these shock waves can result from the most trifling causes,
and how driving habits can contribute to the disruption. Other kinds of
mathematical analysis tell engineers how to time city traffic lights to
minimise delays to traffic.

As planners turn their attention from single, isolated roads to the
complex warp and weft of traffic on a city’s network of streets, mathematical
analysis has to give way to computer simulations. With the aid of such simulations,
researchers have begun to develop a general theory of traffic flow in complex
street networks. But investigators cannot agree on how to study the intractable
problem of traffic congestion.

Traffic flow is simplest on multilane roads such as motorways, where
everyone is going the same way, and there are no traffic lights or crossing
traffic to confuse matters. Here, theorists use three variables to describe
traffic behaviour: vehicle density (the number of cars per mile of road);
vehicle speed; and traffic volume (the number of cars passing a fixed point
per hour), which is equal to the speed multiplied by the vehicle density.

MOTORWAY MANNERS

When the traffic is light, cars do not interfere with one another. Each
driver can choose his or her own speed, and overtake slower vehicles at
will. Under these conditions, increasing the vehicle density by a few cars
per mile of road makes no significant difference to drivers’ speeds. The
extra cars merely boost the traffic volume in proportion to their number.
But as more and more cars enter the motorway, drivers coming up behind a
slower vehicle begin to find themselves boxed in by the traffic in neighbouring
lanes, and cannot overtake until they are clear. This delay slightly reduces
average speeds, with the result that the volume may level off. Eventually,
traffic reaches a maximum volume, usually about 2200 vehicles per lane per
hour, which engineers call the capacity of the roadway.

Once traffic volume has reached capacity, cramming still more cars onto
the motorway slows everyone down sharply – so sharply, in fact, that the
volume of traffic flowing past any given point will decrease, even though
the cars are spaced more closely. Traffic engineers call this sudden drop
in speed a ‘breakdown’ of the flow, and admit that they don’t understand
why it happens at some times and places and not at others where the flow
is equally heavy. If things get really bad, cars end up in a long queue,
creeping forward in a stop-go fashion: in short, a classic traffic jam,
and the cause of frayed tempers and burnt dinners for thousands of hapless
commuters.

When traffic flow breaks down, it generates what the experts term a
shock wave, by analogy with compression waves in a fluid. Watching from
above, you see a wave of stopping traffic sweeping backwards up the motorway
as driver after driver hits the brake and joins the queue. Later, as the
jam unclogs, a similar wave of accelerating traffic similarly spreads backwards
through the queue.

These moving shock waves are the reason why motorway traffic often stops
at places where there is no obstacle to be seen. Think of the similar case
of sound waves. You hear the bang of a firework wherever the sound wave
reaches you, not when you pass the site of the explosion. Similarly, your
car stops on the motorway wherever it meets the shock wave, which may be
far from the broken-down car or burst of traffic from a slip road that caused
the problem.

Most flow breakdowns happen for fairly obvious reasons, says Fred Hall
of McMaster University in Hamilton, Ontario. What drivers have to realise,
he notes, is that the cause may lie far ahead of them. Usually, traffic
backs up because of a bottleneck caused by a slip road feeding traffic onto
an already crowded road, or perhaps by an accident or road repairs.

Now and then, however, breakdowns can happen spontaneously, caused by
nothing more than clumsy driving. In closely spaced traffic, drivers worry
about hitting the car in front of them, and if it slows slightly – to round
a curve, for example, or because the driver is gawping at an accident or
just daydreaming for a moment – the driver behind may play safe and brake
more than is strictly necessary. That can cause the third driver in line
to slow down even more, and so on, in an amplifying wave of deceleration.
If enough drivers overreact, those at the back of the queue could come to
a halt, even though the driver who started the chain reaction barely slowed
at all.

HEAVY RIGHT FOOT

A mathematical analysis conducted more than thirty years ago at the
General Motors Research Laboratories at Warren in Michigan by Robert Herman,
whom many regard as the father of traffic science, reveals what it is about
drivers’ behaviour that causes such pointless fluctuations. Herman devised
a series of equations that describe how drivers speed up and slow down to
maintain their spacing with the car in front of them. Whether one driver’s
braking is amplified by each successive driver should depend, he predicted,
on how quickly drivers respond to what happens ahead of them and how hard
they brake. Drivers who anticipate well and change speed gradually can use
the gap between cars as a shock absorber to damp out small changes in speed;
those who react slowly or have a heavy right foot lose this advantage –
though it is the cars behind them that pay the price. ‘Smooth traffic is
good traffic. The flow is better, and it’s safer,’ says Herman, who has
now retired from the University of Texas in Austin, where he moved after
his spell in industry.

Herman road-tested his analysis by measuring how long drivers took to
respond to changes in the flow of traffic ahead of them. Car drivers, he
found, teetered at the brink of stability, with about half driving well
enough to accommodate the lead car’s speed changes without amplifying them.
Professionals did much better: every one of the bus drivers Herman tested
drove well within the margin of stability.

The way drivers maintain the space between their car and the one in
front can cause traffic flow to break down somewhere totally unexpected
– a mile or so downstream from where a slip road joins, according to a study,
as yet unpublished, by traffic researcher Michael Cassidy of the University
of California at Berkeley. Drivers hate to let cars cut in front of them,
Cassidy says, so those already on the motorway tend to bunch up more tightly
to discourage drivers on the slip road from moving in. Once past the slip
road, these drivers relax and back off to a more comfortable distance behind
the car in front. This deceleration can trigger a flow breakdown, typically
about a mile past the merge – an effect traffic experts have often overlooked,
Cassidy notes. ‘What’s quite amazing about this is that despite the fact
that merges are the most common problem, all the presumptions on how traffic
behaves have been completely misunderstood.’ The dynamics are quite different
from what people thought, he says.

WAITING ROOM

With the exception of Cassidy’s work, however, efforts to understand
the causes of flow breakdown and the stop-go behaviour of traffic are dismissed
by Gordon Newell, another traffic theorist at Berkeley and, like Herman,
a physicist who became interested in traffic flow in the 1950s. All anyone
needs to know to understand motorway flow, claims Newell, is where the bottlenecks
are and how fast cars pass through them. He likens getting through a bottleneck
to sitting in a waiting room: ‘It really doesn’t make any difference how
you move around in the waiting room.’

Newell’s theory, which he published last year in Transportation Research
(vol 27B, p 281), treats motorway traffic much like a stream of water being
poured through a set of nested funnels. Each funnel collects the water flowing
through the previous funnel, plus any added water (traffic joining a motorway),
less any water that is siphoned off (traffic leaving a motorway). Each funnel’s
output is simply its net input or the funnel’s maximum flow rate, whichever
is less. Any excess water simply stays in the funnel until it has room to
pass through.

Newell claims that this simple model allows him to analyse flow on a
short section of road using just pencil and paper. He ignores how one driver
follows another, the spacing between vehicles, the fluctuating speed in
a flow, and all the other arcana of most traffic models, and so is able
to model the entire freeway system of the Bay Area around San Francisco
on a desktop computer. Most other simulations of such a road network require
much more powerful computers.

Concise as Newell’s formulation may be, some traffic practitioners say
they don’t find it very useful. ‘Newell must be asked a whole lot different
questions than I usually get asked,’ suggests Rick Donnelly, a transport
consultant based in Albuquerque, New Mexico. Down in the trenches of the
war against traffic congestion, he says, planners need to make tactical
decisions such as whether to install traffic lights to regulate the flow
of traffic entering a motorway and, if so, what rate of switching is most
efficient. Questions like these demand more detailed analyses, Donnelly
insists.

Once a driver turns off the motorway and onto city streets, traffic
planners’ most obvious role – and the one that probably provokes the most
complaints – is to coordinate the timing of traffic lights to keep delays
to a minimum. To do this, the engineers must juggle the duration of each
complete cycle, from green to red and back to green again, and the synchronisation
of a series of lights, so that as many drivers as possible hit a succession
of greens. Paradoxically for the planners, the most rational strategies
for doing this seem perfectly designed to irritate rush-hour drivers.

Take the question of cycle length. Every time the lights change, there
is a period of a few seconds when no one is using the intersection; one
traffic stream has stopped but the next one has not yet started. If the
signal changes often, this wasted time amounts to a relatively large fraction
of the total time, which reduces the total flow through the intersection.
Signals that change less frequently have a greater traffic capacity, but
cars that miss the green have to wait longer before they can move again.

Planners must balance these two demands. According to Nathan Gartner,
a traffic-light expert at the University of Massachusetts at Lowell, they
tend to favour long cycles for high capacity at busy intersections and shorter
cycles to reduce waiting time where traffic is lighter. However, this solution
happens to be the one that leaves commuters, who travel at the busiest
times of day, with the most time to grumble as they sit at red lights.

On streets with a succession of traffic lights, synchronisation of the
entire chain can make an enormous difference to the traffic flow. Deep down,
most motorists harbour the conviction that each set of lights should turn
green as they approach, allowing them to sweep down the street without slowing.
Such a scheme could indeed be designed for a lucky few, says Gartner, but
at some cost to everyone else. ‘That works well if you coordinate lights
in only one direction, but if you also want to do it in the other direction,
you will have to compromise and will not obtain such good coordination,’
he says.

SEEING RED

To minimise the delay to drivers in both directions down a particular
street, the best synchronisation scheme is for all the lights to change
at the same time, says Gartner. A little thought shows why, when you consider
just two junctions a short distance apart. The optimum solution is for both
sets of lights to turn green at the same time. Delaying one set by a few
seconds will make things better for the traffic moving in one direction,
but at the cost of hindering those going in the opposite direction. However,
the opposite applies for two-way flow through lights that are spaced too
widely for drivers to have much chance of making both lights on a single
green. Here the best compromise is for one to turn green just as the next
one turns red.

Few traffic-light systems follow this simple rule, however, because
reality intrudes, and knocks the symmetry of this idealised model askew.
Traffic flow will not usually be equal in the two directions, and planners
may want to favour flow in the busier direction. Also, flows may well differ
at every junction, as traffic joins and leaves the street. Pedestrians,
filter lanes for turning traffic, crossing traffic and other complications
such as tram lines only worsen the problem. ‘When you get a fairly large
system that is operating under different peaking characteristics at different
places, and you’re trying to maintain a steady flow in more than one direction,
optimising becomes a very difficult mathematical problem,’ says Donnelly.

Complexities like these have led some traffic theorists, and the vast
majority of planners, to downplay mathematical analysis and turn instead
to computer simulations of traffic flow in road networks. Such simulations
allow researchers to juggle myriad details about the behaviour of drivers
and their cars, including the way one vehicle follows another, lane changing
and the speed preferences of individual drivers. Theoreticians often describe
these different behaviours using totally different mathematical forms, Donnelly
notes, which makes a pencil-and-paper solution almost impossible. ‘But if
you can see the result of the simulation,’ he says, ‘you start to get an
²¹±è±è°ù±ð³¦¾±²¹³Ù¾±´Ç²Ô.’

For designers of traffic systems, simulations offer another precious
advantage over reality: the chance to experiment. ‘One of the neat things
about simulation is that you can make a change to a problem spot and look
for unanticipated changes to the sur-rounding system,’ says Donnelly. For
example, traffic lights to limit the flow of vehicles onto a city freeway
might cause a queue to back up into an intersection on a surface street.
‘Our profession is characterised by solving one problem and creating other
problems. The beauty of simulation is you can avoid doing that.’

While most traffic experts use simulations to solve problems about specific
street layouts, a few are chasing more elusive generalities about the way
traffic behaves in any complex urban network. In the late 1980s, Hani Mahmassani
and colleagues at the University of Texas at Austin used computers to mimic
simple, idealised traffic networks and the larger, messier reality of downtown
Austin’s streets. As they watched what happened to traffic flow when they
added more vehicles, or modified streets and traffic lights, they were pleased
to find the relationships between speed, density and volume that they already
knew applied to motorway flow began to emerge for the more complex flow
in city streets.

In particular, the simulations show traffic volume increasing as vehicles
crowd more densely onto the roads – but only up to a point. As still more
cars hit the streets, speeds and flow volumes plummet until the network
reaches gridlock – the citywide equivalent of flow breakdown on a motorway.
‘It’s really a very powerful result in traffic theory,’ Mahmassani says.
City street networks are too complex for researchers to derive these results
mathematically, as they can for a motorway. ‘But through simulation, we
are able to verify that these relations are similar,’ he says.

Mahmassani recognises that there are some important differences, as
well as similiarities. For example, some streets carry more traffic than
others, so densities at the busiest intersections can be much higher than
the average for the whole city. As a result, networks start to become congested
at much lower average densities – as low as 30 cars per lane per mile averaged
over the network, as compared to about 80 for single motorways, Mahmassani
says. In short, the traffic capacity of a city is less than the sum of its
parts.

The complexity of street networks can also contribute to gridlock directly.
‘Gridlock typically occurs when traffic patterns are complicated and conflicting,’
says Mahmassani. ‘Because you have turns and intersections (with crossing
traffic) and so on, things can just lock up completely.’ Planners fight
gridlock by simplifying traffic patterns – prohibiting turns during rush
hours, for example – as well as by trying to divert cars away from the
problem area. On the bright side, Mahmassani notes, networks also offer
drivers the chance to escape congestion by choosing alternative, less congested
routes.

But beyond these crude generalities, Mahmassani says that gridlock,
like its more straightforward cousin flow breakdown, remains somewhat mysterious
to traffic analysts. Researchers can calculate the flow volumes and densities
that cause congestion, but they still know little about exactly where, when
or why traffic flow ultimately collapses. And, until they do, traffic planners
can never be certain they are following the best strategies to get commuters
off the roads and home to supper.