Headway is a measurement of the distance between
vehicles in a transit system. The precise definition varies
depending on the application, but it is most commonly measured as
the distance from the tip of one vehicle to the tip of the next one
behind it, expressed as the time it will take for the trailing
vehicle to cover that distance. A "shorter" headway signifies a
more frequent service. Freight trains might have headways measured
in parts of an hour,
metro systems
operate with headways on the order of 1 to 5 minutes, and cars on a
freeway have as little as 2 seconds of
headway between them.
Headway is a key input in calculating the overall capacity of any
transit system. A system that requires large headways has more
empty space than passenger capacity, which lowers the total number
of passengers or cargo quantity being transported for a given
length of line (railroad or highway, for instance). In this case
the capacity has to be improved through the use of larger vehicles.
On the other end of the scale a system with short headways, like
cars on a freeway, can offer very large capacities even though the
vehicles carry few passengers.
The term is most frequently applied to
rail transport, where the number of tracks is
limited and signalling capabilities require long headways between
trains. Newer signalling systems and moving block controls have
dramatically reduced headways in modern systems compared to the
same lines only a few years ago. Fully automated
personal rapid transit systems
further reduce headways, in some cases to fractions of a second,
creating a
flocking of vehicles.
Description
Different measures
There are a number of different ways to measure and express the
same concept, the distance between vehicles. The differences are
largely due to historical development in different countries or
fields.
The term developed from railway use, where the distance between the
trains was very great compared to the length of the train itself.
Measuring headway from the front of one train to the front of the
next was simple and consistent with timetable scheduling of trains,
but constraining tip-to-tip headway does not always ensure safety.
In the case of a metro system, train lengths are uniformly short
and the headway allowed for stopping is much longer, so tip-to-tip
headway may be used with a minor safety factor. Where vehicle size
varies and may be longer than their stopping distances or spacing,
as with freight trains and highway applications, tip-to-tail
measurements are more common.
The units of measure also vary. The most common terminology is to
use the time of passing from one vehicle to the next, which closely
mirrors the way the headways were measured in the past. A timer is
started when one train passes a point, and then measures time until
the next one passes, giving the tip-to-tip time. This same measure
can also be expressed in terms of vehicles-per-hour, which is used
on the
Moscow Metro for instance.
Distance measurements are somewhat common in non-train
applications, like vehicles on a road, but time measurements are
common here as well.
Railway examples
Trains take a very long time to stop, covering long stretches of
ground in the process. The amount of ground covered is often much
longer than the range of the driver's vision. If a train is
stopping on the tracks in front, it is highly likely that the train
behind it will see it far too late to avoid a collision. A key
safety factor of train operations is to space the trains out by at
least this distance, the "brick-wall stop" criterion. In order to
signal the trains in time to allow them to stop, the railways
placed workmen on the lines who timed the passing of a train, and
then signalled any following trains if a certain elapsed time had
not passed. This is why train headways are normally measured as
tip-to-tip times, because the clock was reset as the engine passed
the workman.
As remote signalling systems were invented, the workmen were
replaced with signal towers at set locations along the track. This
broke the track into a series of "blocks" between the towers.
Trains were not allowed to enter a block until the signal said it
was clear, thereby guaranteeing a minimum of one block's headway
between the trains. This had the side-effect of limiting the
maximum speed of the trains to the speed where they could stop in
the distance of one block.
This was an important consideration for the
Advanced Passenger Train in
the United
Kingdom, where the block sizes limited speeds and demanded
a new braking system be developed.
The downside to the block-control approach is that there are two
competing needs that means there is no perfect block size. One is
to use as few signals as possible, as they are expensive and might
fail, as well as the fact that longer blocks gives the trains more
time to stop and thus allow for higher speeds. On the other hand,
spreading the signals out over greater distances increases the
headway, and thus reduces the overall capacity of the line. These
needs have to be balanced on a case-by-case basis.
Other examples
In the case of automobile traffic, the key consideration in braking
performance is the user's reaction time. Unlike the train case, the
stopping distance is generally much shorter than the spotting
distance. That means that the driver will be matching their speed
to the vehicle in front before they reach it, eliminating the
"brick-wall" effect.
Widely used numbers are that a car traveling at 60 mph will
require about 225 feet to stop, a distance it will cover just
under 6 seconds. Nevertheless, highway travel often occurs with
considerable safety with tip-to-tail headways on the order of 2
seconds. That's because the user's reaction time is about 1.5
seconds, so 2 seconds allows for a slight overlap that makes up for
any difference in braking performance between the two cars.
Various
personal rapid
transit systems in the 1970s, and more recent experiments in
car trains and
flocking, reduce the
headways considerably. Under computer control, reaction times can
be reduced to fractions of a second. Whether traditional headway
regulations should apply to PRT and car train technology is
debatable.
In the case of the Cabinentaxi system developed in Germany, headways
were set to 1.9 seconds because the developers were forced to
adhere to the brick-wall criterion. In experiments they
demonstrated headways on the order of half of a second. More
recently,
Volkswagen has carried out
experiments with auto-driving cars that maintain their position
from the car in front to produce "car trains", with distances of
only a few centimetres between them.
Headway and route capacity
Route capacity is defined by three figures; the number of
passengers (or weight of cargo) per vehicle, the maximum safe speed
of the vehicles, and the number of vehicles per unit time. Since
the headway factors into the second of these two inputs, it is a
primary consideration in capacity calculations. The headway, in
turn, is defined by the breaking performance, or some external
factor based on it, like block sizes. Following the methods in
Anderson:
Minimum safe headway
The minimum safe headway measured tip-to-tip is defined by the
braking performance:
T_{min} = t_r + \frac{kV}{2} \left(\frac{1}{a_f} - \frac{1}{a_l}
\right)
where:
- T_{min} is the minimum safe headway, in
seconds
- V is the speed of the vehicles
- t_{r} is the reaction time, the maximum time
it takes for a following vehicle to detect a malfunction in the
leader, and to fully apply the emergency brakes.
- a_{f} is the maximum braking deceleration of
the follower.
- a_{l} is the maximum braking deceleration of
the leader. For brick-wall considerations, a_{l}
is infinte, so \frac{1}{a_l} is zero.
- k is an arbitrary safety factor, greater than or equal
to 1.
The tip-to-tip headway is simply the tip-to-tail headway plus the
length of the vehicle, expressed in time:
T_{tot} = \frac{L}{V} + t_r + \frac{kV}{2} \left(\frac{1}{a_f} -
\frac{1}{a_l} \right)
where:
- T_{tot} time for vehicle and headway to pass a
point
- L is the vehicle length
Capacity
The vehicular capacity of a single lane of vehicles is simply the
inverse of the tip-to-tip headway. This is most often expressed in
vehicles-per-hour:
n_{veh} = \frac{3600}{T_{min}}
where:
- n_{veh} is the number of vehicles per
hour
- T_{min} is the minimum safe headway, in
seconds
The passenger capacity of the lane is simply the product of vehicle
capacity and the passenger capacity of the vehicles.
n_{pas} = P \frac{3600}{T_{min}}where:
- n_{pas} is the number of vehicles per
hour
- P is the maximum passenger capacity per vehicle
- T_{min} is the minimum safe headway, in
seconds
Examples
Consider these examples:
1) freeway traffic, per lane: 100 km/h (~28 m/s) speeds, 4
passengers per vehicle, 4 meter vehicle length, 2.5 m/s braking
(1/4
gee), 2 second reaction time, brick-wall stop,
k of 1.5;
- T_{tot} = \frac{4}{28} + 2 + \frac{1.5 \times 28}{2}
\left(\frac{1}{2.5} \right)
- n_{pas} = {4}\times \frac{3600}{2}
- n_{pas} = 7,200 passengers per hour
For comparison, the County of Marin states that peak flow on the
three-lane
Highway 101 is about 7,200
vehicles per hour. This is about the same number of
passengers per lane.
2) metro system, per line: 40 km/h (~11 m/s) speeds, 1000
passengers, 100 meter vehicle length, 0.5 m/s braking, 2 second
reaction time, brick-wall stop,
k of 1.5;
- T_{tot} = \frac{100}{11} + 2 + \frac{1.5 \times 11}{2}
\left(\frac{1}{0.5} \right)
- n_{pas} = {1000}\times \frac{3600}{20}
- n_{pas} = 180,000 passengers per hour
Note that most signalling systems used on metros place an
artificial limit on headway that is not dependent on braking
performance. Using a typical figure of 2 minutes (120
seconds):
- n_{pas} = {1000}\times \frac{3600}{120}
- n_{pas} = 30,000 passengers per hour
Since the headway of a metro is constrained by signalling
considerations, not vehicle performance, reductions in headway
through improved signalling have a direct impact on passenger
capacity. For this reason, the
London
Underground system has been spending a considerable amount of
money on upgrading the
Jubilee and
Central lines with new signalling to
reduce the headway from about 3 minutes to 1, in preparation for
the
2012 Olympics.
3) automated
personal rapid
transit system, 30 km/h (~8 m/s) speeds, 3 passengers, 3
meter vehicle length, 2.5 m/s braking (1/4
gee), 0.01
second reaction time, brake-failure on lead vehicle for 1 m/s
slowing,
k of 1.1;
- T_{tot} = \frac{3}{8} + 0.01 + \frac{1.1 \times 8}{2}
\left(\frac{1}{2.5} - \frac{1}{1} \right)
- n_{pas} = {3}\times \frac{3600}{0.2}
- n_{pas} = 54,000 passengers per hour
This
number is similar to the ones proposed by the Cabinentaxi system, although they predicted that actual use
would be much lower. Although PRTs have less passenger
seating and speeds, their shorter headways dramatically improve
passenger capacity. However, these systems are often constrained by
brick-wall considerations for legal reasons, which limits their
performance to a car-like 2 seconds. In this case:
- n_{pas} = {3}\times \frac{3600}{2}
- n_{pas} = 5,400 passengers per hour
Headways and ridership
Headways have an enormous impact on ridership levels above a
certain critical waiting time. Following Boyle, the effect of
changes in headway are directly proportional to changes in
ridership by a simple conversion factor of 1.5. That is, if a
headway is reduced from 12 to 10 minutes, the average rider wait
time will decrease by 1 minute, the overall trip time by the same
one minute, so the ridership increase will be on the order of 1 x
1.5 + 1 or about 2.5%. Also see Ceder for an extensive
discussion.
References
Notes
- The Metro normally states their best headway as 142 trains per
hour, but their english page uses the more familiar units.
- Parkinson and Fisher, pg 17
- For a links to a variety of sources on the brick-wall stop in
public transit planning, see Richard Gronning, "Brick-Wall Stops and PRT", June 2009
- Leonard Hugh Williams, "Advanced Passenger Train: A Promise
Unfulfilled", Ian Allan, 1985, ISBN 0711014744
- Parkinson and Fisher, pg 18-19
- W van Winsum and W Brouwer, "Time headway in car following and
operational performance during unexpected braking.", Perceptual
and Motor Skills, Volume 84 Issue 3 part 2 (1997), pg.
1247-57
- General Atomics, "Investigation #6 – Driving Safety", 2003
- Carnegie, Appendix 1
- "Traffic Analysis Toolbox", US Department of
Transit, FHWA-HRT-04-040
- Anderson, pg. 47-48
- "How a Freeway Breaks Down", Marin County
Public Works
- railway-technology.com, "London Olympics Transport Upgrade"
- Communications on research ained at improving
transport conditions in cities, towns and other built-up
areas", Forschung Stadtverkehr, Issue 25 (1979)
- Boyle, pg. 13
- Ceder, pg. 537-542
Bibliography
- John Edward Anderson, "Transit Systems Theory", Lexington
Books, 1978
- John Edward Anderson, "The Capacity of a Personal Rapid Transit System", 13
May 1997
- Daniel Boyle, "Fixed Route Transit Ridership Forecasting and
Service Planning Methods", Synthesis of Transit Practice,
Volume 66 (2006), Transportation Research Board, ISBN
030909772X
- Jon Carnegie, Alan Voorhees and Paul Hoffman, "Viability of Personal Rapid Transit In New
Jersey", February 2007
- Avishai Ceder, "Public transit planning and operation: theory, modelling
and practice", Butterworth-Heinemann, 2007, ISBN
0750661666
- Tom Parkinson and Ian Fisher, "Rail Transit Capacity", Transportation Research
Board, 1996, ISBN 0309057183