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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

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 considerably reduced the headways compared to earlier rail systems. 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. In 2017, in the UK, 66% of cars and Light Commercial Vehicles, and 60% of motorcycles left the recommended two-second gap between themselves and other vehicles.

Headway spacing is selected by various safety criteria, but the basic concept remains the same – leave enough time for the vehicle to safely stop behind the vehicle in front of it. The "safely stop" criterion has a non-obvious solution, however; if a vehicle follows immediately behind the one in front, the vehicle in front simply cannot stop quickly enough to damage the vehicle behind it. An example would be a conventional train, where the vehicles are held together and have only a few millimetres of "play" in the couplings. Even when the locomotive applies emergency braking, the cars following do not suffer any damage because they quickly close the gap in the couplings before the speed difference can build up. There have been many experiments with automated driving systems that follow this logic and greatly decrease headways to tenths or hundredths of a second in order to improve safety. Today, modern CBTC railway signalling systems are able to significantly reduce headway between trains in the operation. Using automated "car follower" cruise control systems, vehicles can be formed into flocks that approximate the capacity of conventional trains. These systems were first employed as part of personal rapid transit research, but later using conventional cars with autopilot-like systems.

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 two of the three inputs, it is a primary consideration in capacity calculations. The headway, in turn, is defined by the braking performance, or some external factor based on it, like block sizes. Following the methods in Anderson:

The minimum safe headway measured tip-to-tail is defined by the braking performance: $T_ = t_r + \frac \left\left(\frac - \frac \right\right)$ where: * $T_$ 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 minimum braking deceleration of the follower. * $a_l$ is the maximum braking deceleration of the leader. For brick-wall considerations, $a_l$ is infinite and this consideration is eliminated. * $k$ 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: $T_ = \frac + t_r + \frac \left\left(\frac - \frac \right\right)$ where: * $T_$ 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: $n_ = \frac$ where: * $n_$ is the number of vehicles per hour * $T_$ 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: $n_ = P \frac$ where: * $n_$ is the number of passengers per hour * $P$ is the maximum passenger capacity per vehicle * $T_$ 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; : $T_ = \frac + 2 + \frac \left\left(\frac \right\right)$ : $n_ = \times \frac$ : $T_$ = 10.5 seconds ; $n_$ = 7,200 passengers per hour if 4 people per car and 2 seconds headway is assumed, or 342 passengers per hour if 1 person per car and 10,5 seconds headway is assumed. The headway used in reality is much less than 10.5 seconds, since the brick-wall principle is not used on freeways. In reality, 1.5 persons per car and 2 seconds headway can be assumed, giving 1800 cars or 2700 passengers per lane and hour. For comparison, the Marin County, California (near San Francisco) 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. Notwithstanding these formulas it is widely known that reducing headway increases risk of collision in standard private automobile settings and is often referred to as tailgating. 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; : $T_ = \frac + 2 + \frac \left\left(\frac \right\right)$ : $n_ = \times \frac$ : $T_$ = 28 seconds ; $n_$ = 130,000 passengers per hour Note that most signalling systems used on metros place an artificial limit on headway that is not dependent on braking performance. Also the time needed for station stops limits the headway. Using a typical figure of 2 minutes (120 seconds): : $n_ = \times \frac$ : $n_$ = 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 spent a considerable amount of money on upgrading the SSR Network,Bombardier to Deliver Major London Underground Signallin

Press release, Bombardier Transportation Media Center, 2011. Accessed June 2011
Jubilee line|Jubilee and Central lines with new CBTC signalling to reduce the headway from about 3 minutes to 1, while preparing 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, bot 2.5, m/s if lead vehicle breaks. $k$ of 1.1; : $T_ = \frac + 0.01 + \frac \left\left(\frac - \frac \right\right)$ : $n_ = \times \frac$ : $T_$ = 3 seconds ; $n_$ = 28,000 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: : $n_ = \times \frac$ : $n_$ = 5,400 passengers per hour

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.Ceder, pg. 537–542

References

Notes

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, * 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, * Tom Parkinson and Ian Fisher
"Rail Transit Capacity"
Transportation Research Board, 1996, {{Use dmy dates|date=August 2019 Category:Rail technologies Category:Public transport Category:Transportation planning Category:Scheduling (transportation)