Mixed-speed rail capacity

Why does running one fast express down a busy line cost the timetable several slower trains?

A railway proposes one extra express each hour on an existing line. The planners reply that it will cost six or seven stopping services. No track has been removed, and the express occupies the rails for less time than the trains it displaces, since it is travelling faster.

That last observation is the one to sit with. If speed means less time on the track, the fast train ought to be the cheap one. Something about how a railway is shared must not work the way the intuition assumes — so what is actually scarce here?

Start with what separates trains. A train cannot stop within sight, so the line is divided into blocks and only one train may occupy a block at a time. What a following train needs, therefore, is not distance but time — the minimum headway, the shortest interval at which one train may follow another without being brought to a signal check. Call it h. On a homogeneous line, capacity is simply how many of those intervals fit in an hour: at a three-minute headway, twenty paths.

Now insert a faster train into that stream and follow it along. It leaves behind a slower train and immediately begins closing the gap on the one ahead. If it is not to catch that train, the gap ahead of it at the start of the section must be large enough to have shrunk only to h by the end. How much larger? Exactly the amount of time it gains.

That is computable. Take a forty-kilometre section. The stopping service averages sixty kilometres per hour over it, calling at stations, so it takes forty minutes. The express averages a hundred and twenty, so it takes twenty. The express gains twenty minutes. With a three-minute headway, the gap that must open ahead of it is twenty-three minutes where an ordinary stopping train would have needed three. The difference, twenty minutes, is a little under seven headways — so the express has consumed roughly seven slow paths and returned one. Six or seven services, as the planners said, and the number came out of arithmetic rather than pessimism.

Look now at what that expression contains. The paths lost are the time gained divided by the headway, and the time gained is a difference between two journey times. Not the express's speed. The difference.

Push that as far as it goes, because it is the point. Suppose every train on the line ran at a hundred and twenty. The difference is zero, no gap needs opening, and the line carries the same twenty paths per hour — while moving everyone twice as fast. Suppose every train ran at sixty: again twenty paths per hour. A homogeneous fast railway and a homogeneous slow railway have, to a first approximation, the same capacity in trains per hour. It is not speed that is expensive. It is heterogeneity of speed.

Which reframes the complaint. The express is not greedy; the express and the stopping service are jointly incompatible, and the cost belongs to the pair. You could as truthfully say the stopping service costs the express six paths — and if the line's purpose were long-distance travel, that is exactly how it would be written up.

Once you see that, the standard remedies stop looking like tricks. A passing loop lets the express overtake rather than wait. Flighting — expresses in a bunch, then stopping services in a bunch — confines the closing-up to the seam between groups. Most commonly of all, the express is simply slowed to a schedule that never quite catches the train in front. And the option that removes the problem rather than managing it is a separate line for one of the two speeds: the argument for dedicated high-speed railways in its least romantic form.

The arithmetic above is a clean idealisation, and real railways are not clean.

The claim that homogeneous-fast equals homogeneous-slow is only approximately true. Braking distance grows faster than speed, so with fixed blocks sized for the quickest train a genuinely fast railway needs longer blocks, and its minimum headway rises somewhat. Moving-block signalling narrows this considerably — one of the main reasons resignalling buys capacity without new track.

The second qualification matters more in practice. In most real timetables the binding constraint is not the following headway at all. It is structure: clockface patterns repeating every hour, connections held at junction stations, platform occupancy and dwell at busy stops, and junctions themselves, where crossing movements conflict in ways plain following does not. Much of what is described as "the express taking paths" is a periodic timetable refusing a service that does not fit its pattern. My arithmetic gives the mechanism, not the number for any particular railway; serious capacity work — the UIC 406 compression method is the standard European approach, as I recall it — measures the actual timetable rather than a formula. Single-track lines are a different problem in similar clothes, constrained by crossing rather than following.

How would you falsify the account? It makes an unusually sharp prediction: capacity in paths per hour should fall with the variance of speeds and be near-indifferent to the mean. So take a line and raise every train's speed together — paths per hour should hold roughly steady rather than move. If a uniformly faster railway with equivalent signalling carried materially fewer trains per hour than a uniformly slower one, heterogeneity would not be the mechanism, and the case for separating traffic by speed would weaken with it.

Every term the collection defines is gathered in the glossary.