Module 5, lesson 3 · 90 minutes

Plan the start: the wave planner

Plan the start of a 40,000-runner marathon and watch, on a space–time map, what waves, gaps, release rates and pace assignment do to density at every pinch point.

You’ll be able to

predict how start-line throughput, waves, gaps and pace assignment change density at a pinch point, and say why;
read a space–time density map and the busiest-five-minutes metric at the finish;
find a start plan that meets a density target within a start-window limit, and explain what it costs.

You have 40,000 runners, one start line and a road that narrows to 5 m at the 6 km mark. Every one of those runners will pass through that 5 m in the first two hours of the race. Nothing you do on race morning changes how many people go through it, or how wide it is.

What you can change is when each of them arrives. Hold a wave back by five minutes, release runners a little more slowly, put the fast ones first, and the same 40,000 people arrive at the pinch in a different pattern. Some patterns are comfortable and some are a crush. This lesson gives you a planner to find out which is which, and shows you why the start line is the cheapest place on the whole course to buy space.

The model, in words

No technical description of Marcel Altenburg’s Start Right model has been published. What follows is our own model of the kind of calculation it must make, built only from what has been reported about its inputs and outputs: start times, course layouts and runners’ pacing in; counts at points on the course and runners per five-minute finish window out. Treat every number the planner produces as an illustration of the mechanism, not a prediction for any real race.

The model has four steps, and you could write each in a few lines of code.

A field. Each runner gets a predicted finish time. Ours come from a lognormal distribution with a median of 4:15 and a spread that puts 80% of the field between about 3:13 and 5:38. That is illustrative, not any real race’s field. A runner’s pace is their time per kilometre, in min/km: predicted finish time divided by 42.195 km. We hold it constant for the whole race.
Waves. Runners are sorted (or not) and cut into waves: groups released across the start line one after another, each starting a set time after the previous one has cleared the line. Each wave crosses the line at the start-line throughput, the number of runners per minute who cross the start line while a wave is being released.
Positions. A runner who crosses the line at time s with pace p is at distance x = (t − s) / p at time t.
Counts. Every minute, count the runners in each 100 m stretch of road and divide by the stretch’s area. That is the density, in runners per square metre.

$x_i(t) = \frac{t - s_i}{p_i} \qquad \rho(x, t) = \frac{n(x, t)}{\Delta x \; w(x)}$

xᵢ(t) = (t − sᵢ) / pᵢ ρ(x, t) = n(x, t) / (Δx · w(x))

sᵢ: time runner i crosses the start line, min
pᵢ: runner i’s pace, min/km
n(x, t): runners in the 100 m stretch at x at minute t
Δx: stretch length, 100 m
w(x): road width there, m
ρ: density, runners/m²

That is the whole model. It is a macroscopic, space–time calculation of the kind you met in Module 4: nobody steers, nobody bumps into anyone, and runners pass through each other like ghosts. For 40,000 runners over seven hours it is about ten million small additions, which a browser does in well under a second.

The one equation to remember

Before you touch the planner, work out what a wave does just after it crosses the line. Runners leave at c per minute. Each covers 1,000 / p metres in a minute. So every metre of road behind the line holds c × p / 1,000 runners, and every square metre holds that divided by the width.

$\rho_{\text{release}} \approx \frac{c \, p}{1000 \, w}$

ρ(release) ≈ c · p / (1000 · w)

c: start-line throughput, runners/min
p: the wave’s pace, min/km
w: road width, m

Worked example: a wave of runners at 8 min/km released at 1,000 a minute reaches a 5 m pinch. ρ ≈ 1,000 × 8 / (1,000 × 5) = 1.6 runners/m², about 0.6 m² each. A wave at 4.6 min/km under the same conditions gives 0.9 runners/m². Same release, same road, nearly double the density. Slow runners take up more road per minute of release, and that fact drives most of what you are about to see.

A real wave spreads out as it goes, because its runners do not all share one pace, so the density at the pinch comes out a little lower than this. How much lower depends on how mixed the wave is.

How to read a space–time map

The planner’s main output is a space–time diagram: a map with distance along the route on one axis and time on the other, so that each runner’s journey is a line and a crowd is a coloured band. Distance runs left to right and time runs downwards, like a timetable. Figure 1 shows how to read one.

Reading the map. A horizontal slice is a photograph of the whole course at one moment; the planner draws it as the ribbon above the map. A vertical slice is what a marshal at one point sees over the morning; the planner draws it as the point chart. Steeper lines are slower runners. Schematic, not to scale.

A runner is a line. It starts at the left edge when they cross the start line and slopes down to the right. A fast runner’s line is shallow, a slow runner’s steep.
A wave is a band that fans out as fast runners pull ahead of slow ones. The wider the fan, the more the wave has thinned out.
A pinch is a column. The same flow on a narrower road is denser, so bands darken as they cross it.
Trouble is where bands overlap: the fast runners of a later wave catching the slow runners of an earlier one.

The planner

The defaults are a plausible big-city plan: four equal waves assigned by predicted finish time, as London assigns its waves, with the fastest first, released at 1,000 runners a minute with 5-minute gaps. The target is 0.33 runners/m², the “three square meters for each runner” design target reported for New York. The course is invented: wide at the start, a 5 m pinch at 6 km, a 7 m bridge at 12 km, a 6 m pinch at 20 km and a funnel to the finish.

Keep the defaults. In the table, find the peak at Pinch A. Drag the time cursor (or click the map) to that moment. Which wave is passing?
Press Optimise gaps and read the message. Then set the gap to 15 minutes yourself. What changes, and what does not?
Switch “Who goes in which wave” to slowest first. Where on the map do the peaks move, and what happens to the busiest five minutes at the finish?
Now try at random. The peaks fall. Before you celebrate, compare the readout “Overtakes per runner, first 10 km” with the fastest-first plan.
Go back to fastest first and tick Slower waves cross the line more slowly. Then try 8 waves with no gap. Compare Pinch A and the start window with step 1.
Halve the start-line throughput to 500. Which pinch improves most, and what does it cost?
Open “Edit the course”, widen Pinch A from 5 m to 8 m and apply. How much did that one change buy?

Gaps buy space only when a later wave can catch an earlier one. With the fastest runners first, none ever does, so the densest point on the course sits inside a single wave, usually the slowest, and its density is set by how fast you release runners for their pace and the width of the road: ρ ≈ c·p / (1000·w). Put the slow runners first and the fast waves plough through them mid-course, and the peaks there double or triple. The start line is the cheapest place to buy space: every runner a minute you don’t release becomes room at every pinch downstream.

Our own illustrative model, not Start Right. Runners keep a constant pace and pass through each other; the congestion option is a crude, uncalibrated feedback, there to show that slowing in crowds turns peaks into queues. The map covers the first seven hours; the slowest runners in this field are still out after that. Elite and wheelchair starts are ignored. The colours use the site’s Level of Service scale for walkers; runners need far more room than walkers, which is why the target sits at the A/B boundary. The design target is from New York; the course and field are invented.

Four levers and what each one buys

Table 1 shows what the planner gives for a handful of plans with the same field (seed 1). The numbers are outputs of our model, not measurements of any race; the patterns are what matter.

Same 40,000 runners, different startsPeak density in runners/m² at each watched point, from the planner (our model, seed 1). Target 0.33.
Plan	Start window (min)	Pinch A	Bridge	Pinch B	Busiest 5 min at finish	Overtakes per runner, first 10 km
Defaults: 4 waves, fastest first, 1,000/min, 5-min gaps	55	1.39	0.83	0.64	1,517	1,424
Same, 15-min gaps	85	1.39	0.83	0.64	1,517	1,424
Slowest first	55	1.99	2.09	1.98	2,705	4,535
At random	55	0.95	0.61	0.56	1,415	4,119
Fastest first, slower waves cross more slowly	69	1.00	0.71	0.64	1,576	1,231
Same, 8 waves, no gaps	58	0.94	0.66	0.64	1,450	587
Defaults at 500/min	95	0.81	0.52	0.56	1,470	936

1. Start-line throughput sets the density inside a wave

This is the main dial. Inside a wave that has not yet spread out, density is c·p / (1000·w), so halving c roughly halves the peak at the first pinch. Its cost is time: the start window is the field divided by the throughput, plus the gaps. At 500 a minute, 40,000 runners take 80 minutes just to cross the line.

2. Gaps only work if waves would otherwise meet

With the fastest wave first, every later wave is slower than the one ahead, so it never catches up. The bands on the map never overlap and a gap between them changes nothing at any pinch. It only makes the morning longer. That is why the optimiser reports that gaps cannot fix the default plan.

Gaps start to earn their keep when waves contain a mix of paces, as in the random plan: the fast runners of wave 2 would otherwise catch the slow runners of wave 1 a few kilometres in. Then a few extra minutes at the start keep the bands apart, which is the effect the optimiser searches for.

3. Who goes in which wave

Slowest first is the disaster you should be able to predict from the map: each faster wave ploughs into the slower one ahead, and the overlaps land squarely on the bridge and Pinch B. Peaks at the bridge and Pinch B rise two- to threefold, and because the whole field converges, the busiest five minutes at the finish jumps by three-quarters.

Random assignment looks better than fastest first on density, and that is a genuine result of the model, not a bug. A mixed wave fans out quickly, so it is thinner by the time it reaches a pinch. The price is that the field spends the race overtaking itself: about three times as many overtakes in the first 10 km. Our model lets runners pass through each other at no cost; real runners have to find a gap, change line or slow down, and every pass in a narrow, crowded road is a chance to clip someone’s heels. This is a good example of a model answering only the question it was built for. On density alone, mixing wins. On anything to do with passing, it loses.

4. Release slower waves more slowly

The equation says the slowest wave is the densest when every wave crosses the line at the same rate. Ticking “Slower waves cross the line more slowly” releases each wave at the same packing instead, so a wave at 8 min/km leaves at a little over half the rate of one at 4.6. The densest wave gets thinner, the fast waves are untouched, and the start window grows only by the extra time the slow waves take. Combine it with more, smaller waves and you get densities close to the random plan with about a seventh of its passing.

The start decides how many runners per minute enter the course and in what order. It cannot make a 5 m road wider. On this invented course we searched 1,800 combinations of the levers, and none with a start window of two hours or less held every pinch at 3 m² per runner. Widening Pinch A to 8 m, with the default start, does more for it (1.39 down to 0.87 runners/m²) than the best 55-minute start plan we found (0.90). Some problems are design problems, and the planner’s job is to show you which ones.

In the default plan, why does the optimiser say gaps cannot help? What would you change instead, and what would it cost?

The waves go fastest first, so every wave is slower than the one ahead and never catches it; the waves never overlap. The peak at Pinch A therefore comes from inside one wave (mostly the slowest), where density is set by throughput, pace and width: ρ ≈ c·p / (1000·w). Lower the throughput, or let slower waves cross more slowly, or widen the pinch. The first two cost start-window time; the third costs money and may not be possible on a public road.

The busiest five minutes at the finish

The finish chart counts runners per five-minute window, the output reported for Start Right in New York. It matters because everything at the finish is sized to a peak rate rather than a total: the funnel, the medal tables, the baggage trucks and the medical teams. The World Marathon Majors magazine says the model’s predictions are used to place aid stations, toilets and medical staff.

Notice how little the start plan changes it, unless you do something drastic like slowest first. By 42 km the spread of paces dominates: a runner at 4:00 and one at 5:00 finish an hour apart, and a few minutes of difference at the start hardly matter. The finish peak is mostly a property of the field: how many runners, and how their predicted times are distributed. Try the spread-of-finish-times slider to see this.

For scale: in New York, the busiest finish window Altenburg projected held 1,366 runners and the real one held 1,367. Our default field happens to produce a peak of the same order, about 1,500, which tells you nothing about Start Right but does tell you the invented field is not absurd. Lesson 2 looks at what that famous comparison does and does not show.

When crowding slows runners down

So far every runner holds their pace whatever is around them. Real runners slow down in a crowd, and that changes the picture: a pinch where runners slow down holds more of them, which slows them more. Tick “Congestion slows runners” to switch on a deliberately simple version: speed is unchanged up to 0.33 runners/m², falls in a straight line to 30% of normal at 1.5 runners/m², and nobody can move into a stretch of road already at 2.0. Runners who cannot get in wait at the end of the stretch behind.

With the default plan, Pinch A jams. The peak hits the cap and stays there, the dense region spreads backwards up the road, and the bridge and Pinch B fill too. This is the bottleneck queue of Module 1, lesson 5 appearing in a running race. The numbers in this mode are invented for teaching and have not been calibrated against any race. The lesson is qualitative: in a model where crowding slows people down, a plan that is merely dense in the simple model becomes a queue.

Mini-capstone: grow the field

The race wants to grow from 40,000 to 52,000 runners on the same course. Your brief: do it without raising the peak density at Pinch A above the default plan’s 1.39 runners/m², and say what else changes. Work it out in the planner before you open the model answer.

Plan a 52,000-runner start that keeps Pinch A’s peak at or below 1.39 runners/m². What did you change, and what did it cost?

First, set 52,000 and change nothing else. Pinch A barely moves (1.41 runners/m²), because with fastest-first waves the peak sits inside the slowest wave and depends on throughput, pace and width, not on how many runners there are. What does change: the start window grows from 55 to 67 minutes, Pinch A spends 72 minutes over the target instead of 60, and the bridge and Pinch B get busier (0.97 and 0.84 runners/m²) because there is more overlap mid-course. The busiest five minutes at the finish rises from about 1,520 to about 1,920.

Two plans meet the brief. Dropping the throughput slightly, to 980 a minute, brings Pinch A back to 1.38 for one extra minute of start window. Ticking “Slower waves cross the line more slowly” takes it down to 1.00, a better answer than the brief demands, for a start window of about 85 minutes.

The point of the exercise: field size mainly buys duration at a pinch, not peak density. And a brief that names one pinch can be met while others get worse, so always read the whole table.

“Gaps between waves are free space.” They buy space only when a later wave could catch an earlier one. With waves sorted fastest first, a gap only lengthens the morning.
“More waves means less crowding.” Not by themselves. Eight equal waves released at the same rate give the same or slightly higher peaks than four, because each wave is more uniform in pace and spreads out less. More waves help when each one is released more slowly or when the gaps keep mixed waves apart.
“Lower density means the plan is safer.” Density is one hazard. A random plan is less dense in our model but asks the field to overtake itself about three times as often. A model only answers the question it was built to answer.
“The planner shows what Start Right does.” It shows what a model of this kind does. Start Right’s method is unpublished.