Race four aircraft boarding methods against each other and one of them wins. Race them again and a different one might. Boarding is a process made of almost nothing but chance, which makes it an unusually honest way to show what a single run of any model is actually worth — including yours. We built the race as an ordinary Excel workbook: you can download it and run it yourself.
One run of a stochastic model produces one sample from a distribution of possible outcomes, not the answer. It looks like a result because it is a specific number, but it carries no information about how much of that number was luck. Only repeated runs reveal the shape underneath, and therefore whether a difference is real.
Our boarding model makes the point sharply. Two of the four methods we tested differ by 11 seconds on a job that takes about 21 minutes. Yet in a single boarding, the gap between them exceeds a whole minute 42% of the time — in whichever direction the dice happen to fall. Watch one race and you will see a decisive-looking result that is very nearly a coin toss.
Here is one boarding, rendered from the model itself rather than screen-captured — four cabins filling at once, each seat shaded by the minute it was taken:
Watch it to the end and WilMA loses to random boarding. That is not a mistake and it is not a rigged demonstration — WilMA is genuinely about two and a half minutes faster on average. It is one race, and one race is entitled to any result it likes.
An A320-style cabin: 30 rows, three seats either side of a single aisle, one door, and 122 passengers by default. Each passenger walks to their row at a fixed pace, cannot overtake the person in front, stops to stow a bag for a lognormally distributed time, and climbs over anyone already seated between them and the aisle. On top of that sit the two things that make boarding messy in real life:
It is a simplification, and the numbers below are outputs of this model rather than measurements of real aircraft. What matters is not the absolute minutes but what happens to them when you run the thing thousands of times.
Each of the 3,000 iterations is paired: every method faces the identical passengers, identical bags and identical stow times, so the comparison isolates the method rather than the luck. This is the technique known as common random numbers, and it is the reason the confidence intervals below are as tight as they are.
Read the back-to-front row carefully, because it is the most useful line in the table. The difference is real — 3,000 iterations put the confidence interval comfortably below zero, so we can say with confidence that back-to-front beats random. It is also worth eleven seconds, and it wins barely more often than it loses.
Real and worth having are two different questions. A simulation answers both. A single race answers neither, while appearing to answer the first.
Re-run the same comparison with everyone queueing exactly as instructed and nobody travelling in groups, and back-to-front stops beating random at all — it comes in 19 seconds slower.
The effect does not merely shrink under different assumptions. It changes sign. The mechanism is not mysterious: boarding by zone deliberately concentrates passengers into the same few rows, where they queue behind each other, and the more faithfully people obey the call the more concentrated they get. The small advantage the default settings show comes largely from passengers ignoring the zones and spreading themselves out.
A result that reverses when you change an assumption you were not thinking about is not a finding you should act on. Testing that is called robustness analysis, and it is only possible once you can run the model more than once.
Enough that the confidence interval around the quantity you care about is narrower than the difference you are trying to detect. That is the only honest rule, and it means the number is set by your question, not by convention.
Our boarding model shows the whole range in one place. Steffen's eight-minute advantage is beyond doubt inside about twenty runs. The eleven-second one needs this many:
So a difference roughly forty times smaller needs roughly a hundred times the runs. That is the shape of the trade in general: the iterations you need scale with the square of the precision you want, which is why halving your error bar costs four times the computation.
This is also why fixed advice like "run 10,000 iterations" is only half useful. It is usually enough, which is why people say it. But if you are choosing between two options whose values differ by 1% while the spread of outcomes is 20%, no round number saves you — you need enough iterations for the uncertainty in the difference to shrink below that 1%.
The workbook shows this happening live. Its Simulate sheet keeps a running 95% confidence interval and a verdict line that reads "cannot tell it apart from random" until the evidence justifies changing it. Hold the F9 key and watch the interval close.
Because its advantage is not a property of the method alone — it is the method multiplied by how well you can enforce it.
Steffen's ordering is genuinely fast in our model, saving over eight minutes. But it requires 122 people to queue in an exact sequence. Drop the compliance setting and the advantage drains away:
Group travel does the same thing. Raising the share of seat blocks travelling together from 0% to 50% costs Steffen about four minutes, because any scheme that separates people who intend to board together decays toward the thing it replaced.
Put both at their ideal — everyone in the right slot, nobody travelling together — and Steffen finishes in 7:27 against random's 21:17, a 65% saving. Set them to something more like a real gate and two thirds of that is gone. The method did not change. Only the assumptions about people did.
This is not our discovery. Steffen himself noted that his method would separate those boarding in groups and would require a perfectly organised line and uniform stowing times, and identified that as the likely reason no airline has adopted it. What simulation adds is the ability to put a number on the fragility — to say how much of the advantage survives at 85% compliance rather than merely observing that some of it does not.
That question — how much of a modelled benefit survives contact with reality — is the one worth carrying into your own models.
Not the thing everyone argues about. Here is what happens to random boarding when only the average time to stow a bag changes:
Cutting the average stow from 15 seconds to 10 saves 5 minutes 39 seconds. Switching from random boarding to WilMA saves 2 minutes 37. The operational variable nobody measures is worth more than twice the best boarding policy an airline could realistically run — and about two thirds of what even the theoretically optimal order achieves.
This is the most transferable finding in the whole exercise, and it generalises well beyond aircraft. Sensitivity analysis routinely shows that the input a team argues about is not the input driving the output. You cannot see that from a single scenario, because a single scenario holds everything else still. It falls out of a simulation almost for free.
Yes — and the workbook is deliberately built that way, with no add-in and no macros, so anyone can open it, and every intermediate value is visible on the Engine sheet for anyone who wants to audit the model rather than trust it.
Download the boarding race workbook (.xlsx) — free, no registration.
It has six sheets. The race fills four cabins seat by seat, colour-coded by the minute each seat was taken, with a clock over each; press F9 and it boards again. Simulate accumulates every press into running distributions, head-to-head win rates and that narrowing confidence interval. Findings holds 24 pre-computed scenarios the live model reproduces. Thirteen input cells are yours to change.
It also demonstrates, honestly, what simulation costs without a simulation engine: plain Excel adds one boarding per keypress. Reaching a stable answer takes several hundred. That is not a flaw in the model, it is the arithmetic of doing Monte Carlo by hand. Laying the model out one iteration per row gets you thousands of iterations per keypress, as our guide to Monte Carlo simulation in Excel shows, but only for models that fit on a row.
ModelRisk removes exactly that constraint. The same model runs 10,000 iterations in seconds; VoseLognormal replaces the hand-built stow-time draw and ModelRisk's distribution fitting fits it to real turnaround data instead of a guess; exact percentiles replace the approximate histogram; and a tornado chart identifies the stow-time finding above automatically, in one run, instead of by hand-built scenario sweep. The workbook's With ModelRisk sheet lists the exact cell swaps.
VoseLognormal
The alternating-row method is due to Jason Steffen, who published it in 2008 in the Journal of Air Transport Management. That paper is a simulation study, and reports that the optimal order could cut boarding time by a factor of four or more, depending on the dimensions of the aircraft (arXiv:0802.0733, retrieved 17 August 2026). Steffen and Hotchkiss later tested the methods for real, with 72 volunteers in a mock 757 fuselage, and the margins came in smaller: Steffen 3:36, WilMA 4:13, random 4:44, back-to-front 6:11 (arXiv:1108.5211, retrieved 17 August 2026). The authors put the uncertainty on any one of those times at roughly 10%.
One caution when reading the two side by side: what that experiment calls blocks — boarding in groups of four rows — is the method our model calls back-to-front, while its back-to-front is a stricter row-by-row order. Names in this field are not standardised, which is its own small lesson about comparing published results.
On the experiment's figures the Steffen order boarded about 24% faster than random. Our model is considerably more favourable to it than that — about 39% faster at the default settings, and 65% faster with perfect compliance and no group travel. We have not tuned the model to match the literature, and the gap is worth stating plainly: our model is generous to Steffen throughout, which makes the fragility findings above conservative rather than overstated. A model tuned to reproduce the published figures would show the advantage collapsing faster, not slower.
Enough that the confidence interval on your output is narrower than the difference you need to detect. The iterations required scale with the square of the precision you want, so halving the error bar costs four times the runs. In our boarding model a large difference settles within about twenty runs while a difference forty times smaller needs about two hundred. Watch the interval rather than fixing a number in advance.
Three scenarios give you three numbers with no probabilities attached, and the combination of every input at its worst is usually far less likely than people assume. A simulation samples the inputs together thousands of times and returns a distribution, so you can state how likely an outcome actually is.
A technique for comparing options fairly by giving each of them the identical set of random draws in each iteration. Because the options face the same luck, the difference between them is measured much more precisely, so far fewer iterations are needed to detect a real effect.
Yes. Excel's RAND function plus a recalculation loop is enough in principle, and the workbook here does exactly that. The practical limits appear quickly: one iteration per keypress, approximate percentiles, no sensitivity analysis and no way to fit a distribution to data. An add-in removes those limits rather than adding capability Excel lacks entirely.
It is a deliberate simplification: a single-aisle cabin, one door, no premium boarding, no gate-checked bags. It reproduces the qualitative results reported in the boarding literature — structured methods beat random, and the theoretically optimal method is fragile — but the minute figures are outputs of this model, not measurements of real aircraft.
In our model, Steffen's ordering, by a wide margin, followed by window-middle-aisle. But the ranking is less interesting than its fragility: most of Steffen's advantage depends on passengers queueing in an exact order, which is precisely what does not happen at a real gate.
The boarding race is a toy with a serious point: the answer you get from one run of any uncertain process is a sample, not a conclusion. If your forecasts, cost estimates or schedules are single numbers today, the same demonstration applies to them — the cost-contingency worked example is the same idea applied to real money, and the Monte Carlo simulation explainer covers the method underneath.
Start the free 15-day ModelRisk trial — fully functional, no restrictions. If you are moving from another Monte Carlo add-in, the built-in converter will bring your existing models across.
When one run is not enough — and it rarely is — ModelRisk runs your existing Excel model thousands of times, fits distributions to your data, and reports the confidence intervals this article says you should be asking for. A fully functional 15-day trial is free.