Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

Are arrivals at the school canteen random? A Poisson model of a queue

AI HLAI SL Solid Also in: Statistics, Simulation

Research question

Do arrivals at my school canteen in one-minute intervals follow a Poisson distribution, and would opening a second till reduce the expected queue length at peak time?

Make it yours, then agree it with your teacher.

Next step: plan this IA free — the planner opens with this question filled in. For feedback on your own draft, the IA package (€39 once) gives you 5 AI feedback credits and every annotated exemplar in full.

Why it makes a good exploration

Queues are everywhere and queueing theory starts with the Poisson distribution. Counting arrivals yourself gives personal data, and the second-till question is a real decision you could put to your school.

The mathematics you'll need

Poisson distribution (AI HL)
Chi-squared goodness-of-fit test
Mean and variance comparison
Simple queue simulation in a spreadsheet

The statistics, step by step

Worked with every number shown, with what examiners look for and the common mistakes: Chi-squared goodness of fit · Binomial and Poisson models. Then run the same steps on your own data in Analyse my data, or start from the statistics workflow.

Where the data comes from

Count arrivals per minute over several lunchtimes and time how long each service takes.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

A possible outline

Explain why arrivals might be Poisson (independent, random).
Collect counts; compare mean and variance.
Test the fit with chi-squared.
Simulate the queue with one and two tills.
Reflect: groups of friends arrive together, bells cause surges.

Pitfalls that cost marks

Counting across the bell when the rate changes.
Expected frequencies of 5 or less in the tail.
Service times assumed constant without evidence.

Showing personal engagement

Present your findings to the canteen staff.
Compare two days with different menus.
Explain the groups-of-friends effect statistically.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Not a natural fit	The core technique sits in the AI course or at HL; an AA SL student could use it only as clearly explained new mathematics.
AA HL	Not a natural fit	The mathematics is mainly from the AI course; at AA HL the exploration would need an AA-level approach (calculus, proof or probability theory) to reach the top of Criterion E.
AI SL	Good fit	Chi-squared goodness-of-fit test; Mean and variance comparison
AI HL	Good fit	Poisson distribution; Chi-squared goodness-of-fit test

Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Reflection (D)

For this idea, start with the weak points: counting across the bell when the rate changes; and expected frequencies of 5 or less in the tail. Say how each one affects your answer.

Use of mathematics (E)

SL: chi-squared goodness-of-fit test; mean and variance comparison; simple queue simulation in a spreadsheet, each explained in your own words and carried out correctly.

HL adds: poisson distribution, each explained in your own words and carried out correctly.

How probability explorations reach the top bands in general: Probability & chance IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Use the M/M/1 queue formulas (new maths — explain them) and compare with your simulation.

Extending it for HL

This idea already has HL mathematics in it: Poisson distribution. Make that the centre of the exploration and use it with full rigour.

See a complete IA, marked

Are arrivals at the school canteen random? A mid-band AI HL draft, annotated to show how to improve (AI HL) explores a question like this one, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AI HL and AI SL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Poisson distribution (AI HL), chi-squared goodness-of-fit test, mean and variance comparison and simple queue simulation in a spreadsheet.

Where can I get data for this IA?

Count arrivals per minute over several lunchtimes and time how long each service takes. Checked sources and tools: Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “Are arrivals at the school canteen random? A Poisson model of a queue” filled in: the question, the maths and the data sources.

Plan this IA free

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