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

AI HLAI SL Solid Also in: Statistics

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?

Adapt it: change the place, the data or the comparison until the question is yours.

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

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

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.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

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 below 5 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.

Taking it further

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

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Desmos workspacePlot your data, fit models and keep your graphs with your plan. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, a practising IA examiner, reads your draft and talks it through with you (€100).