IA idea · Simulation & Monte Carlo methods

One snake queue or a queue at every till? Simulating supermarket checkouts

AI SLAI HLAA SLAA HL Solid Also in: Probability, Statistics

Research question

For the arrival rate and service times you measure at a real shop, does one shared queue feeding several tills give shorter and less variable waits than a separate queue at each till, and by how much?

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

Free: the A–E checklist an examiner uses, by email ↓

Why it makes a good exploration

Everyone has an opinion about queues; a simulation built on your own timings settles it. You collect the inputs, build the model, and compare both the average wait and its spread.

The mathematics you'll need

Modelling arrivals (Poisson counts, exponential gaps) from your own data
Service-time distributions fitted to timings
Simulation in a spreadsheet or short program, explained step by step
Mean and standard deviation of waiting times; comparing distributions
HL: confidence intervals for the mean wait

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.

Modelling the data, step by step

This idea usually needs an exponential model. See it worked step by step, with a criterion tip at every step: Exponential y = a e^(kx) + c and y = a·bˣ.

Model your own data Paste it from Desmos, GeoGebra or a spreadsheet and get the same play-by-play with your numbers. New to modelling? Start with the modelling workflow. Writing it up? The IA modelling planner comments on each paragraph as you draft — it never writes it for you.

The statistics, step by step

Worked with every number shown, with what examiners look for and the common mistakes: Confidence intervals for a mean · Binomial and Poisson models · Descriptive statistics and box plots. Then run the same steps on your own data in Analyse my data, or start from the statistics workflow.

Where the data comes from

Time customer arrivals and service at a shop or canteen for several busy periods (with permission, no personal data).

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

Collect arrival and service data; fit distributions and check the fit.
Simulate a single till and compare with your observations.
Simulate both queue designs for the same arrivals.
Compare averages, spreads and the longest waits.
Reflect on queue-jumping, choosing the 'wrong' queue and staff breaks.

Pitfalls that cost marks

Assuming arrivals are Poisson without checking.
Running too few simulations to tell the designs apart.
Comparing only averages, not the spread that customers notice.

Showing personal engagement

Time the queues yourself at a place you use.
Ask the manager why they chose their layout.
Test a 'join the shortest queue' rule against random choice.

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

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Modelling arrivals (Poisson counts, exponential gaps) from your own data; Service-time distributions fitted to timings
AA HL	Good fit	Modelling arrivals (Poisson counts, exponential gaps) from your own data; Service-time distributions fitted to timings
AI SL	Good fit	Modelling arrivals (Poisson counts, exponential gaps) from your own data; Service-time distributions fitted to timings
AI HL	Good fit	Modelling arrivals (Poisson counts, exponential gaps) from your own data; Service-time distributions fitted to timings

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

Personal engagement (C)

Design the simulation yourself: choose the rules, test them on a small case you can check by hand, and change one rule at a time to answer a question you care about.

Reflection (D)

Compare simulation with exact theory or real data, say how many runs you used and how much the answer varies between batches, and question the random-number assumptions. For this idea, start with: assuming arrivals are poisson without checking — say how it affects your answer.

Use of mathematics (E)

SL: A probability model described precisely, simulated correctly, with the simulated answer compared with an exact calculation for at least one simple case and the results summarised with appropriate statistics.

HL: An estimate of the simulation's error (standard error, or a confidence interval for the estimate), a distribution fitted to the results and tested, or an exact result proved for the general case that the simulation confirms.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Add customers who leave if the queue is too long, or find the number of tills needed to keep the average wait under two minutes.

Extending it for HL

This idea already has HL mathematics in it: confidence intervals for the mean wait. Give a confidence interval for each simulated estimate and show how it narrows as the number of runs grows, or prove a general result that the simulation confirms.

See a complete IA, marked

Our annotated exemplar Should I lease or buy my first car? (AI SL) asks a different question, but shows how a complete simulation exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

Before you start: the checklist an examiner uses

Every check for Criteria A–E in a 4-page PDF, the mistakes that cost the most marks and a self-assessment grid. We'll email it with a short IA tip every few days, timed to your deadline if you give it. Free — no account, no payment.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).