IA idea · Simulation & Monte Carlo methods

Why do buses come in bunches? A simulation of a bus route

AI HLAI SLAA HL Ambitious Also in: Probability, Kinematics

Research question

On a route where buses leave at regular intervals but dwell longer at stops with more waiting passengers, how quickly do small delays grow into bunches, and which simple rule (holding buses, limiting dwell time) breaks the bunches up best?

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

Bus bunching is a famous feedback effect: a late bus picks up more passengers and gets later still. A simulation lets you watch it happen and test fixes that transport planners actually use.

The mathematics you'll need

A deterministic model of headways and dwell times
Random delays added with chosen distributions
Simulation over many runs; summary statistics of headways
Positive feedback in a recurrence
Comparing control rules with clear measures

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

Time buses at a stop you use on several days, or use timetables and observed arrivals; estimate boarding times by watching (no personal data).

Transport for London open data — Timetables, station entry/exit counts, cycle-hire journeys and air quality for London.
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

Build a simple recurrence for headways.
Show the feedback effect with no randomness.
Add random delays and simulate many days.
Test holding rules and compare waits.
Reflect on what your simulation leaves out (traffic lights, overtaking).

Pitfalls that cost marks

Building a model too complicated to explain.
Inventing parameters without any observation behind them.
Measuring success only by average wait, not reliability.

Showing personal engagement

Time your own bus route.
Ask a driver or the operator how they try to prevent bunching.
Design your own rule and test it.

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	Good fit	A deterministic model of headways and dwell times; Random delays added with chosen distributions
AI SL	Fits — ambitious at SL	A deterministic model of headways and dwell times; Random delays added with chosen distributions
AI HL	Good fit	A deterministic model of headways and dwell times; Random delays added with chosen distributions

Level: Ambitious. Suits confident students; expect to learn some mathematics on your own. 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: building a model too complicated to explain — 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

Find the holding time that minimises the average total journey time, or compare your simulation with a full day of observed arrivals.

Extending it for HL

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 How long does a skydiver take to reach terminal velocity? (AA HL) 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).