Maths IA · Modelling · topic 40 of 40

Shop queue: predict its length from arrivals · IB Maths IA idea: guide, sample graph and step-by-step plan

stretch topic: observe first, model second

You stand in a queue and wonder why it's never short. You're going to count it, model it, and find out why a busy till is worse than it looks.

Difficulty●●● Stretch
KitA stopwatch, a real queue, a spreadsheet or Python
TimeAbout 14 to 18 hours
FitsAI HL
  • ✓ you like probability with real counting
  • ✓ you can observe a queue and keep tallies
  • ✓ you want a model and a simulation to argue it out

The big idea

Customers arrive at random and each takes a random time to serve. If arrivals come at average rate λ and service at rate μ, the ratio ρ = λ/μ decides almost everything: as ρ nears 1 the queue grows without limit. Collect real arrival and service data, test which distributions fit (Poisson arrivals? exponential service?), predict the queue by a formula and by simulation, and compare both with what you saw. HL probability or Markov chains carry criterion E, so do not stop at counting people. Queueing formulas and the exponential distribution are in neither course, so introduce and justify them. Count, don't identify: no names or faces.

Make it yours: pick one twist

Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.

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Count a real queue

A canteen, a bus stop or a till (with permission): arrival times and service times.

→ Real data to test your distributions. Privacy: record counts and times only.
Test the distributions

Fit Poisson to arrivals per minute and exponential to gaps; check with a chi-squared test.

→ A hypothesis test with conditions to check, so you test the model, not just use it.
Use a Markov chain

Treat the queue length as states and find the long-run distribution.

→ Matrices and eigenvectors; the steady state is compared with what you observed.
One fast till or two slow

Compare one quick server with two slower ones sharing a line.

→ A design question with a clear answer to test by simulation.
Vary the load

Simulate several values of ρ and find where the queue passes a limit you choose.

→ A graph of queue length against ρ, and a view of where small changes matter.
Replay your data

Feed the observed arrival times through the simulation and compare with the observed queue.

→ A fair test of the service model, and a strong evaluation.

Your Maths IA research question: run the check

write it FIRST, then check it

Write your own question first. Then make sure it ticks every box:

  • Names the real queue (where, when, for how long) and exactly what you count.
  • Names the quantities to predict (mean queue length, waiting time) and compare with observation.
  • Says which probability models you will test, and how.
  • Says how you will deal with the simulation's randomness: number of runs and run length.
  • Is possible to observe in your time, with enough customers (at least 100 arrivals) from a similar period.

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The model

Customers arrive at rate λ, wait, and are served at rate μ. The tally strip is your real data: arrival times that test the distribution or feed the simulation.

λμarrivals: how many per minute?queue: those waitingservice time: stopwatch ityour tally of arrival gaps feeds the simulation

How your Maths IA could be laid out

The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.

Introduction and aim~1.5 pages
Your question; why this queue; what you expect before any modelling.
Data collection and ethics~1.5 pages
What, when and how you counted; permission; why your time window is representative; one clear table.
Probability models~3 pages
The distributions you test, their parameters from your data, and the conditions for using them.
Testing the models~2 pages
Hypotheses, expected counts, test statistic, conclusion; conditions checked.
The simulation~2 pages
Rules in words; run length; number of repeats; how you start and stop. Code in an appendix.
Predicting queue length~2.5 pages
A formula or a Markov chain. HL: derive it, for example with a geometric series.
Comparing with observation~2 pages
Prediction, simulation and observation together, with numbers for the differences.
Conclusion~1 page
Answer the question and say how far the model can be trusted.
Reflectionbuilt in, plus ~0.5 page
Not one paragraph at the end: a comment after each result or choice (a poor fit, a surprise, a decision you changed), then a specific evaluation of the model's limits.
Bibliographynot counted
Every source for data, theory and any code or software used.

What your Maths IA graph could look like

define 'queue length' ✓

Made-up values to show the form of a strong graph, not the answer: one server, random arrivals and service. Yours will look different. example only

0.30.40.50.60.70.80.9012345678910Utilisation ρ = λ/μ (no units)Mean number L (people)waiting in the queuein the shop (waiting + served)simulation: mean of repeated runsqueue explodes as the tillnears full useILLUSTRATIVE · MADE-UP DATA
  • ρ is a ratio, so it has no unit: say so.
  • Two curves because 'queue length' can mean waiting or in the shop: define yours.
  • Simulation points are means of many runs. The bars show the run-to-run spread.
  • The curve steepens near ρ = 1, so small errors in λ or μ matter most when the shop is busiest.
  • Another graph to consider: observed arrivals per minute as bars, with the fitted Poisson probabilities as dots.

How long must the simulation run?

how long is long enough?

A queue simulation starts empty and wanders. Short runs are noisy and read low. Repeat each run length many times and watch the typical error fall: that tells you how long yours must be.

50 customers62.6%200 customers54%1000 customers32.6%5000 customers14.6%short runs misleadroot-mean-square error in the simulated mean wait, % of the theory value (ρ = 0.8, starting empty)ILLUSTRATIVE · MADE-UP NUMBERS

Your game plan

trial observation saves weeks

Tick steps off as you go. We'll remember on this device.

0 of 10 done

  1. Read and predictLearn the basic queue idea. Predict how the queue grows as the till gets busier, and why.
  2. Get permission and do a trial (1 lesson)Observe for 15 minutes. Can you record arrival times and service times reliably? What gets in the way?
  3. Plan the observationChoose the queue, the times, the length of each session and what you will not record. Write down why.
  4. Collect the dataRecord arrival times and service times. Note unusual events (a rush, a till closing).
  5. Fit and test the distributionsEstimate λ and μ. Test whether arrivals and service times follow your chosen distributions, checking the conditions.
  6. Build the simulationWrite the rules, choose run length and repeats, and keep a log of the settings.
  7. Do the theoryPredict the queue with a formula or Markov chain. HL: derive it and say which assumptions it needs.
  8. ComparePut prediction, simulation and observation together and quantify the differences.
  9. Write, then cutWrite in layout order. Rules in words and mathematics; code in an appendix.
  10. Check before your teacher's draftCheck every assumption is stated, symbols defined, graphs titled with units, and reflection runs through the piece.

Maths you'll need

Traps to avoid in this Maths IA

  • Assuming Poisson arrivals without testing them: queues at bell times or lunch are not random.
  • Mixing 'waiting' and 'in the shop' when you compare with a formula.
  • Simulating a handful of customers once and calling it a result.
  • Using a steady-state formula when your observation never reached a steady state.
  • Recording people, not counts: no names, no faces, no filming without permission.
  • Ignoring that service time depends on the items bought, the cashier or the time of day.

More Maths IA ideas like this

Make sure it's yours

This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.

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