Mathematics 11–12 · Year 12

How many people can leave per minute: network flow through measured doorways

Network flow (Mathematics Standard 2, Year 12)

Practical, model not builtLow risk

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The idea

The largest flow through a network of doorways and corridors is limited by its narrowest cut, so widening the wrong doorway changes nothing.

What you need

Tally counters and stopwatches for four or five observers
A floor plan of one building wing with its doorways and corridors
A spreadsheet

How to do it

At a normal class change, observers count people passing each doorway and corridor section minute by minute (counts only, no names or photographs) and note the minutes in which a queue forms behind it.
Draw a directed network from the classrooms (source) to the playground (sink). A doorway that queued was passing as many people as it could, so its count in a queued minute is its capacity; a doorway that never queued passed everyone who arrived, so its busiest-minute count is only a lower bound on its capacity.
List every cut that separates source from sink and compute its capacity; the smallest is the maximum flow.
Find a flow that achieves that maximum and mark the saturated edges.
Increase the capacity of one edge in the minimum cut and one edge outside it, and recompute the maximum flow each time.

What you should see

Worked check on the model's test network (people per minute: S to A 60, S to B 40, A to B 15, A to T 45, B to T 50): the cuts have capacities 100, 100, 110 and 95, so the maximum flow is 95 per minute, reached with 45 through A to T and 50 through B to T. Raising A to T to 60 lifts the maximum to only 100, because the cut at the source then becomes the smallest; raising S to A instead keeps it at 95. In the learner's measured network the maximum flow is only a lower bound for the building unless every edge in the minimum cut queued, and the busiest minutes at different doorways need not balance at each junction, so a bottleneck is confirmed only by a cut made of queued edges. The learner knows it worked when the flow found equals the smallest cut capacity and no edge carries more than its capacity.

What changes

What you change: the capacity given to each doorway or corridor edge, and which edge is then raised
What you measure: the maximum flow from the classrooms to the playground, in people per minute
What you keep the same: all counts taken during the same class change
one observer and one counter at each doorway for the whole period
counts only, with no names and no photographs

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

The maximum flow is the sum of the capacities leaving the source; a narrower cut further along limits it.
Widening any doorway helps; only edges in the minimum cut raise the maximum flow.

Safety card

Low riskLearners carry it out

Hazards

observers standing in busy corridors

Controls

observers stand to the side of doorways and never stand in anyone's way
count during an ordinary class change, not during an emergency drill

Note

No chemicals and no heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.

Curriculum references

The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.

Mathematics Standard 11–12 Syllabus (2024), Year 12 Standard 2 focus area Network flow; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MST-12-S2-06
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

All Concept Studio activities