How Mathematics Improves The World | Designing an Evacuation Before the Emergency
The worst time to discover that a staircase is too narrow is when everyone needs it.
The worst time to discover that two escape routes merge into one bottleneck is when smoke is already moving.
The worst time to discover that people do not begin evacuating the instant an alarm sounds is during the emergency the plan assumed they would.
So fire safety and emergency planning try to learn earlier.
Before the building is occupied.
Before the event is crowded.
Before the emergency becomes real.
That is where Mathematics enters.
Not to promise that every person will behave predictably.
Not to replace fire engineers, emergency services, building codes or human judgement.
But to ask disciplined questions about time, capacity, movement, uncertainty and behaviour while there is still time to redesign the system.
Quick Read
Evacuation analysis asks whether occupants can reach a sufficiently safe location before conditions become untenable. In performance-based fire safety, this is often expressed through two quantities: Available Safe Egress Time (ASET), the time before hazardous conditions exceed defined tenability limits, and Required Safe Egress Time (RSET), the time occupants need to perceive cues, decide to act and travel to safety. A safety margin requires ASET to exceed RSET under the design scenarios being assessed.
Mathematics helps estimate travel times, stair and door capacities, queue formation, density, flow, route interaction and uncertainty. Computational egress models can represent individuals moving through buildings and can test alternative layouts or procedures. But movement is only part of evacuation. NIST research has repeatedly emphasised that human behaviour—recognition, interpretation, information seeking, helping others, route familiarity, social influence and delayed initiation—materially affects outcomes and is often more difficult to model than walking speed.
A good evacuation model is therefore not a video-game crowd animation that tells us the future. It is a structured hypothesis about how people, geometry, hazards and procedures may interact. Its assumptions need evidence, sensitivity testing and professional interpretation.
One-sentence answer: Mathematics improves the world by letting designers test whether escape routes, capacities, timings and emergency procedures remain workable under plausible human behaviour before a real emergency forces the building to reveal its weaknesses.
The Evacuation Problem Is Really Two Clocks
Imagine a fire starts.
One clock begins counting how long the environment remains tenable.
Smoke spreads.
Heat rises.
Visibility falls.
Toxic gases accumulate.
That is the available-time side.
A second clock measures how long people need.
Detect the cue.
Understand it.
Decide what to do.
Begin moving.
Travel through the building.
Wait at bottlenecks.
Reach a safer location.
That is the required-time side.
Safety exists in the margin between the clocks.
ASET: How Long Does the Environment Remain Survivable?
Available Safe Egress Time is estimated from fire and smoke modelling together with tenability criteria.
The model may track:
- smoke-layer height;
- temperature;
- visibility;
- carbon monoxide or other toxic species;
- radiant heat;
- sprinkler activation;
- smoke-control systems;
- fire growth.
Different occupants may tolerate conditions differently.
A tenability criterion appropriate for one building or population may not be appropriate for another.
ASET therefore depends on scenario definitions and safety standards.
It is not one universal clock attached to every fire.
RSET: People Need More Than Walking Time
Required Safe Egress Time is often decomposed into stages.
- detection and alarm time;
- pre-movement or pre-evacuation time;
- movement time;
- queueing and interaction delays;
- sometimes an explicit safety factor or margin depending on the methodology.
The dangerous simplification is to calculate only distance divided by walking speed.
A person 40 metres from an exit walking at 1.2 metres per second does not necessarily leave in 33 seconds.
They may spend a minute deciding whether the alarm is real.
They may collect a child.
They may look for colleagues.
They may encounter a queue.
They may choose a familiar route that is longer.
Human time begins before feet move.
Pre-Movement Time: The Missing Minute That Can Matter More Than the Staircase
Many evacuation models historically treated pre-movement time as a simple user-entered delay.
NIST research has highlighted why that can be inadequate.
People interpret cues.
An alarm may be ambiguous.
Someone smells smoke.
A colleague says there is a problem.
A public-address message provides information.
People seek confirmation.
They observe what others are doing.
They may undertake protective or preparatory actions before moving toward an exit.
The delay is not simply laziness.
It is decision-making under uncertain information.
The Myth of Instant Panic
Popular films often portray emergencies as immediate mass panic.
Real behavioural research is more complicated.
People frequently seek information.
Help others.
Follow familiar social groups.
Delay because cues are ambiguous.
Use familiar routes rather than geometrically shortest exits.
NIST’s work on human behaviour in fire argues that evacuation models need behavioural theory rather than simplistic assumptions about occupants as identical particles.
This is a crucial mathematical lesson.
The easiest model of a human is often the least human model.
Movement: Distance Is Only the Beginning
Once people begin moving, geometry matters.
How far to the exit?
How wide is the corridor?
How many doors?
How wide are the stairs?
Where do routes merge?
Does one exit attract far more occupants than another?
Is an exit visible?
Are signs clear?
Does smoke alter route choice?
Movement becomes a network-flow problem with human decisions inside it.
A Door Has Capacity
If one person can pass through a door every second, one hundred people need at least one hundred seconds even if all are standing directly in front of it.
Real flow depends on width, crowd density, behaviour and the geometry around the doorway.
The central idea is simple:
a bottleneck converts a movement problem into a service-rate problem.
Arrival rate exceeds exit capacity.
A queue forms.
The same Mathematics appears at traffic lights, airport security and computer servers.
Different stakes.
Same queueing logic.
Wider Is Not Always Proportionally Faster
Double a doorway width.
Does evacuation time halve?
Not necessarily.
The doorway may not be the controlling bottleneck.
The staircase downstream may have lower capacity.
The corridor feeding the door may be narrow.
Route choice may leave the wider door underused.
Two streams may conflict after the door.
This is another network principle:
improving one component does not improve the whole system if another component becomes the bottleneck.
Density Changes Walking Speed
In an empty corridor, people walk at their preferred speeds.
As density rises, interactions increase.
People cannot choose arbitrary paths.
They shorten steps.
Speeds fall.
At extreme density, flow can become unstable or dangerously constrained.
Pedestrian-flow theory uses relationships among speed, density and flow similar in spirit to traffic-flow diagrams.
The maximum number of people passing per metre of width does not occur at zero density or infinite density.
It occurs in between.
The Shortest Exit May Be the Slowest Exit
Suppose Exit A is 20 metres away and Exit B is 40 metres away.
Everyone chooses A.
A queue forms.
B remains almost empty.
The longer walking route may become the faster evacuation route.
This is why route choice and load balancing matter.
Emergency signage, staff instructions and building familiarity influence whether occupants distribute themselves efficiently across exits.
Mathematics can identify the potential benefit of using alternative routes.
Human factors determine whether people actually use them.
Familiarity: People Often Leave the Way They Came In
An emergency exit can satisfy code and remain psychologically invisible.
People know the main entrance.
They used it ten minutes ago.
They understand where it leads.
A green EXIT sign points to a door they have never opened.
Which route feels safer?
Not necessarily the mathematically shortest one.
This is why evacuation modelling cannot treat route choice as a pure shortest-path algorithm.
Shortest Path Is a Graph Problem, But Human Path Choice Is More
Model the building as a graph.
Rooms and junctions become nodes.
Corridors and stairs become edges.
Dijkstra’s algorithm can find a shortest route.
Useful.
But people may value:
- familiarity;
- visibility;
- social following;
- perceived smoke;
- staff instructions;
- mobility constraints;
- route congestion.
The graph can include these costs.
But only if the model has evidence for how people weigh them.
People Move in Groups
A family does not always evacuate as five independent optimisation agents.
Parents look for children.
Friends wait for one another.
Workers help colleagues.
People follow a person who seems knowledgeable.
Social groups can slow individual travel and improve collective safety at the same time.
A model that rewards only fastest individual exit may misrepresent what humans actually value.
Again, optimisation needs a correct objective before it can be meaningful.
Mobility Is a Distribution, Not One Walking Speed
People vary.
Age.
Mobility.
Health.
Vision.
Hearing.
Familiarity.
Luggage.
Footwear.
Stress.
A credible model uses distributions or scenario-specific populations rather than one “average human”.
Designing only for the average can fail people in the tails.
Safety systems should be especially interested in the tails.
Stairs: Capacity Changes With Direction, Density and Geometry
Stair movement differs from level walking.
Speeds change.
People use handrails.
Landings create interactions.
Flows from multiple floors merge.
Firefighters may be moving in the opposite direction.
A high-rise evacuation is therefore not simply forty identical floor evacuations stacked vertically.
Lower floors receive people from above.
Stair density can increase as streams merge.
The building behaves as one vertical network.
Phased Evacuation: Sometimes Everyone Leaving at Once Is Worse
In very tall buildings, simultaneous evacuation of every floor can overload stairs.
Some emergency strategies therefore use phased or staged evacuation, depending on building design, fire strategy and local regulations.
People closer to the hazard may move first.
Others wait temporarily under controlled conditions.
This is an allocation problem:
who should enter the scarce stair capacity at what time?
The policy must be supported by fire-resistance, communication systems and emergency procedures.
Mathematics can test queue consequences.
Fire-safety engineering determines whether the strategy is acceptable.
Evacuation Lifts: A Resource That Must Be Designed Into the Strategy
Ordinary advice often says not to use lifts during a fire.
That remains appropriate for ordinary lifts unless the building’s emergency procedures state otherwise.
However, some modern buildings include specially designed occupant-evacuation elevators or firefighting lifts as part of a regulated fire-safety strategy.
Modelling them introduces a scheduling problem.
Which floor should the lift serve next?
How many people can it carry?
How long do doors remain open?
How does the lift interact with stair evacuation?
The system must be designed and approved as a whole.
Fire and Evacuation Models Have to Talk to Each Other
A crowd model cannot know which corridor is safe unless it receives hazard information.
A fire model can predict smoke but does not know where people choose to go.
Coupled modelling links the two.
Smoke reduces visibility.
Lower visibility slows movement.
A route becomes untenable.
Occupants reroute.
That changes crowd density elsewhere.
The combined system has feedback.
This is why high-resolution safety modelling becomes computationally expensive.
We are not simulating one process.
We are simulating interacting processes.
Agent-Based Models: Give Each Person a State
Many evacuation simulators use agent-based methods.
Each occupant has attributes.
Position.
Speed.
Destination.
Route preference.
Pre-movement delay.
Perhaps group membership or mobility characteristics.
The simulation advances time step by step.
Agents choose movements under behavioural rules.
Emergent queues appear from local interactions.
The advantage is detail.
The danger is false realism.
A crowd animation looks human even when the behavioural rules underneath are simplistic.
Cellular Automata: Humans on a Grid
Some models divide space into cells.
An occupant moves from one cell to a neighbouring cell according to rules.
This creates a discrete spatial model.
It can be computationally efficient.
But cell size constrains movement resolution.
Diagonal movement requires conventions.
Fine-grained geometry can be difficult.
Once again, the representation shapes the behaviour the model can express.
Social Force Models: Movement as Virtual Forces
Another family models pedestrians through force-like interactions.
An individual is attracted toward a destination.
Repelled from walls.
Repelled from other people when personal space is compressed.
The resulting equations produce continuous trajectories.
The metaphor is powerful.
But humans are not molecules.
The “forces” are modelling devices, not literal physical forces of intention.
Use the analogy to model spatial interaction.
Do not let it erase cognition.
NIST’s Warning: Movement Models Can Be Better Than Behaviour Models
NIST researcher Erica Kuligowski and colleagues have repeatedly made a crucial point.
Evacuation models often represent physical movement more convincingly than the human decision processes that precede and shape movement.
Walking speeds can be measured.
Door widths can be measured.
Stair geometry can be measured.
Human interpretation is harder.
Will the alarm be believed?
Will a person investigate?
Will they follow colleagues?
Will they stop to help someone?
Will staff instructions override route familiarity?
These variables matter enormously and are difficult to reduce to universal constants.
Defaults Are Dangerous Because They Look Professional
Software needs default values to run easily.
Walking speed: default.
Pre-movement delay: default.
Route choice: default.
Flow coefficient: default.
NIST research has warned that convenient defaults can be optimistic or unrealistic if users do not understand what populations and situations they represent.
A default is not evidence.
It is a placeholder until evidence justifies it.
Sensitivity Analysis: Which Assumption Controls the Answer?
Suppose a model predicts total evacuation in 11 minutes.
Change average walking speed by 10%.
Maybe the result becomes 10.5 minutes.
Change pre-movement time by two minutes.
Maybe total RSET becomes 13 minutes.
Now you know where the model is sensitive.
Sensitivity analysis varies assumptions deliberately to see which ones materially change outcomes.
This is one of the best protections against false precision.
If the safety conclusion flips when one uncertain assumption changes slightly, the conclusion is fragile.
Monte Carlo Simulation: One Evacuation Is Not a Distribution
People have different walking speeds.
Different pre-movement times.
Different route choices.
If a model contains random variables, running it once gives one possible evacuation.
Run it hundreds or thousands of times with values sampled from justified distributions.
Now we get a distribution of outcomes.
Median evacuation time.
95th percentile.
Worst observed scenarios.
Probability a route becomes congested.
Safety becomes a probability question rather than a single animation.
Validation: Does the Model Reproduce Real Evacuations?
Evacuation models should be compared with experiments, drills and real incidents where data exists.
Can the model reproduce measured door flows?
Stair descent speeds?
Queue formation?
Pre-movement distributions?
Route choices?
NIST’s analysis of the World Trade Center evacuation used survivor interviews and other evidence to study evacuation initiation delays and stair travel.
Real incidents are painful data sources.
The ethical responsibility is to learn from them carefully so future designs improve.
Evacuation Drills Are Experiments—But Not Perfect Ones
Drills provide valuable data.
How long did notification take?
Which routes were used?
Where did queues form?
Did staff instructions work?
But participants know it is a drill.
There is no smoke.
No real fear.
People may behave more compliantly.
Or less urgently.
Drill data therefore needs interpretation.
A drill is evidence about one condition, not a perfect reproduction of an emergency.
Communication Can Be More Valuable Than Another Metre of Staircase
Suppose a building’s geometry is adequate but occupants wait five minutes because the alarm is ambiguous.
Widening a staircase may save seconds.
Clearer emergency communication may save minutes.
This is why human behaviour is not a decorative addition to egress engineering.
Emergency messages can influence:
- whether people believe the threat;
- how quickly they begin moving;
- which route they choose;
- whether they wait for more information.
Sometimes the best evacuation improvement is informational rather than architectural.
Wayfinding: A Door Is Useless If People Cannot Find It
Exit capacity exists only if occupants use the exit.
Wayfinding design uses:
- sign placement;
- lighting;
- visibility;
- spatial layout;
- colour;
- symbols;
- staff guidance;
- familiarity.
Mathematical route capacity and human route discovery must meet.
A perfectly sized hidden exit contributes less than a slightly smaller exit people reliably recognise and use.
Crowd Pressure: Density Becomes Safety-Critical
At high densities, crowd interaction changes qualitatively.
Individuals lose freedom of movement.
Forces can propagate through bodies.
A small disturbance can create collective motion.
These conditions are dangerous and require specialist crowd-safety analysis.
Mathematics can estimate density and flow.
But ethical design tries to avoid regimes where individual control is lost rather than merely simulating them accurately.
The Fastest Evacuation Is Not Always the Safest Evacuation
Suppose every occupant is directed to run.
Individual travel speeds rise.
But falls may increase.
Stair interactions become more dangerous.
People with lower mobility are pressured.
Doorway flow may not increase proportionally.
The objective is not simply minimum mean evacuation time.
The objective is safe egress for the population under realistic conditions.
This is why optimisation cannot be separated from safety constraints.
Accessibility: A Safety System Is Only as Good as Its Least Served Occupant
Evacuation planning must include people with disabilities, mobility limitations, sensory impairments and other needs.
A staircase-only model can make an entire population mathematically disappear.
Accessible evacuation strategies may use areas of refuge, evacuation chairs, protected lifts, staff assistance or other building-specific measures under applicable codes and procedures.
The mathematical question is not:
How quickly can the average occupant leave?
It is:
Does the safety strategy work for the people who are actually expected to be here?
Evacuation Is a Network Reliability Problem
An exit route can fail.
Smoke blocks a corridor.
A door jams.
Debris obstructs a stair.
People choose the wrong path.
Good design includes alternative routes and redundancy where required.
Mathematics can test route loss:
remove Exit A.
Where do people go?
Do other exits overload?
Does RSET still remain below ASET?
Safety improves when the system can survive one route becoming unavailable.
A Classroom Thought Experiment: The Two Doors
Put thirty counters on a paper floor plan.
Door A is close but lets one counter leave per turn.
Door B is farther away but lets two leave per turn.
Ask students to evacuate all counters as quickly as possible.
Most initially send everyone to A.
The queue grows.
Then they discover load balancing.
Now remove Door B halfway through.
They discover contingency.
Now make five counters move every second turn.
They discover population diversity.
The Mathematics becomes human without becoming frightening.
Secondary Mathematics: Egress Becomes Rates, Graphs and Probability
Secondary students can explore:
- distance ÷ speed;
- door flow rates;
- queue growth;
- network shortest paths;
- probability distributions of delay;
- graphs of cumulative evacuation;
- simple Monte Carlo simulation.
Plot the number evacuated against time.
The slope is evacuation flow.
A flat section reveals a bottleneck or delay.
A changed door width changes the curve.
A changed pre-movement distribution shifts the whole curve.
Students can see systems thinking emerge from familiar Mathematics.
Advanced Mathematics: Safety as a Coupled Dynamic System
Advanced evacuation modelling draws on:
- queueing theory;
- graph theory;
- probability;
- stochastic processes;
- agent-based modelling;
- cellular automata;
- operations research;
- partial differential equations;
- optimisation;
- behavioural science;
- fire dynamics.
Fire models predict hazard growth.
Behaviour models predict decision stages.
Movement models predict crowd dynamics.
Monte Carlo methods propagate uncertainty.
Safety engineering combines the pieces into scenarios that can be checked against acceptance criteria.
Why This Improves the World
1. It discovers bottlenecks before emergencies
Door, stair and corridor capacities can be tested during design instead of after an incident.
2. It makes time margins explicit
ASET/RSET analysis forces designers to compare available safe time with the time people actually need.
3. It reveals the cost of delayed response
Behavioural models show that clearer warnings and better procedures can matter as much as physical movement speed.
4. It lets alternatives be compared cheaply
Designers can test different stair widths, exit placements, signage and procedures before construction.
5. It makes uncertainty visible
Sensitivity and probabilistic analysis expose fragile safety conclusions that depend too heavily on one uncertain assumption.
6. It forces the model to include the actual population
Good modelling asks whether the strategy works for diverse occupants rather than only an idealised average person.
What Mathematics Does Not Do
An evacuation simulation does not predict exactly what every person will do.
It does not replace fire codes or professional fire-safety engineering.
It does not make a building safe merely because a model completed successfully.
It does not justify unrealistic behavioural defaults.
It does not remove uncertainty.
It does not replace drills, inspections, maintenance or clear emergency communication.
And during a real emergency, occupants should follow the building’s emergency procedures and instructions from authorised personnel—not an educational model or article.
Frequently Asked Questions
What is ASET?
Available Safe Egress Time is the estimated time from the start of a design fire scenario until conditions along the relevant escape route become untenable according to defined criteria.
What is RSET?
Required Safe Egress Time is the estimated time occupants need to detect and interpret the emergency, begin protective action and travel to a safe location, including delays caused by queues and interactions.
Why do people not evacuate immediately?
People may need to interpret ambiguous cues, seek information, confirm the threat, gather others, help someone or decide which route to use. Behavioural research shows that pre-movement decision processes can be a major part of evacuation time.
Can a wider exit always solve evacuation problems?
No. Another bottleneck may control the system, occupants may not use the exit, or widening one component may shift congestion elsewhere. The whole route network must be analysed.
Are evacuation models reliable?
They can be valuable when appropriate models, evidence-based inputs, verification, validation and sensitivity analysis are used. Their results should be interpreted as scenario-based estimates, not guaranteed predictions of individual behaviour.
Sources and Further Reading
- NIST, Guidance for the Model User on Representing Human Behavior in Egress Models.
- NIST, The Need for Behavioral Theory in Evacuation Modeling.
- NIST, A Multi-disciplinary Perspective on Representing Human Behavior in Evacuation Models.
- NIST, Modeling the Evacuation of the World Trade Center Towers on September 11, 2001.
- NIST, Bounding Defaults in Egress Models.
Continue Through eduKateSG
Continue with How Mathematics Works. Batch 003 also includes Making Traffic Lights Cooperate Instead of Compete, Finding Useful Pages in a Web Too Large to Read, and Letting Many Wireless Systems Share the Same Air.
Final Thought: The Model Exists So the Emergency Does Not Have to Teach Us First
Every evacuation model is a rehearsal made from Mathematics.
The staircase fills.
The queue forms.
The exit is blocked.
The alarm is delayed.
A person walks slowly.
A group chooses the familiar door.
The design fails.
Then we reset.
Move the exit.
Change the procedure.
Improve the communication.
Test again.
That reset button is one of Mathematics’ quiet gifts to safety.
We can make a design fail a thousand times in a model so the real building has a better chance of never needing to.