How Mathematics Improves The World | Making Elevators Decide Who to Pick Up Next
Press an elevator button in a quiet building and the problem looks trivial.
There is one passenger.
One call.
One elevator nearby.
Send it.
Now make the building sixty storeys high.
Add six elevators.
At 8:45 a.m., fifty people enter the lobby.
Three people call from Floor 12.
Someone on Floor 37 wants to go down.
One car is nearly full.
Another is travelling upward but will pass the caller’s floor.
A third is closer in distance but moving the wrong way.
A fourth has fewer passengers but more stops.
Which elevator should answer the next call?
“The nearest one” is suddenly a poor rule.
Elevator dispatching is a dynamic scheduling problem where every choice changes the future state of the whole group.
Mathematics is what makes the group behave like one transportation system instead of six independent boxes moving up and down.
Quick Read
Elevator group control decides which car should serve each passenger request. The objective can include average waiting time, total journey time, long-wait risk, energy use, capacity utilisation and fairness. Modern tall buildings often use several elevators coordinated by a central group controller because one car cannot serve peak passenger demand efficiently.
The mathematical problem is difficult because passenger arrivals are random, cars are moving, destination information may be incomplete, each car has finite capacity and every assignment affects later calls. Research in elevator group control treats dispatching as a constrained multi-objective optimisation problem. Recent work also studies destination control, occupancy-aware dispatching and reinforcement learning.
Traditional hall-call systems tell the controller only “up” or “down”. Destination-dispatch systems ask passengers to enter their destination before boarding. That extra information allows people travelling to similar floors to be grouped, reducing unnecessary stops and improving handling capacity under appropriate traffic conditions. The system becomes better because it knows more about future demand before committing a car.
Efficiency is not the only objective. Research has shown that passengers needing more space—such as wheelchair users or people with strollers or bulky luggage—can experience systematically longer waits when the controller optimises only average performance. Fairness can therefore be added explicitly to the dispatch objective.
One-sentence answer: Mathematics improves the world by turning elevator groups into coordinated real-time transportation systems that estimate future waiting, assign cars to calls, balance capacity and travel time, and continuously revise decisions as new passengers arrive.
The Elevator Is a Queueing System
Passengers arrive.
They wait.
A car arrives.
They board.
The car serves destinations.
This is a queueing system.
But unlike a supermarket checkout, the server moves physically between customers.
Service time depends on where the next passenger is.
And serving one passenger changes the route for everyone already inside.
Queueing and routing are coupled.
Waiting Time and Journey Time Are Different
A passenger experiences at least two delays.
Waiting time: from registering the call until the car arrives.
Travel time: from boarding until reaching the destination.
Optimise only waiting time and a controller might send the nearest car even if it then makes eight stops.
Optimise only in-car travel and the controller may hold a passenger waiting for a more direct car.
Many elevator studies therefore use total journey time or multiple objectives.
The metric defines what “good service” means.
The Nearest-Car Rule Fails Quickly
Car A is two floors away, travelling upward, almost full and already assigned four stops.
Car B is six floors away, empty and travelling toward the caller.
Which arrives first?
Distance alone cannot answer.
The controller needs predicted arrival time, not geometric closeness.
That prediction includes:
- current position;
- direction;
- speed profile;
- door time;
- assigned stops;
- expected boarding;
- capacity.
One simple distance becomes a small simulation of the car’s future route.
Door Time Matters More Than It Looks
An elevator spends time accelerating and travelling.
It also spends time stopped.
Doors decelerate the system.
Open.
Passengers leave.
Passengers enter.
Doors close.
A trip with ten stops can take far longer than a trip with two even if vertical distance is similar.
Destination grouping is valuable partly because it removes stops.
Stop Cost Is Nonlinear
Add one extra stop.
Everyone already inside is delayed.
The car’s next pickup is delayed.
That passenger’s destination service is delayed.
The car arrives later back at the lobby.
Future passengers then wait longer.
One stop propagates through the schedule.
The marginal cost of a stop depends on who is already in the car and what traffic arrives later.
Passenger Arrival Is Random
At 10:13:00, no one is waiting.
At 10:13:05, six people arrive.
Elevator demand is stochastic.
Poisson arrival models are often used as a first approximation, especially when arrivals are independent.
Real buildings violate the assumption.
A train arrives and releases a group.
A meeting ends.
School dismissal creates a burst.
Passenger arrival distribution depends on the building’s social timetable.
Traffic Patterns Change Through the Day
Elevator engineers traditionally describe several dominant patterns.
Up-peak
Most passengers enter at the lobby and travel upward, typical of office arrival.
Down-peak
Many passengers leave upper floors and travel to the lobby, typical at day’s end.
Two-way or interfloor traffic
Passengers move among many floors, common during lunch or mixed-use activity.
A control rule excellent for up-peak can be poor during interfloor traffic.
The system needs a traffic model, not one permanent trick.
Up-Peak Is a Batching Problem
During morning arrival, a lobby fills.
Suppose passengers want Floors 5, 6, 7, 18, 19, 20, 34 and 35.
Randomly load cars and each may stop across the whole building.
Group similar destinations and one car can serve low floors, another mid floors, another high floors.
The dispatch problem becomes clustering.
Which passengers should share a car so total stopping and waiting cost is low?
Destination Dispatch: Ask Where the Passenger Is Going Before They Board
Traditional hall buttons provide only direction.
Up.
Down.
The destination is revealed after boarding.
Destination-dispatch systems collect the floor first.
The controller can then assign passengers to cars based on destination clusters.
This is an information advantage.
Better prediction is possible because future stops are known earlier.
Information Can Create Capacity
No new elevator shaft is built.
No car becomes faster.
Yet destination control can increase effective handling capacity by reducing unnecessary stops and grouping trips.
This is a recurring theme in infrastructure.
Better information can create useful capacity from the same physical hardware.
ADS-B creates airspace capacity.
Grid state estimation creates operational confidence.
Destination information creates elevator capacity.
Car Assignment: A Cost Function for Every Possible Elevator
A hall call appears on Floor 22.
For each car, estimate a cost.
One simple cost might combine:
- predicted pickup delay;
- additional delay to existing passengers;
- energy cost;
- risk of capacity failure;
- future positioning cost.
Then assign the lowest-cost car.
The difficulty is that the true future cost depends on calls that have not arrived yet.
Dispatch is an online optimisation problem.
Greedy Control: Best Now Can Be Bad Later
A greedy dispatcher minimises immediate cost.
Send the car that reaches the caller fastest.
That may leave all cars clustered on high floors just before a lobby surge.
A slightly worse decision now can position the fleet better for likely future demand.
Optimal control therefore values future state.
The controller is not only serving passengers.
It is continuously arranging where the elevators will be when the next passenger appears.
Parking Strategy: Empty Elevators Still Need Decisions
No calls are active.
Where should empty cars wait?
All at the lobby?
One every ten floors?
Near historically busy floors?
Parking strategy anticipates demand.
During up-peak, lobby parking is valuable.
During mixed traffic, distributing idle cars can reduce response time.
Idle time becomes preparation.
Zoning: Divide the Building Into Vertical Territories
One method for reducing interference is zoning.
Cars are assigned primarily to ranges of floors.
Low zone.
Mid zone.
High zone.
This reduces travel and stop count.
But rigid zoning wastes capacity when one zone is quiet and another overloaded.
Dynamic zoning changes boundaries as traffic changes.
The building becomes a partitioning problem.
Sky Lobbies: Change the Network Topology
Very tall buildings sometimes use express elevators to sky lobbies.
Passengers transfer to local elevators serving upper zones.
This resembles public transport.
Express train to a hub.
Local service from there.
The design reduces the number of shafts required to serve every floor directly.
Architecture becomes network topology.
Double-Deck Elevators: Two Floors at Once
Some skyscrapers use double-deck elevator cars stacked vertically.
One deck may serve odd floors and the other even floors, or paired lobby levels.
This increases passenger capacity per shaft.
It also complicates dispatch because two passenger groups share one car movement.
A stop that serves one deck may be unnecessary for the other.
Mechanical coupling creates scheduling coupling.
Capacity: The Car Can Arrive and Still Fail to Pick You Up
The elevator arrives.
Doors open.
It is full.
Your waiting time continues.
Traditional dispatch models can underestimate this if they model cars without realistic occupancy.
Occupancy-aware systems use load sensors, cameras or inferred passenger counts to estimate remaining capacity.
Research has shown that including occupancy can reduce failed pickups and improve dispatch quality.
A vehicle’s location is insufficient.
Its available capacity is part of the state.
Fairness: Average Waiting Time Can Hide Very Long Waits
Suppose nine passengers wait 10 seconds and one waits 200.
Average wait:
(9×10 + 200)/10 = 29 seconds
Twenty-nine sounds acceptable.
One person experienced a terrible service failure.
Modern service design therefore looks at percentiles and tail waiting:
- 95th-percentile wait;
- maximum wait;
- fraction waiting over 60 seconds.
Averages compress inequality.
Fairness needs distributional metrics.
Accessibility: Space Requirement Can Create Structural Unfairness
A wheelchair user may need more free floor area than a standing passenger.
So can a parent with a stroller or someone moving bulky luggage.
If cars repeatedly arrive nearly full, these passengers may be unable to board even while others can squeeze in.
Research on fair elevator dispatch has explicitly modelled such passengers and shown that occupancy information and fairness-aware assignment can reduce unequal waits.
The optimisation objective matters.
Minimum mean delay does not automatically imply equal access.
Energy: Empty Vertical Travel Costs Power
Elevators consume energy moving cars and counterweights, accelerating, operating doors and powering controls.
Regenerative drives can return some energy during favourable motion.
Dispatch strategy changes energy use.
Sending two cars when one could serve both calls wastes movement.
Parking every car at the lobby may improve waiting and increase empty repositioning.
Energy and service quality can conflict.
Elevator group control is therefore naturally multi-objective.
Regenerative Braking: Gravity Can Help Pay the Energy Bill
An elevator system is balanced by counterweights.
When a heavily loaded car descends or a lightly loaded car rises, gravity can drive the motor as a generator.
Regenerative drives feed energy back into the building electrical system rather than dissipating it as heat.
A sophisticated dispatch controller can consider which assignments create regenerative opportunities.
But passenger service generally remains the primary operational constraint.
Predictive Control: The Next Passenger Has Not Pressed Yet
Historical data says Floors 4–8 generate many down calls at 12:00.
At 11:58, should an idle car move there pre-emptively?
Predictive control says maybe.
Forecast demand distribution.
Evaluate future waiting cost.
Position cars accordingly.
The risk is false prediction.
Move a car unnecessarily and a real call appears elsewhere.
The controller must balance anticipation with flexibility.
Markov Decision Processes: The Elevator Group as a State Machine
A state can include:
- each car’s floor;
- direction;
- speed;
- occupancy;
- assigned stops;
- hall-call queues;
- time of day.
An action assigns a car or changes a policy.
A cost measures waiting, travel, energy or unfairness.
Passenger arrivals make the next state stochastic.
This is a Markov decision-process formulation.
In principle, dynamic programming can find optimal policies.
In practice, the state space becomes enormous.
The Curse of Dimensionality
Six cars.
Fifty floors.
Each car has many possible positions, directions, loads and stop lists.
Each floor can have up and down calls.
The number of possible states explodes combinatorially.
Exact optimal control becomes computationally impossible in real time.
Practical systems therefore use heuristics, simulation, approximate dynamic programming, metaheuristics and machine learning.
The mathematically perfect policy may be less useful than a robust near-optimal policy that decides in milliseconds.
Genetic Algorithms: Evolve Better Assignments
Research has used genetic algorithms for elevator dispatch.
Encode a candidate assignment as a chromosome.
Evaluate predicted waiting and travel.
Select better candidates.
Recombine and mutate.
Repeat.
Metaheuristics search difficult combinatorial spaces without guaranteeing the exact global optimum.
The value is speed and flexibility.
The limitation is that performance depends on fitness design and tuning.
Simulated Annealing: Occasionally Accept a Worse Move
Greedy search can get trapped in a local optimum.
Simulated annealing sometimes accepts a worse candidate early in the search.
This lets it escape local minima.
As the “temperature” falls, bad moves become less likely.
Elevator research compares such methods with genetic algorithms and rule-based heuristics.
The optimisation method is less important than the discipline:
define the cost correctly, test realistic traffic and compare against strong baselines.
Reinforcement Learning: Let the Controller Learn From Simulated Buildings
A reinforcement-learning agent observes the group state.
Assigns a car.
Receives reward based on passenger delay, energy or other objectives.
Across millions of simulated trips, it learns a policy.
Recent research studies traffic-pattern-aware deep reinforcement learning and safe reinforcement-learning formulations for elevator group control.
The promise is adaptation to complicated traffic.
The danger is simulation mismatch.
If the virtual passengers behave unrealistically, the learned controller becomes expert at the wrong building.
Simulation Is the Elevator Controller’s Wind Tunnel
You cannot experiment freely on a real office tower at 8:45 a.m.
Passengers will not appreciate being made to wait because researchers want data.
So control strategies are tested in simulation.
Create realistic traffic profiles.
Run thousands of days.
Measure average and tail waiting.
Compare energy use.
Stress the system with unusual demand.
A controller can fail cheaply in simulation before being deployed.
Simulation Needs Calibration
Passengers do not all walk at the same speed.
Boarding time depends on crowding.
Door holding behaviour varies.
People press buttons twice.
Some wait for friends.
Elevator simulation therefore needs field data.
Compare predicted waiting distributions with real building logs.
Compare car loading.
Compare traffic by time of day.
A simulated building that feels plausible is not necessarily statistically correct.
Mean Waiting Time Is Not Enough
A research controller lowers average wait from 28 seconds to 24.
Success?
Maybe.
What happened to:
- 95th percentile?
- worst 1%?
- wheelchair users?
- energy?
- crowding?
- failed pickups?
Recent elevator research increasingly studies tail service and resilience because mean-based metrics can hide rare but severe delays.
Quality is a distribution.
Resilience: What Happens When One Elevator Fails?
Six cars become five.
Capacity falls.
Waiting rises nonlinearly during peak demand.
A resilient controller changes zoning, parking and assignment priorities.
It may sacrifice average travel time to prevent extreme queues.
Failure changes the objective.
The system moves from efficiency mode to degraded-service mode.
Fire Mode: Normal Optimisation Stops
During fire emergencies, ordinary passenger dispatch is overridden by life-safety procedures and applicable codes.
Elevators may return to designated floors, become unavailable to ordinary passengers or operate under firefighter control depending on system design.
Specially designed occupant-evacuation elevators can form part of approved high-rise evacuation strategies.
The key boundary is clear:
an optimisation algorithm does not improvise fire-safety policy.
Safety modes sit above ordinary efficiency objectives.
Security Access Changes the Graph
In secure buildings, not every passenger may access every floor.
Badge credentials constrain destination assignments.
Some elevators may serve restricted zones.
The controller optimises inside an access-control graph.
A physically reachable floor can be mathematically infeasible for one passenger.
Operational constraints are part of route planning.
Privacy: Destination Data Reveals Behaviour
Destination-dispatch systems know where passengers are going.
Badge integration may identify who they are.
Historical movement data can reveal work patterns.
That data can improve forecasts.
It also creates privacy risk.
Good system design minimises stored identity when aggregate traffic is sufficient.
Optimisation does not require surveillance by default.
Digital Twins: A Virtual Elevator Group That Learns the Building
Modern building systems can stream:
- car positions;
- door cycles;
- motor current;
- passenger calls;
- fault codes;
- load estimates.
A digital twin uses this data to maintain a live model of the elevator group.
It can simulate new policies, predict component wear and detect unusual performance.
The useful twin is not the animated 3D elevator.
It is the calibrated state model that remains answerable to real logs.
Predictive Maintenance: A Slow Door Can Become a Queue Problem
A door motor weakens.
Opening time increases by one second.
One second sounds trivial.
Across thousands of stops per day, it reduces handling capacity.
Passenger queues rise.
Condition-monitoring data can detect drift before complete failure.
Maintenance becomes part of service optimisation.
Elevator Planning: How Many Shafts Does the Building Need?
Dispatch mathematics operates after a building exists.
Planning mathematics decides the system before construction.
How many cars?
What capacity?
What speed?
Which floors?
Sky lobby or direct service?
Building traffic simulation tests alternatives against target handling capacity and waiting criteria.
Recent research on vertical transportation uses simulation and optimisation to reduce shaft area while maintaining passenger service.
Mathematics can therefore create rentable floor area by improving vertical transport design.
Handling Capacity: How Many People Can the System Move in Five Minutes?
Elevator planning often uses handling capacity over a defined period, commonly a five-minute window under specified traffic assumptions.
One car’s round-trip time depends on:
- travel height;
- speed;
- acceleration;
- number of stops;
- door time;
- passenger transfer time.
Reduce expected stops through destination grouping and round-trip time falls.
Handling capacity rises.
The building can serve more people without simply making the motor faster.
A Classroom Thought Experiment: Three Elevators, One Call
Draw a 20-floor building.
Elevator A is at Floor 5 going up with stops at 8, 12 and 17.
Elevator B is at Floor 10 going down with one stop at 3.
Elevator C is at Floor 1 idle.
A passenger on Floor 9 presses up.
Which car should answer?
Students estimate travel and stop times.
Then add five passengers waiting in the lobby.
The best assignment may change.
The exercise teaches that decisions depend on future demand, not only current distance.
A Second Thought Experiment: Optimise the Average, Hurt One Passenger
Create ten passenger cards.
Give nine 10-second waits and one 200-second wait.
Average: 29 seconds.
Now compare with everyone waiting 35 seconds.
Average is worse.
Fairness is better.
Which policy should the building choose?
The answer is not contained in arithmetic alone.
The metric is a social choice.
Primary Mathematics: Elevators Begin With Time, Distance and Capacity
Primary students already know:
- floor numbers;
- distance;
- time;
- capacity;
- ordering;
- averages;
- simple graphs.
“Which elevator arrives first?” is a time problem.
“How many passengers fit?” is capacity.
“Who has waited longest?” is ordering.
One lift lobby contains a whole elementary mathematics lesson.
Secondary Mathematics: The Building Becomes a Queueing Network
Secondary students add:
- probability;
- expected value;
- functions;
- optimisation;
- graphs and networks;
- statistics;
- simulation.
Passenger arrivals become random variables.
Average waiting becomes an expectation.
Car assignment becomes an optimisation.
Tail waiting becomes a percentile.
Advanced Mathematics: Vertical Transportation as Stochastic Control
Elevator group control draws on:
- queueing theory;
- Markov decision processes;
- dynamic programming;
- combinatorial optimisation;
- stochastic processes;
- simulation;
- multi-objective optimisation;
- reinforcement learning;
- human factors.
The state changes continuously.
Future demand is uncertain.
Each decision changes later options.
That is the definition of sequential decision-making under uncertainty.
Why This Improves the World
1. It reduces waiting without adding shafts
Better group control extracts more service from existing elevator hardware.
2. It increases effective handling capacity
Destination grouping and zoning can reduce stops and improve round-trip times.
3. It makes tall buildings practical
Efficient vertical transportation reduces the floor area that must be devoted to elevator cores.
4. It can improve accessibility and fairness
Fairness-aware dispatch can reduce extreme waits for passengers who need extra boarding space.
5. It reduces energy waste
Coordinated car assignments reduce unnecessary empty travel and can make better use of regenerative operation.
6. It lets failure happen in simulation first
Controllers can be stress-tested against peak traffic, car outages and unusual demand before deployment in occupied buildings.
What Mathematics Does Not Do
Mathematics does not make a shaft larger.
It does not make a full car accept another passenger safely.
It does not decide whether average speed matters more than fairness.
It does not override fire-safety codes.
It does not guarantee a reinforcement-learning controller will generalise outside simulation.
It does not repair a failing door motor.
And no dispatch algorithm can compensate indefinitely for too few elevators serving too many people.
Frequently Asked Questions
How does an elevator decide which car to send?
Group-control systems estimate the cost of assigning each available car to a call using information such as car position, direction, existing stops, passenger demand, occupancy and predicted waiting. The exact algorithm varies by manufacturer and building.
Why is the nearest elevator not always sent?
A nearby car may be moving away, full or scheduled for many stops. A more distant car can sometimes reach the passenger sooner and create less delay for other passengers.
What is destination dispatch?
Destination dispatch asks passengers to enter their destination before boarding. The system assigns a particular car, allowing passengers with similar destinations to be grouped and reducing unnecessary stops.
What is elevator handling capacity?
Handling capacity describes how many passengers an elevator group can transport over a defined time under specified traffic assumptions. It depends on car capacity, speed, round-trip time, stops and door/boarding time.
Can AI control elevators?
Research uses machine learning and reinforcement learning for traffic prediction and dispatch. Real deployment requires rigorous safety constraints, validation, fallback logic and integration with certified elevator control systems.
Sources and Further Reading
- Gharbi, Exploring Heuristic and Optimization Approaches for Elevator Group Control Systems, Applied Sciences, 2024.
- Elevator Group Control as a Constrained Multiobjective Optimization Problem, Applied Soft Computing, 2022.
- Fair and Effective Elevator Car Dispatching Method in Elevator Group Control System Using Cameras, Procedia Computer Science, 2019, on efficiency and fairness for passengers requiring more car space.
- Current and Future Trends in Vertical Transportation, European Journal of Operational Research, reviewing mathematical methods, simulation and tall-building elevator planning.
- Energy-Efficient Resilience Scheduling for Elevator Group Control via Queueing-Based Planning and Safe Reinforcement Learning, 2026, on tail waiting and resilience objectives.
Continue Through eduKateSG
Continue with How Mathematics Works. Compare this article with Making Traffic Lights Cooperate Instead of Compete: both allocate movement capacity through time and both can fail when local optimisation damages the larger network. It also connects to Keeping Aircraft Safely Apart in a Crowded Sky, where scheduling and queueing again turn moving vehicles into a live control problem.
Final Thought: The Elevator Is Deciding About the Future Before the Doors Open
You press the button.
The system already knows six cars are moving.
One is almost full.
One is close but committed.
One is empty but farther away.
The lobby is beginning to fill.
Lunch traffic is likely in seven minutes.
A controller estimates several futures.
Assigns one car.
Then a new passenger arrives and the future changes again.
The ride feels ordinary because the decision happens invisibly.
That is one of Mathematics’ quiet achievements.
It makes a building move people vertically without making every passenger solve the scheduling problem themselves.