Elevator: Mechanics, Control & Optimization

Updated 3 July 2026

Elevators are vertical transportation systems that operate via classical mechanics, enabling smooth, measurable transitions between floors.
Queueing theory applied to elevators models stochastic passenger arrivals and system capacity to optimize waiting times and dispatch synchrony.
Modern elevator control integrates combinatorial optimization and machine learning to significantly reduce average waiting times and improve service efficiency.

An elevator is a vertical transportation system designed to efficiently move people or goods between floors in a building. Modern elevator technology spans from fundamental mechanics and control-theoretic queueing models to advanced optimization and machine learning-based scheduling algorithms. Elevators play a pivotal role in the architecture, accessibility, and operational efficiency of multistory and high-rise structures.

1. Physical Modeling and Elevator Dynamics

The physical motion of an elevator is fundamentally governed by classical mechanics, with observable quantities such as acceleration, velocity, and apparent weight being accessible through direct measurement or sensor instrumentation. When an elevator accelerates, the apparent weight of an object measured by a static scale changes according to $W' = m(g - a)$ , where $m$ is mass and $a$ is the elevator’s acceleration. Thus, the apparent mass $m'$ reads $m' = m(1 - \frac{a}{g})$ (Shi et al., 2024). Typical elevator trips consist of distinct kinematic phases: periods of variable acceleration, constant acceleration, uniform motion, variable and constant deceleration, and transitionary ramps. The exact number and extent of these intervals depend on the distance traveled and whether intermediate stops occur. For example, with short travel distances, constant-velocity and constant-acceleration phases may shrink or vanish (Shi et al., 2024).

Instrumentation using smartphone barometric sensors enables analysis of elevator altitude and vertical velocity profiles by relating atmospheric pressure $p(h)$ to height $h$ via the hydrostatic approximation. For $h \lesssim 100$  m,

$h \approx \frac{p_0 - p}{p_0} \frac{RT_0}{Mg},$

with $p_0$ and $m$ 0 reference pressure and temperature, $m$ 1 the gas constant, and $m$ 2 air molar mass. The barometric method achieves submeter precision and is superior to accelerometer integration and GPS in indoor scenarios, with measured elevator speeds for modern skyscrapers routinely reaching $m$ 3–6 m/s in high-rise applications (Monteiro et al., 2016).

2. Fundamental Queueing Theory and Minimalist Models

The operation of elevators in a building can be abstracted as a queueing system, with stochastic passenger arrivals and discrete-capacity transporters. In the start-of-day uppeak regime, the system is governed by a Poisson arrival process of rate $m$ 4 at the lobby and $m$ 5 identical cars of capacity $m$ 6 serving $m$ 7 floors. Each car cycles through: loading at the lobby, traveling to destination floors in ascending order, returning empty, and repeating. Cycle time for a departure with $m$ 8 passengers is

$m$ 9

with $a$ 0 the time per floor, $a$ 1 dwell time per passenger. The steady-state occupancy $a$ 2 for an infinite-capacity car satisfies $a$ 3; for finite capacity, system stability requires $a$ 4 (Feng et al., 2020). When $a$ 5, elevators undergo a synchrony transition analogous to bus bunching, with all cars clustering their arrivals and departures.

Key analytic results include:

The distribution of cycle times and individual waiting times, with waiting time density

$a$ 6

where $a$ 7 is the steady-state cycle time distribution.

The emergence of stepwise, plateaued waiting-time distributions and submodular effects in the lobby line clearance problem.
Transition-to-synchrony thresholds: for $a$ 8 elevators, synchrony onset occurs when $a$ 9.

These predictions are validated by event-driven simulations, demonstrating nearly Gaussian cycle-time distributions at low arrival rates, Fermi-Dirac-shaped waiting-time densities, and the appearance of $m'$ 0-fold synchrony and multi-modal car inter-arrival statistics near critical load (Feng et al., 2020).

3. Optimization and Control: Group Scheduling Algorithms

Group elevator scheduling is a combinatorial optimization problem. The goal is typically to minimize average waiting time (AWT) under system and matroid constraints: each passenger (hall-call) must be assigned to exactly one car, and the transport capacity must not be exceeded. One approach formulates the problem as a submodular function maximization under a matroid, modeling waiting times with a quadratic Boolean function

$m'$ 1

where $m'$ 2 are assignment indicators, $m'$ 3 the individual waiting times, and $m'$ 4 the pairwise interactions (positive if joint dispatch increases wait). The corresponding negative objective $m'$ 5 is monotone submodular and can be efficiently approximated to within $m'$ 6 of optimum by greedy selection (Ramalingam et al., 2017). Experimental evaluation shows empirical AWT reductions of 8–11% over industrial group-collective and ETA-control baselines.

Recent advances exploit machine learning and reinforcement learning (RL) paradigms. In one RL formulation, the group elevator control system is modeled as a Markov Decision Process (MDP) with state space $m'$ 7, action space $m'$ 8 of elevator-subset responders, transition kernel $m'$ 9, reward $m' = m(1 - \frac{a}{g})$ 0, and discount $m' = m(1 - \frac{a}{g})$ 1 (Vaartjes et al., 12 Jun 2025). Key innovations include:

Infra-step simulation to decouple controller actions from high-frequency events.
Combinatorial action encoding: for 6 elevators and maximum 3 responders, the action set has cardinality 41.
A dense, composite reward encompassing waiting, travel, boarding, and energy penalties.
Dueling Double DQN architecture with fixed decision-step discounting, outperforming traditional rule-based ETD controllers by 10% in average passenger travel time.

Deep Q-Learning has been explored for elevator optimization, though on single-elevator abstractions DQN offers no statistically significant improvement over naive rules. The critical bottleneck is the violation of the true MDP assumption due to exogenous, stochastic hall-call arrivals, which impedes analytical RL convergence (Cao et al., 2022).

4. Predictive and Learning-Based Control Architectures

The integration of sequence modeling for anticipatory elevator group dispatch has been advanced through predictive architectures. The Predictive Group Elevator Scheduler (PGES) incorporates a Transformer-based destination predictor, trained on partial passenger trajectories, yielding destination probability vectors, most notably the Predicted Probability of Going to Elevator (PPGE). If a person’s PPGE crosses a calibrated threshold ( $m' = m(1 - \frac{a}{g})$ 2), a linear regression model is used to predict arrival time at the elevator, supplying a "certainty-equivalent" future arrival to the scheduler.

At runtime, the PGES simulates alternative car assignments including imminent future arrivals, then chooses the assignment minimizing expected AWT. Empirically, this strategy yields up to 50% AWT reduction in light down-peak traffic and ~15% for medium loads, far outperforming myopic or nearest-car baselines. Key design elements are the transformation of position sequences into grid indices for the Transformer, linear regression for real-time time-to-arrival estimation, and efficient single-continuation simulation to maintain $m' = m(1 - \frac{a}{g})$ 3 per-decision complexity. The approach is robust to false-positive arrivals and generalizes to large-scale installations with only moderate sensing infrastructure (Zhang et al., 2022).

5. Human–Robot Interaction and Multimodal Elevator Use

Elevators are not only infrastructure for human passengers but also critical environments for service robots and mobile agents. Robots operating in elevators require advanced navigation methods to safely and efficiently enter occupied cabins, often crowded with people. A hybrid method blends RL for navigation with “clear-path” voice prompts in an MDP framework, augmenting geometric action sets with a discrete beep (“clear path”) flag. The reward function balances time efficiency, collision avoidance, and beep cost. A transformer-augmented value network achieves significant gains in success rate (+15–20 pp over baselines) in 8-person elevator crowds, maintaining collision rates below 2% and making human-acceptable, minimal use of voice prompts. Real-world deployment on platforms such as Turtlebot demonstrates the practical viability of RL-voice hybrid policies for elevator entry in dynamic, non-cooperative settings (Ma et al., 2022).

Autonomous robotic mapping and navigation further depend on robust multi-floor localization through elevator rides. The Elevator-LIO framework introduces a decoupled state-estimation model combining standard LiDAR-inertial odometry (LIO) with explicitly modeled elevator vertical displacement, velocity, and acceleration. An event-triggered zero-velocity and zero-acceleration update, along with adaptive voxel downsampling, yields subcentimeter terminal height accuracy across diverse multi-floor real-world sequences, filling a long-standing gap in high-precision, elevator-resilient SLAM and odometry (Zhang et al., 23 May 2026).

6. Space Elevators and Advanced Theoretical Constructs

The concept of a space elevator involves an ultra-high-strength tether from the Earth's surface to geosynchronous orbit, with a cross-section following an exponential taper profile: $m' = m(1 - \frac{a}{g})$ 4. This profile enables reel-to-reel deployment in which new ribbon is paid out at the earth’s surface while the old cable is reeled in at the counterweight (or across a GEO pulley in the “pull-down” mode). The entire tether profile effectively shifts in altitude, thickening uniformly and expediting construction relative to conventional climber-lift build-up, with doubling times potentially below one month for $m' = m(1 - \frac{a}{g})$ 5 km/h and feasible $m' = m(1 - \frac{a}{g})$ 6 m $m' = m(1 - \frac{a}{g})$ 7. However, supporting an inverse-taper initial seed requires a minimum material strength $m' = m(1 - \frac{a}{g})$ 8 GPa. These methods dramatically accelerate elevator build-up and cloning, contingent on breakthroughs in material science and operationalizing large-scale in-space deployment hardware (Gassend, 2024).

7. Practical Design, Measurement, and Limitations

Elevator performance fundamentally depends on the interplay of passenger arrival statistics, capacity constraints, number of elevators, and service speed. Critical system limits, such as the maximum stable arrival rate $m' = m(1 - \frac{a}{g})$ 9, are dictated by loading/unloading dwell times and shaft travel times. Measurement and validation of elevator motion and system performance employ a range of techniques:

Apparent weight recordings (via electronic scales) to infer acceleration profiles and operational phases.
Barometric altitude and velocity extraction for speed benchmarking and fault/failure detection.
Large-scale, event-driven simulations to validate theoretical and algorithmic predictions.
Analysis of clearing probabilities, synchrony, and group inter-arrival statistics to calibrate models.

Limitations of current modeling and control approaches include imperfect Markovianity (for RL), omitting higher-order interactions in quadratic/Boolean scheduling, sensitivity to real-world sensing delays, elevator hardware constraints, and unpredictable human behavior in dense, mixed-use environments. Recent research addresses some of these challenges by combining classical queueing and submodular optimization with predictive learning, adaptive reward shaping, and robust sensor/data fusion.

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