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From Queues to Crowd Flows: Reflected Diffusion Limits for Controlled Agents

Thoa Thieu, Roderick Melnik

TL;DR

The paper develops a diffusion-approximation framework for multi-agent constrained queueing systems with reflection and inter-agent interactions, proving convergence to a system of interacting reflected Ornstein-Uhlenbeck processes under diffusion scaling. The limiting dynamics capture goal-directed drift, repulsive interactions, and boundary reflections, enabling tractable analysis of large-scale constrained networks and crowd-like systems. The authors establish a Skorokhod-reflection-based existence and convergence theory and provide a constructive interpretation of the queueing limit as fluctuations around nominal targets with Gaussian noise. Two numerical experiments—crowd dynamics and neural population models—validate the diffusion limit and illustrate how discrete stochastic, constrained systems converge to their continuous reflected OU counterparts, with explicit mean-squared-error convergence consistent with $O(n^{-1/2})$ scaling.

Abstract

We establish a diffusion approximation for a class of multi-agent controlled queueing systems, demonstrating their convergence to a system of interacting reflected Ornstein--Uhlenbeck (OU) processes. The limiting process captures essential behavioral features of the underlying stochastic dynamics, including goal-directed motion, inter-agent repulsion, and reflection at domain boundaries. This result provides a rigorous analytical framework for approximating constrained queueing networks and crowd motion models, offering tractable characterizations of their steady-state behavior and transient dynamics under large-scale regimes. We further illustrate the theoretical findings through two numerical examples. The first example considers a crowd dynamics scenario, modeling interacting agents navigating within a confined domain, while the second focuses on a neural population model that describes stochastic activity evolution under competition and bounded constraints. These experiments validate the convergence predicted by the diffusion approximation and demonstrate how discrete stochastic systems with reflection and interaction mechanisms approach their continuous reflected OU limits, offering both physical and biosocial interpretations of the theory.

From Queues to Crowd Flows: Reflected Diffusion Limits for Controlled Agents

TL;DR

The paper develops a diffusion-approximation framework for multi-agent constrained queueing systems with reflection and inter-agent interactions, proving convergence to a system of interacting reflected Ornstein-Uhlenbeck processes under diffusion scaling. The limiting dynamics capture goal-directed drift, repulsive interactions, and boundary reflections, enabling tractable analysis of large-scale constrained networks and crowd-like systems. The authors establish a Skorokhod-reflection-based existence and convergence theory and provide a constructive interpretation of the queueing limit as fluctuations around nominal targets with Gaussian noise. Two numerical experiments—crowd dynamics and neural population models—validate the diffusion limit and illustrate how discrete stochastic, constrained systems converge to their continuous reflected OU counterparts, with explicit mean-squared-error convergence consistent with scaling.

Abstract

We establish a diffusion approximation for a class of multi-agent controlled queueing systems, demonstrating their convergence to a system of interacting reflected Ornstein--Uhlenbeck (OU) processes. The limiting process captures essential behavioral features of the underlying stochastic dynamics, including goal-directed motion, inter-agent repulsion, and reflection at domain boundaries. This result provides a rigorous analytical framework for approximating constrained queueing networks and crowd motion models, offering tractable characterizations of their steady-state behavior and transient dynamics under large-scale regimes. We further illustrate the theoretical findings through two numerical examples. The first example considers a crowd dynamics scenario, modeling interacting agents navigating within a confined domain, while the second focuses on a neural population model that describes stochastic activity evolution under competition and bounded constraints. These experiments validate the convergence predicted by the diffusion approximation and demonstrate how discrete stochastic systems with reflection and interaction mechanisms approach their continuous reflected OU limits, offering both physical and biosocial interpretations of the theory.
Paper Structure (10 sections, 3 theorems, 52 equations, 2 figures)

This paper contains 10 sections, 3 theorems, 52 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setting of the model equations
  3. Preliminaries and assumptions
  4. Concept of Solution
  5. Statement of the main results
  6. Queueing Interpretation of the Limit.
  7. Numerical examples
  8. Human crowd dynamics
  9. Diffusion scaling in neural network populations
  10. Conclusions and discussion

Key Result

Lemma 1

Suppose that for each $i=1,\ldots,N$, the arrival process $A_t^{(n), i}$ and the departure process $D_t^{(n), i}$ are independent Poisson processes with intensities $n\lambda_i$ and $n\mu_i$, respectively. Then, the centered and scaled net input process satisfies where $W_t^i$ is a standard Brownian motion and $\sigma_i^2 = \lambda_i + \mu_i$.

Figures (2)

  • Figure 1: [Color online] Comparison between the mean trajectories and mean squared errors (MSE) of a 2D reflected OU limit process and scaled queueing approximations for three interacting agents. (Top) The mean $x$-position trajectories show how the scaled queueing processes with different system sizes ($n=50,200,800$) converge toward the continuous OU limit. (Bottom) The corresponding MSE curves illustrate the decreasing deviation between the scaled queueing dynamics and the OU limit as $n$ increases, confirming the diffusion approximation convergence.
  • Figure 2: [Color online] Mean firing-rate dynamics and diffusion scaling of interacting neural populations. The top panel shows the average firing-rate trajectories of five interacting neural populations evolving under reflected OU dynamics (dotted lines) compared with their corresponding finite-population (queueing-like) stochastic counterparts at different network sizes $n=50,200,800$. Each color represents a distinct neural population with unique initial activity. The system exhibits convergence of finite-size dynamics to the mean-field reflected OU limit under diffusion scaling. The bottom panel shows the corresponding MSE between the finite-population and OU trajectories over time, illustrating decreasing stochastic deviation as population size increases. Reflective boundaries constrain activity within the bounded cortical domain $[0,5]^2$, while mutual inhibition and goal-directed drift shape coordinated dynamics.

Theorems & Definitions (6)

  • Lemma 1: Functional central limit theorem (FCLT) for net input process
  • proof
  • Theorem 1: cf. Theorem 4.1 in Saisho1987
  • Definition 1: Reflected SDE Solution
  • Theorem 2: Diffusion Limit of the Scaled Queueing System
  • proof