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

TL;DR

This paper develops a diffusion approximation for multi-agent queueing systems, showing their convergence to reflected Ornstein-Uhlenbeck processes, which effectively model crowd dynamics and neural activity under constraints.

Contribution

It introduces a rigorous analytical framework for approximating complex constrained queueing and crowd systems using reflected diffusion limits, specifically interacting OU processes.

Findings

Convergence of multi-agent queueing systems to reflected OU processes.

Validation through numerical examples in crowd and neural models.

Demonstration of the approximation's accuracy in large-scale regimes.

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.