# A roundabout model with on-ramp queues: exact results and scaling approximations

## TL;DR

This paper models a single-lane roundabout with on-ramp queues using a Markovian lattice framework, providing exact stationary distributions and scalable Gaussian and Poisson approximations for traffic performance metrics.

## Contribution

It introduces a detailed Markovian model of roundabout traffic dynamics and derives explicit stationary distributions along with novel scaling approximations for large systems.

## Key findings

- Explicit stationary distribution for each cell on the lattice.
- Scaling limits approximating joint distributions of segments as Gaussian and Poisson.
- A new empirical method to assess convergence in distribution.

## Abstract

This paper introduces a general model of a single-lane roundabout, represented as a circular lattice that consists of $L$ cells, with Markovian traffic dynamics. Vehicles enter the roundabout via on-ramp queues that have stochastic arrival processes, remain on the roundabout a random number of cells, and depart via off-ramps. Importantly, the model does not oversimplify the dynamics of traffic on roundabouts, while various performance-related quantities (such as delay and queue length) allow an analytical characterization. In particular, we present an explicit expression for the marginal stationary distribution of each cell on the lattice. Moreover, we derive results that give insight on the dependencies between parts of the roundabout, and on the queue distribution. Finally, we find scaling limits that allow, for every partition of the roundabout in segments, to approximate 1) the joint distribution of the occupation of these segments by a multivariate Gaussian distribution; and 2) the joint distribution of their total queue lengths by a collection of independent Poisson random variables. To verify the scaling limit statements, we develop a novel way to empirically assess convergence in distribution of random variables.