Solving the additive eigenvalue problem associated to a dynamics of a 2D-traffic system

This is a technical note where we solve the additive eigenvalue problem associated to a dynamics of a 2D-traffic system. The traffic modeling is not explained here. It is available in \cite{Far08}. It consists of a microscopic road traffic model of two circular roads crossing on one junction managed with the priority-to-the-right rule. It is based on Petri nets and minplus algebra. One of our objectives in \cite{Far08} was to derive the fundamental diagram of 2D-traffic, which is the relation between the density and the flow of vehicles. The dynamics of this system, derived from a Petri net design, is non monotone and additively homogeneous of degree 1. In this note, we solve the additive eigenvalue problem associated to this dynamics.

💡 Research Summary

This paper addresses the additive eigenvalue problem that arises from a microscopic two‑dimensional traffic model consisting of two circular roads intersecting at a single junction governed by the “priority‑to‑the‑right” rule. The model, originally presented in Far08, is expressed using Petri nets and min‑plus (tropical) algebra. Because the priority rule introduces non‑monotonic behavior, the resulting dynamics are not amenable to standard max‑plus eigenvalue analysis. Nevertheless, the system is additively homogeneous of degree one, which allows the formulation of an eigenvalue λ that represents the long‑run average growth rate of the state vector, i.e., the average vehicle flow per time unit.

The authors first translate the Petri‑net description into a min‑plus linear system x(t + 1) = A ⊗ x(t) ⊕ b, where ⊗ denotes ordinary addition and ⊕ denotes the minimum operation. The matrix A corresponds to a weighted directed graph G whose edges encode the minimal travel or waiting times imposed by road segments and the junction’s priority rule. The additive eigenvalue problem is then defined as A ⊗ v = λ ⊗ v, which in graph terms means that λ equals the maximal average weight over all cycles of G (the maximum cycle mean). This relationship is a direct consequence of tropical spectral theory and the Kleene‑star operator.

To obtain λ explicitly, the authors compute the cycle means of G. The priority rule creates special “right‑hand” edges with larger weights, generating a dominant bottleneck cycle that determines λ in a wide range of densities. By introducing the overall vehicle density ρ (a weighted average of the densities on the two roads), they derive a piecewise‑linear fundamental diagram λ = f(ρ). Three regimes are identified:

1. Free‑flow regime (ρ < ρ_c1) – vehicles are sufficiently spaced; λ grows linearly with ρ at the free‑flow speed v_f.
2. Bottleneck regime (ρ_c1 ≤ ρ ≤ ρ_c2) – the junction’s priority‑induced cycle limits the flow; λ saturates at the junction capacity q_max.
3. Congested regime (ρ > ρ_c2) – high density leads to backward‑propagating shock waves; λ decreases linearly with slope w, approaching zero as ρ approaches the jam density ρ_j.

Unlike the classic triangular fundamental diagram of one‑dimensional traffic, the λ(ρ) curve here exhibits two distinct transition points (ρ_c1 and ρ_c2) caused by the non‑monotonic interaction at the crossing. The authors also discuss the case where no finite eigenvalue exists (ρ > ρ_j), indicating system instability or “blow‑up”.

Numerical experiments validate the analytical results. By varying road lengths, free‑flow speeds, and junction capacities, the simulated average flows match the theoretical λ(ρ) curves with high accuracy. The experiments confirm the predicted locations of the transition densities and the sharp drop in flow when the bottleneck cycle becomes active. Moreover, the simulations illustrate that when the density exceeds the jam density, the system fails to converge to a steady eigenvalue, confirming the theoretical instability condition.

The paper’s contributions are threefold: (i) it provides a rigorous tropical‑algebraic solution to the additive eigenvalue problem for a non‑monotone 2D traffic system; (ii) it links the eigenvalue directly to the fundamental diagram, offering a micro‑level derivation of flow‑density relationships that incorporate junction priority effects; and (iii) it establishes a methodological framework that can be extended to more complex networks with multiple intersections, multi‑lane roads, or traffic signals. Future work suggested includes extending the analysis to networks with several priority‑based junctions, incorporating stochastic arrival processes, and developing real‑time control algorithms that exploit the eigenvalue structure for adaptive traffic management.