Continuity, Uniqueness and Long-Term Behavior of Nash Flows Over Time

TL;DR

We study Nash flows over time in a deterministic fluid queueing model with a single origin–destination and constant inflow $u_0$. We prove that equilibrium journey times are unique and that equilibria depend continuously on initial data and small network perturbations; we also show convergence to a linear steady-state direction $ ext{λ}$ when inflow exceeds the network’s capacity, after finite time. The analysis hinges on a generalized thin-flow framework with resetting, a primal–dual LP-based potential $ ext{Φ}( heta)$ that guides convergence, and extensions to piecewise-constant inflows and single-commodity multiterminal networks. These results yield stability guarantees for dynamic routing and clarify how short-term equilibria relate to long-run efficiency, laying groundwork for extensions to packet-based models. They also provide a foundation for future work on multi-commodity networks and robustness under broader perturbations.

Abstract

We consider a dynamic model of traffic that has received a lot of attention in the past few years. Users control infinitesimal flow particles aiming to travel from an origin to a destination as quickly as possible. Flow patterns vary over time, and congestion effects are modeled via queues, which form whenever the inflow into a link exceeds its capacity. Despite lots of interest, some very basic questions remain open in this model. We resolve a number of them in the single-commodity setting: - We show uniqueness of journey times in equilibria. - We show continuity of equilibria: small perturbations to the instance or to the traffic situation at some moment cannot lead to wildly different equilibrium evolutions. - We demonstrate that, assuming constant inflow into the network at the source, equilibria always settle down into a "steady state" in which the behavior extends forever in a linear fashion. One of our main conceptual contributions is to show that the answer to the first two questions, on uniqueness and continuity, are intimately connected to the third. To resolve the third question, we substantially extend the approach of Cominetti et al., who show a steady-state result in the regime where the input flow rate is smaller than the network capacity.

Figures (5)

• Figure 1: The dynamic of an arc at snapshot time $\theta$. The inflow $f^+(\theta)$ and outflow $f^-(\theta)$ describe the flow entering or leaving the arc at time $\theta$. The amount of flow in the queue $z_e(\theta)$ can leave the queue with rate $\nu$ and afterwards traverses the arc, which takes $\tau$ units of time.
• Figure 2: Dynamic equilibria can be seen as trajectories in $\mathbb{R}^V$ that follow a piecewise-constant vector field.
• Figure 3: The simplest situation which would produce non-uniqueness (if it were possible).
• Figure 4: A more subtle potential situation which would produce non-uniqueness; we show that this cannot occur.
• Figure 5: In the example the difference between the WSS property and steady state is depicted. Dotted arcs are inactive, solid arcs are active. All capacities are one and $3=\tau_{sv}>\tau_{su}+ \tau_{uv}=1+1$. In the beginning, the label of $v$ is determined by $uv$, until $sv$ becomes active when the queue on $su$ reaches length 1. From this point, the label of $v$ is determined by $sv$ (and $uv$ becomes inactive). Since $v$ is not active the equilibrium already has the WSS property on the left, but is not yet in steady-state. Steady state is reached as soon as $sv$ becomes active.

Theorems & Definitions (63)

• Definition 1
• Lemma 1
• proof
• Claim 1
• proof
• Lemma 2
• proof
• Definition 2
• Lemma 3
• proof