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

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 a source to 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:• 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. Our result also shows very clearly that resolving uniqueness and continuity, despite initial appearances, cannot be resolved by analytic techniques, but are related to very combinatorial aspects of the model. To resolve the third question, we substantially extend the approach of [CCO21], who show a steady-state result in the regime where the input flow rate is smaller than the network capacity.* N.O. is partially supported by NWO Vidi grant 016.Vidi.189.087.

Convergence of a Packet Routing Model to Flows over Time

“…A much weaker notion of stability would be that if we slightly perturb the equilibrium at a single moment in time, or some bounded number of moments in time, by (say) perturbing some queue lengths or transit times by a small amount, that the equilibrium in the once-perturbed instance stays close to the unperturbed equilibrium. We demonstrated this quite recently for the deterministic queueing model [OSVK22]. Our earlier result can be seen as a precursor to this one, and we will rely on it in a number of places in our proof.…”

“…Given the vector ℓ of earliest arrival labels of a dynamic equilibrium, we will follow [OSVK22] in calling ℓ an equilibrium trajectory. We will discuss properties of dynamic equilibria and equilibrium trajectories in more detail in Section 2.5. ε-equilibria.…”

“…We now briefly summarize some useful facts about the structure of (exact) equilibria. For more details, we refer to [CCL15] and [KS11] on thin flows and the piecewise-linear structure; to [CCO21] and [OSVK22] for long-term behavior; and to [OSVK22] for the vector-field view and uniqueness and continuity of equilibria.…”

“…Generalized subnetworks. A further generalization we will need in our arguments is the notion of a generalized subnetwork, following [OSVK22]. A generalized subnetwork of our network G is defined by valid configuration ( Ẽ, E ∞ ).…”

Side-Constrained Dynamic Traffic Equilibria

“…Finally, it is worth mentioning that our model and the subsequent characterization and existence results require only mild continuity properties of the walk-delay operator and the edge load functions, respectively, and thus apply for several realistic and well-studied network loading models including the Vickrey queueing model with point queues [6,7,30,37,44], with spillback [41] with departure-time choice [12,23], the Lighthill-Whitham-Richards (LWR) model [16], the LWR model with spillback [25] and the classical link-delay model of Friesz et al [13].…”