Side-Constrained Dynamic Traffic Equilibria

Abstract

We study dynamic traffic assignment with side-constraints. We first give a counter-example to a key result from the literature regarding the existence of dynamic equilibria for volume-constrained traffic models in the classical edge-delay model. Our counter-example shows that the feasible flow space need not be convex and it further reveals that classical infinite dimensional variational inequalities are not suited for the definition of side-constrained dynamic equilibria. We propose a new framework for side-constrained dynamic equilibria based on the concept of feasible $γ$-deviations of flow particles in space and time. Under natural assumptions, we characterize the resulting equilibria by means of quasi-variational and variational inequalities, respectively. Finally, we establish first existence results for side-constrained dynamic equilibria for the non-convex setting of volume-constraints.

Figures (10)

• Figure 1: An instance with constant volume constraint on one edge where the constraint set $S$ defined by edge capacities is not convex when using linear edge delays.
• Figure 2: The flow volume $x_{e_1}(h,t)$ on edge $e_1$ given a constant inflow rate of $2$ (left) or $1$ (right) during the interval $[0,2]$. See CareyMcCartney for more details on computing the flow dynamics with linear edge delays.
• Figure 3: A counterexample to zhong11.
• Figure 4: An example for how an admissible $\gamma$-deviation $(q,\gamma,\Delta) \in A_{p}(h)$ changes the path inflow rates on the involved paths $p$ and $q$ from the original flow $h$ (left) to the new flow $h' \coloneqq H_{p\to q}(h,\gamma,\Delta)$ (right).
• Figure 5: A three commodity network with fixed network inflow rates. All values of $\tau_e$ not explicitly given in the figure are $1$ and all $\nu_e$ not given are infinity. Using the Vickrey point queue model for the edge dynamics and the capacity constraint on edge $e$ as volume or inflow rate constraint, this network has a unique feasible flow which is a strict CDE but neither a weak BSDE nor a weak LPDE.

Theorems & Definitions (87)

• Definition 2.1
• Lemma 2.2
• Definition 2.3
• Lemma 2.4: cf. Dafermos dafermos1980traffic
• Definition 2.5
• Lemma 2.6
• Definition 2.7
• Lemma 2.8
• Definition 3.1
• Lemma 3.2