Model (static)
-
directed graph $G=(V,E)$
-
commodities $i \in I$ with fixed flow volume $Q_i \in \IR_{\geq 0}$
and finite set $\Pc_i$ of source,sink-paths
-
route choices of all agents described by vector
\[(x_{p})_{p} \in X \coloneqq \Big\{x \in \IR_{\geq 0}^{\sum_{i \in I}\Pc_i} \,\Big|\, \sum_{p \in \Pc_i}x_p = Q_i \text{ f.a. } i \in I\Big\}\]
-
path costs $\ell: X \to \IR_{\geq 0}^{\sum_{i \in I}\Pc_i}$
$x^\ast \in X$ a Wardrop equilibrium if $x^\ast_{p}>0 \implies \ell_{p}(x^\ast) \leq \ell_{q}(x^\ast)$ f.a. $p,q \in \Pc_i$.
-
additional side-constraints $Z \subseteq X$ (e.g. $Z \coloneqq \Set{x \in X \Mid x_e \leq c_e}$)
- TEXTTEXTTEXT
$x^\ast \in\;$$Z$ a side-constrained equilibrium if $x^\ast_{p}>0 \implies \ell_{p}(x^\ast) \leq \ell_{q}(x^\ast)$ f.a. $p \in \Pc_i$ and unsaturated $q \in \Pc_i$.
Studied e.g. by [Bernstein, Smith 94], [Larsson, Patriksson 95], [Marcotte, Nguyen, Schoeb 04], [Correa, Schulz, Stier-Moses 04], ...
Model (dynamic)
-
directed graph $G=(V,E)$
-
commodities $i \in I$ with fixed flow volume $Q_i \in \IR_{\geq 0}$
and finite set $\Pc_i$ of source,sink-paths
-
route and departure time choices of all agents described by
\[(h_{p})_{p} \in \Lambda(Q) \coloneqq \Big\{h \in \left(L^2_+([0,T])\right)^{\sum_{i \in I}\Pc_i} \,\Big|\, \sum_{p \in \Pc_i}\int_0^T h_{p}(t)dt=Q_i \text{ for all } i \in I\Big\}\]
-
effective path delay operator $\Psi: \Lambda(Q) \to C([0,T])^{\sum_{i \in I}\Pc_i}$
$h^\ast \in \Lambda(Q)$ a dynamic equilibrium if $h^\ast_{p}(t)>0 \implies \Psi_{p}(h^\ast,\;$$t$$) \leq \Psi_{q}(h^\ast,\;$$t'$$)$ f.a. $p,q \in \Pc_i$ and $t,t' \in [0,T]$.
-
constraint set $S \subseteq \Lambda(Q)$
-
admissible $\varepsilon$-deviations $A_p$$:\;S \rightrightarrows \Pc_i \times \mathcal{M}([0,T]) \times \IR_{\geq 0} \times \IR$
$h^\ast \in S$ a SCDE if $h^\ast_{p}(t)>0 \implies \Psi_{p}(h^\ast,t) \leq \Psi_{q}(h^\ast,t+\Delta)$ f.a. $p \in \Pc_i$, $(q,\Delta) \in U_p(h^\ast,t)$ $\;\subseteq \Pc_i$, $t \in [0,T]$.
where $U_p(h,t) \coloneqq \Set{(q,\Delta) \in \Pc_i \,|\, \forall \delta > 0, \varepsilon > 0: \exists J \subseteq [t-\delta,t+\delta], \varepsilon' \leq \varepsilon: (q,J,\varepsilon',\Delta) \in A_p(h)}$