= Solution
A useful starting model treats a road network as a directed <flow network> and travelers as a divisible population in a <nonatomic congestion game>: an individual traveler has negligible effect on link congestion. For each source-sink pair $k$, fix a demand $d_k$ and a finite set $\mathcal P_k$ of allowed routes. Let $f_p\geq0$ be the flow on route $p$, with $\sum_{p\in\mathcal P_k}f_p=d_k$. The <link-route incidence matrix> $A$ gives link loads $v=Af$. If link $e$ has delay $\ell_e(v_e)$, the route travel time is
$$
L_p(f)=\sum_{e\in p}\ell_e(v_e).
$$
A <Wardrop equilibrium> is a feasible route-flow vector for which every used route has minimum travel time among the routes serving the same source-sink pair. Equivalently, there are numbers $\lambda_k$ such that
$$
\boxed{L_p(f)\geq\lambda_k\quad(p\in\mathcal P_k),\qquad f_p>0\Longrightarrow L_p(f)=\lambda_k.}
$$
The condition says that a traveler cannot lower their own delay by changing route. It does not assert that their choice minimizes the total delay experienced by all travelers. At a <Wardrop equilibrium>, unused routes may have exactly the same delay as used ones.
This static <flow network> model requires delay to depend on present link load, with each source-sink demand fixed. Models from <queueing theory> instead describe vehicles or packets as discrete customers whose delay depends on random arrivals, service and accumulated queues; a time-dependent model must also track changes of flow and travel-time propagation. Those models answer different questions from a static <Wardrop equilibrium>. The static model is nevertheless valuable because its route-choice condition has a precise <convex optimization> formulation, as developed next.
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