Abstract
We study dynamic network flows and introduce a notion of instantaneous dynamic equilibrium (IDE) requiring that for any positive inflow into an edge, this edge must lie on a currently shortest path towards the respective sink. We measure current shortest path length by current waiting times in queues plus physical travel times. As our main results, we show: existence and constructive computation of IDE flows for multi-source single-sink networks assuming constant network inflow rates,finite termination of IDE flows for multi-source single-sink networks assuming bounded and finitely lasting inflow rates,the existence of IDE flows for multi-source multi-sink instances assuming general measurable network inflow rates,the existence of a complex single-source multi-sink instance in which any IDE flow is caught in cycles and flow remains forever in the network.
Highlights
Dynamic network flows have been studied for decades in the optimization and transportation literature, see the classical book of Ford and Fulkerson [7] or the more recent surveys of Skutella [22] and Peeta [17]
A directed graph G = (V, E) is given, where edges e ∈ E are associated with a queue with positive rate capacity νe ∈ R>0 and a physical transit time τe ∈ R>0
If the total inflow into an edge e = vw ∈ E exceeds the rate capacity νe, a queue builds up and agents need to wait in the queue before they are forwarded along the edge
Summary
Dynamic network flows have been studied for decades in the optimization and transportation literature, see the classical book of Ford and Fulkerson [7] or the more recent surveys of Skutella [22] and Peeta [17]. The fluid queue model has been mostly studied from a game-theoretic perspective, where it is assumed that agents act selfishly and travel along shortest routes under prevailing conditions This behavioral model is known as dynamic equilibrium and has been analyzed in the transportation science literature for decades, see Friesz et al [8], Meunier and Wagner [16] and Zhu and Marcotte [25]. Complete knowledge requires that a traveler is able to exactly forecast future travel times along the chosen path effectively anticipating the whole evolution of the flow propagation process across the network This assumption has been justified by letting travelers learn good routes over several trips and a dynamic equilibrium corresponds to an attractor of the underlying learning dynamic. We illustrate IDE in comparison to classical dynamic equilibrium with an example
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