Long Term Behavior of Dynamic Equilibria in Fluid Queuing Networks

Abstract

Abstract.A fluid queuing network constitutes one of the simplest models in which to study flow dynamics over a network. In this model we have a single source-sink pair and each link has a per-time-unit capacity and a transit time. A dynamic equilibrium (or equilibrium flow over time) is a flow pattern over time such that no flow particle has incentives to unilaterally change its path. Although the model has been around for almost fifty years, only recently results regarding existence and characterization of equilibria have been obtained. In particular the long term behavior remains poorly understood. Our main result in this paper is to show that, under a natural (and obviously necessary) condition on the queuing capacity, a dynamic equilibrium reaches a steady state (after which queue lengths remain constant) in finite time. Previously, it was not even known that queue lengths would remain bounded. The proof is based on the analysis of a rather non-obvious potential function that turns out to be monotone along the evolution of the equilibrium. Furthermore, we show that the steady state is characterized as an optimal solution of a certain linear program. When this program has a unique solution, which occurs generically, the long term behavior is completely predictable. On the contrary, if the linear program has multiple solutions the steady state is more difficult to identify as it depends on the whole temporal evolution of the equilibrium.

Algorithms for Flows over Time with Scheduling Costs

“…Dynamic equilibria, which is the flow over time equivalent of Wardrop equilibria for static flows, are key objects of study. Existence, uniqueness, structural and algorithmic issues, and much more have been receiving increasing recent interest from the optimization community [4,5,6,7,16,22,23].…”

“…Consider for a moment the model where users cannot choose their departure time, but instead are released from the source at a fixed rate u 0 , and simply wish to reach the destination as early as possible. This is the game-theoretic model that has received the most attention from the flow-overtime perspective [4,6,7,16,22]. Our construction of optimal tolls is applicable to this model as well.…”

On the Price of Anarchy for flows over time

“…time) of a dynamic equilibrium and thus proposed an algorithm to construct a dynamic equilibrium by concatenating static flows. Using this characterization, Cominetti, Correa and Larré [7] gave a constructive proof of existence of equilibria and proved they are essentially unique. Despite these efforts, many fundamental questions remain open, and several apparently obvious properties turn out to be notoriously hard to prove.…”

“…Indeed, until recently, it was not even known whether the size of the queues remains bounded throughout the evolution of the dynamic equilibrium. Along these lines, Cao et al [5] established this property (on a slightly different atomic model that does not influence the result) for series-parallel networks, while Correa et al [8] established the result for general networks by proving that a steady state is always achieved in finite time (naturally, as long as u 0 is at most the capacity of the minimum cut). Quite surprisingly however, the latter results apply only for constant inflow rate u 0 ; if the inflow varies over time, say it is u 0 in all intervals of the form [2i, 2i + 1) and u 0 /2 in all intervals of the form [2i − 1, 2i) for i ∈ N, then the boundedness of the queues is still open.…”

Dynamic Flows with Adaptive Route Choice

“…Subsequently, Cominetti, Correa and Larré [4] derived alternative characterizations and proved existence and uniqueness in terms of experienced travel times of equilibria even for multi-commodity networks. Very recently, Cominetti, Correa and Olver [5] shed light on the behavior of steady state queues assuming single commodity networks and constant inflow rates. Sering and Vargas-Koch [19] analyzed the impact of spillbacks in the fluid queuing model and Bhaskar et al [1] devised Stackelberg strategies in order to improve the efficiency of dynamic equilibria.…”