Dynamic Equilibria in Time-Varying Networks

Pham, Hoang Minh; Sering, Leon

doi:10.1007/978-3-030-57980-7_9

Cited by 4 publications

(5 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Define θ so that ξ = v ( θ) + q e ( θ). Then agrees with the equilibrium of the perturbed instance on the interval [0, θ]; this follows immediately from the sequential construction of dynamic equilibria (as for example described in [PS20] for time-varying capacities). We can then treat ( θ) as the initial conditions for an equilibrium trajectory for the instance with the new capacities.…”

Section: Uniqueness and Continuitymentioning

confidence: 52%

“…Other types of perturbations (e.g., of transit times) can also be dealt with, though we don't discuss this here. Perturbations of this form are related to dynamic equilibria in time-varying networks, where many fundamental results still hold (see [PS20,Ser20]).…”

Section: Model and Preliminariesmentioning

confidence: 99%

“…In this extended abstract, we will only discuss capacity perturbations. Perturbations to transit times can also be handled, with some additional minor complications (in particular, care is required, also in the modeling, when considering decreases in transit times); see [PS20] for a discussion on this topic.…”

Section: Uniqueness and Continuitymentioning

confidence: 99%

See 2 more Smart Citations

Continuity, Uniqueness and Long-Term Behavior of Nash Flows Over Time

Olver¹,

Sering²,

Koch³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We consider a dynamic model of traffic that has received a lot of attention in the past few years. Users control infinitesimal flow particles aiming to travel from a source to destination as quickly as possible. Flow patterns vary over time, and congestion effects are modeled via queues, which form whenever the inflow into a link exceeds its capacity. Despite lots of interest, some very basic questions remain open in this model. We resolve a number of them:• We show uniqueness of journey times in equilibria.• We show continuity of equilibria: small perturbations to the instance or to the traffic situation at some moment cannot lead to wildly different equilibrium evolutions.• We demonstrate that, assuming constant inflow into the network at the source, equilibria always settle down into a "steady state" in which the behavior extends forever in a linear fashion.One of our main conceptual contributions is to show that the answer to the first two questions, on uniqueness and continuity, are intimately connected to the third. Our result also shows very clearly that resolving uniqueness and continuity, despite initial appearances, cannot be resolved by analytic techniques, but are related to very combinatorial aspects of the model. To resolve the third question, we substantially extend the approach of [CCO21], who show a steady-state result in the regime where the input flow rate is smaller than the network capacity.* N.O. is partially supported by NWO Vidi grant 016.Vidi.189.087.

show abstract

Section: Uniqueness and Continuitymentioning

confidence: 52%

Section: Model and Preliminariesmentioning

confidence: 99%

See 1 more Smart Citation

Continuity, Uniqueness and Long-Term Behavior of Nash Flows Over Time

Olver¹,

Sering²,

Koch³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…There are many works investigating properties of equilibria in this model [BFA15], [CCL15], [CCO21], [CCO19], [Kai22], [Koc12], [KS11], [OSVK22], [SS18] and in generalized models [IS20], [PS20], [SVK19], [Ser20]. We will discuss some of these later in Section II-E.…”

Section: Introductionmentioning

confidence: 99%

Convergence of Approximate and Packet Routing Equilibria to Nash Flows Over Time

Olver,

Sering,

Koch

2023

2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)

Self Cite

View full text Add to dashboard Cite

We consider a dynamic model of traffic that has received a lot of attention in the past few years. Infinitesimally small agents aim to travel from a source to a destination as quickly as possible. Flow patterns vary over time, and congestion effects are modeled via queues, which form based on the deterministic queueing model whenever the inflow into a link exceeds its capacity.Are equilibria in this model meaningful as a prediction of traffic behavior? For this to be the case, a certain notion of stability under ongoing perturbations is needed. Real traffic consists of discrete, atomic "packets", rather than being a continuous flow of non-atomic agents. Users may not choose an absolutely quickest route available, if there are multiple routes with very similar travel times. We would hope that in both these situations -a discrete packet model, with packet size going to 0, and ε-equilibria, with ε going to 0 -equilibria converge to dynamic equilibria in the flow over time model. No such convergence results were known.We show that such a convergence result does hold in singlecommodity instances for both of these settings, in a unified way. More precisely, we introduce a notion of "strict" ε-equilibria, and show that these must converge to the exact dynamic equilibrium in the limit as ε → 0. We then show that results for the two settings mentioned can be deduced from this with only moderate further technical effort.

show abstract

“…The flows over time model yields exact user equilibria, called dynamic equilibria or Nash flows over time [17]. They are guaranteed to exist [3,4] and their structure has been studied intensively [5,7,15,16,20,27]. Even though most of the work has been done for single-origin-destination networks, the existence results and further structural insights have been transferred to the multiterminal setting as well [4,26]; see also [25] for an extended analysis on all facets of Nash flows over time.…”

Section: Introductionmentioning

confidence: 99%

Convergence of a Packet Routing Model to Flows Over Time

Sering

Koch

Ziemke

2021

Proceedings of the 22nd ACM Conference on Economics and Computation

Self Cite

View full text Add to dashboard Cite

The mathematical approaches for modeling dynamic traffic can roughly be divided into two categories: discrete packet routing models and continuous flow over time models. Despite very vital research activities on models in both categories, the connection between these approaches was poorly understood so far. In this work we build this connection by specifying a (competitive) packet routing model, which is discrete in terms of flow and time, and by proving its convergence to the intensively studied model of flows over time with deterministic queuing. More precisely, we prove that the limit of the convergence process, when decreasing the packet size and time step length in the packet routing model, constitutes a flow over time with multiple commodities. In addition, we show that the convergence result implies the existence of approximate equilibria in the competitive version of the packet routing model. This is of significant interest as exact pure Nash equilibria, similar to almost all other competitive models, cannot be guaranteed in the multi-commodity setting.Moreover, the introduced packet routing model with deterministic queuing is very application-oriented as it is based on the network loading module of the agent-based transport simulation MATSim. As the present work is the first mathematical formalization of this simulation, it provides a theoretical foundation and an environment for provable mathematical statements for MATSim. CCS Concepts: • Theory of computation → Algorithmic game theory; Network games; Network flows; Exact and approximate computation of equilibria; Routing and network design problems; • Mathematics of computing → Network flows.

show abstract

Dynamic Equilibria in Time-Varying Networks

Cited by 4 publications

References 21 publications

Continuity, Uniqueness and Long-Term Behavior of Nash Flows Over Time

Continuity, Uniqueness and Long-Term Behavior of Nash Flows Over Time

Convergence of Approximate and Packet Routing Equilibria to Nash Flows Over Time

Convergence of a Packet Routing Model to Flows Over Time

Contact Info

Product

Resources

About