# Wardrop Equilibrium in Discrete-Time Selfish Routing with Time-Varying Bounded Delays

Alessandro Giuseppi, Antonio Pietrabissa: Wardrop Equilibrium in Discrete-Time Selfish Routing with Time-Varying Bounded Delays. In: IEEE Transactions on Automatic Control, pp. 1–1, 2020.

## Abstract

This paper presents a multi-commodity, discrete-time, distributed and non-cooperative routing algorithm, which is proved to converge to an equilibrium in the presence of heterogeneous, unknown, time-varying but bounded delays. Under mild assumptions on the latency functions which describe the cost associated to the network paths, two algorithms are proposed: the former assumes that each commodity relies only on measurements of the latencies associated to its own paths; the latter assumes that each commodity has (at least indirectly) access to the measures of the latencies of all the network paths. Both algorithms are proven to drive the system state to an invariant set which approximates and contains the Wardrop equilibrium, defined as a network state in which no traffic flow over the network paths can improve its routing unilaterally, with the latter achieving a better reconstruction of the Wardrop equilibrium. Numerical simulations show the effectiveness of the proposed approach.

```@article{Giuseppi2020a,
title = {Wardrop Equilibrium in Discrete-Time Selfish Routing with Time-Varying Bounded Delays},
author = {Alessandro Giuseppi and Antonio Pietrabissa},
doi = {10.1109/tac.2020.2981906},
year  = {2020},
date = {2020-01-01},
journal = {IEEE Transactions on Automatic Control},
pages = {1--1},
publisher = {Institute of Electrical and Electronics Engineers (IEEE)},
abstract = {This paper presents a multi-commodity, discrete-time, distributed and non-cooperative routing algorithm, which is proved to converge to an equilibrium in the presence of heterogeneous, unknown, time-varying but bounded delays. Under mild assumptions on the latency functions which describe the cost associated to the network paths, two algorithms are proposed: the former assumes that each commodity relies only on measurements of the latencies associated to its own paths; the latter assumes that each commodity has (at least indirectly) access to the measures of the latencies of all the network paths. Both algorithms are proven to drive the system state to an invariant set which approximates and contains the Wardrop equilibrium, defined as a network state in which no traffic flow over the network paths can improve its routing unilaterally, with the latter achieving a better reconstruction of the Wardrop equilibrium. Numerical simulations show the effectiveness of the proposed approach.},
keywords = {LaSalle’s invariance principle, selfish routing, time-delay systems, Wardrop equilibrium},
pubstate = {published},
tppubtype = {article}
}
```