Balancing Traffic in Networks: Redundancy, Learning and the Effect of Stochastic Fluctuations

TL;DR

This paper analyzes how network traffic distribution evolves under learning dynamics, showing convergence to equilibria and stability under stochastic fluctuations, with implications for network redundancy and user patience.

Contribution

It introduces a model linking network redundancy with traffic flow stability and demonstrates convergence and stability properties under stochastic delays.

Findings

Solutions converge to Wardrop equilibria

Interior equilibria are stochastically unstable under fluctuations

Long-term traffic averages approach equilibrium in non-redundant networks

Abstract

We study the distribution of traffic in networks whose users try to minimise their delays by adhering to a simple learning scheme inspired by the replicator dynamics of evolutionary game theory. The stable steady states of these dynamics coincide with the network's Wardrop equilibria and form a convex polytope whose dimension is determined by the network's redundancy (an important concept which measures the "linear dependence" of the users' paths). Despite this abundance of stationary points, the long-term behaviour of the replicator dynamics turns out to be remarkably simple: every solution orbit converges to a Wardrop equilibrium. On the other hand, a major challenge occurs when the users' delays fluctuate unpredictably due to random external factors. In that case, interior equilibria are no longer stationary, but strict equilibria remain stochastically stable irrespective of the fluctuations' magnitude. In fact, if the network has no redundancy and the users are patient enough, we show that the long-term averages of the users' traffic flows converge to the vicinity of an equilibrium, and we also estimate the corresponding invariant measure.