Adaptive Routing with Stale Information 1 , 2

We investigate the behaviour of load-adaptive rerouting policies in the Wardrop model where decisions must be made on the basis of stale information. In this model, an infinite number of agents controls an infinitesimal amount of flow each, thus contributing to a network flow which induces latency. In our dynamic extension of this model, agents are activated in a concurrent and asynchronous fashion and may reroute their flow with the aim of reducing their sustained latency. It is a well-known problem that in settings where latency information is not always up to date such behaviour may lead to oscillation effects which seriously harm network performance. Two quantities determine the difficulty of avoiding oscillation: the steepness of the latency functions and the maximum possible age of the information T . In this work we ask for conditions that the rerouting policies must adhere to in order to converge to an equilibrium despite the information being stale. We consider simple policies which sample another path in a first step and then migrate from the current path to the new one with a probability that is a function of the anticipated latency gain. In fact we can show that our class of policies guarantees convergence if the latter migration probability function satisfies a certain smoothness condition that resembles Lipschitz continuity. It turns out that for smooth adaptation policies where the migration probability is chosen small enough relative to the inverse of the steepness of the latency functions and T , the population actually converges to an equilibrium. In addition, we analyse the speed of convergence towards approximate equilibria of two specific variants of smooth adaptive routing policies, e. g., for a replication policy adopted from evolutionary game theory.