Oblivious routing algorithms for general undirected networks were introduced by Räcke, and this work has led to many subsequent improvements and applications. More precisely, Räcke showed that there is an oblivious routing algorithm with polylogarithmic competitive ratio (w.r.t. edge-congestion) for any undirected graph. Comparatively little positive results are known about oblivious routing in general directed networks. Using a novel approach, we present the first oblivious routing algorithm which is $O(\log^2 n)$-competitive with high probability in directed graphs given that the demands are chosen randomly from a known demand-distribution. On the other hand, we show that no oblivious routing algorithm can be $o(\frac{\log n}{\log \log n})$ competitive even with constant probability in general directed graphs.
Our routing algorithms are not oblivious in the traditional definition, but we add the concept of demand-dependence, i.e., the path chosen for an $s$-$t$ pair may depend on the demand between $s$ and $t$. This concept that still preserves that routing decisions are only based on local information proves very powerful in our randomized demand model.
Finally, we show that our approach for designing competitive oblivious routing algorithms is quite general and has applications in other contexts like stochastic scheduling.