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We analyze a routing scheme for a broad class of networks which converges (in the Cesaro sense) with probability one to the set of approximate Cesaro-Wardrop equilibria, an extension of the notion of a Wardrop equilibrium. The network model allows for wireline networks where delays are caused by flows on links, as well as wireless networks, a primary motivation for us, where delays are caused by other flows in the vicinity of nodes. The routing algorithm is distributed, using only the local information about observed delays by the nodes, and is moreover impervious to clock offsets at nodes. The scheme is also fully asynchronous, since different iterates have their own counters and the orders of packets and their acknowledgments may be scrambled. Finally, the scheme is adaptive to the traffic patterns in the network. The demonstration of convergence in a fully dynamic context involves the treatment of two-time scale distributed asynchronous stochastic iterations. Using an ordinary differential equation approach, the invariant measures are identified. Due to a randomization feature in the algorithm, a direct stochastic analysis shows that the algorithm avoids non-Wardrop equilibria. Finally, some comments on the existence, uniqueness, stability, and other properties of Wardrop equilibria are made.