We analyze networks of mixes used for providing untraceable communication. We consider a network consisting of k mixes working in parallel and exchanging the outputs – which is the most natural architecture for composing mixes of a certain size into networks able to mix a larger number of inputs at once. We prove that after O\mathcal{O}(1) rounds the network considered provides a fair level of privacy protection for any number of messages n. Number of required rounds does not dependent on number of mixes provided that n ≫ k ² . No mathematical proof of this kind has been published before. We show that if at least one of server is corrupted we need substantially more rounds to meet the same requirements of privacy protection.