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In this paper we face the challenging issue of defining and implementing an effective law for load balancing in Content Delivery Networks. We base our proposal on a formal study of a CDN system, carried out through the exploitation of a fluid flow model characterization of the network of servers. Starting from such characterization, we derive and prove a lemma about the network queues equilibrium. This result is then leveraged in order to devise a novel distributed algorithm for load balancing. The overall approach is validated by means of simulations showing the effectiveness of the proposed algorithm in terms of both fair load distribution and limited service time.