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In this paper, we propose a distributed rate allocation algorithm for delay minimal data delivery in overlay networks where multiple sources compete simultaneously for the available network resources. In order to efficiently utilize the network resources, we propose to use inter-session network coding. We devise a distributed algorithm that allows peers to determine the optimal coding combinations and the packet rates for each type of combinations to be requested from the parent peers. The rate allocation problem is formulated as a decoding delay minimization problem, where every peer seeks the rates that minimize the average expected delay of the peer and its children peers. To solve this non-convex optimization problem, we introduce the concept of equivalent packet flows, which permits to estimate the expected number of packets that every peer needs to collect for decoding. We then decompose our original rate allocation problem into a set of convex subproblems, which we eventually combine to obtain the solution to the delay minimization problem. The results demonstrate that the proposed scheme eliminates effectively the bottlenecks and reduces the delay time experienced by users with limited resources.