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In this paper, a delay tolerant network environment is considered where the source is in full control of the two-hop spreading mechanism by setting key parameters such as the number of copies allowed to be spread in the network and the delay bound of the messages. The analysis allows for a differentiation between the source of the message and the intermediate nodes (in terms of e.g. transmission power or speed). Analytical expressions for the cumulative distribution function (cdf) of the delivery delay and the induced overhead are extracted, taking into account the fact that the source node may continue spreading copies after the message delivery. In addition, a fairly accurate approximate expression for the cdf of the delivery delay is also derived and validated through simulations.