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This paper studies fundamental properties of stream-based content distribution services. We assume the presence of an overlay network (such as those built by P2P systems) with limited degree of connectivity, and we develop a mathematical model that captures the essential features of overlay-based streaming protocols and systems. The methodology is based on stochastic graph theory, and models the streaming system as a stochastic process, whose characteristics are related to the streaming protocol. The model captures the elementary properties of the streaming system such as the number of active connections, the different play-out delay of nodes, and the probability of not receiving the stream due to node failures/misbehavior. Besides the static properties, the model is able to capture the transient behavior of the distribution graphs, i.e., the evolution of the structure over time, for instance in the initial phase of the distribution process. Contributions of this paper include a detailed definition of the methodology, its comparison with other analytical approaches and with simulative results, and a discussion of the additional insights enabled by this methodology. Results show that mesh based architectures are able to provide bounds on the receiving delay and maintain rate fluctuations due to system dynamics very low. Additionally, given the tight relationship between the stochastic process and the properties of the distribution protocol, this methodology gives basic guidelines for the design of such protocols and systems.