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In recent years, multimedia content distribution has largely been moved to the Internet, inducing broadcasters, operators and service providers to upgrade with large expenses their infrastructures. In this context, streaming solutions that rely on user devices such as set-top boxes (STBs) to offload dedicated streaming servers are particularly appropriate. In these systems, contents are usually replicated and scattered over the network established by STBs placed at users¿ home, and the video-on-demand (VoD) service is provisioned through streaming sessions established among neighboring STBs following a Peer-to-Peer fashion. Up to now the majority of research works have focused on the design and optimization of content replicas mechanisms to minimize server costs. The optimization of replicas mechanisms has been typically performed either considering very crude system performance indicators or analyzing asymptotic behavior. In this work, instead, we propose an analytical model that complements previous works providing fairly accurate predictions of system performance (i.e., blocking probability). Our model turns out to be a highly scalable, flexible, and extensible tool that may be helpful both for designers and developers to efficiently predict the effect of system design choices in large scale STB-VoD systems.