In peer-to-peer streaming the corresponding file is is divided into many small chunks that are obtained, from other peers, in a fixed order forming a stream of information. We assume a 'video on demand'-type scenario, all peers wish to obtain the entire file. We use a continuous model, where the set of chunks is replaced by a real variable in a finite segment. The file streaming is then depicted as movement of a peer along this segment. The peers form a network, where any peer can download from any other. We consider an open system with continuous inflow of peers. Peers select their targets for downloading in a random fashion, thus forming a random overlay for streaming. We take into account that multiple dowloaders get bandwidth inversely proportional to their number. Our aim is to see whether such strategy can work. We make simulation for somehow realistic system and found that when ignoring possible delay of the search of peers, this system can provide quite good service that seems to scale well with number of peers. Making some simplifications we found a system that can be solved in a 'mean-field' approximation when the characteristics of the contact graph are approximated by some mean values. This solution supports simulations showing linear scaling of the system with arrival rate in the large system size limit. Further, we anylze the effect of multiple contact and suggest that the system has almost ideal scaling, meaning that peers can download with almost maximal speed when having just few simultaneous contacts.