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this paper, we present a chunk scheduling algorithm for a mesh-based peer-to-peer live streaming system and we evaluate it by simulations over large-scale networks. Literature papers typically design chunk scheduling algorithms by considering the chunk delivery ratio as performance metric. We propose a push-based algorithm, which not only tries to maximize the chunk delivery ratio but it also takes into account and tries to minimize the delivery delay of chunks at the peer nodes. This is an important requirement, when dealing with realtime multimedia flows. Another important contribution of this paper is the design and implementation of a simulator able to evaluate the performance of large scale P2P networks (tens of thousands peers). The importance of this contribution lies in the fact that existing simulators and performance studies handle at most hundreds or few thousands of peers, while real-life P2P streaming systems aim at distributing contents to several hundreds of thousands, if not millions, of users. The performance evaluation study aims at providing a comprehensive view of what performance can be expected for mesh-based peer-to-peer streaming systems, both in terms of chunk delivery ratio and delay, for a large range of the number of users. The individual effect of a variety of system parameters, and especially number of partner nodes in the mesh, constrained link bandwidth, node heterogeneity, and network size, has been analyzed. Our results show that performances of the proposed push-based solution are already quite effective even with severely bandwidth constrained large scale networks.