Peer-to-peer (P2P) live video streaming systems have recently received substantial attention, with commercial deployment gaining increased popularity in the internet. It is evident from our practical experiences with real-world systems that, it is not uncommon for hundreds of thousands of users to choose to join a program in the first few minutes of a live broadcast. Such a severe flash crowd phenomenon in live streaming poses significant challenges in the system design. In this paper, for the first time, we develop a mathematical model to: 1) capture the fundamental relationship between time and scale in P2P live streaming systems under a flash crowd, and 2) explore the design principle of population control to alleviate the impact of the flash crowd. We carry out rigorous analysis that brings forth an in-depth understanding on effects of the gossip protocol and peer dynamics. In particular, we demonstrate that there exists an upper bound on the system scale with respect to a time constraint. By trading peer startup delays in the initial stage of a flash crowd for system scale, we design a simple and flexible population control framework that can alleviate the flash crowd without the requirement of otherwise costly server deployment.
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