The recent successes of over-the-top (OTT) video services have intensified the competition between the traditional broadcasting video and OTT video. Such competition has pushed the traditional video service providers to accelerate the transition of their video services from the broadcasting video to the carrier-grade IP video streaming. However, there are significant challenges in providing large-scale carrier-grade IP video streaming services. For a compressed video sequence, central to the guaranteed real-time delivery are the issues of video rate, buffering, and timing as compressed video pictures are transmitted over an IP network from the encoder output to the decoder input. Toward the understanding and eventual resolution of these issues, a mathematical theory of compressed video buffering is developed to address IP video traffic regulation for the end-to-end video network quality of service. In particular, a comprehensive set of theoretical relationships is established for decoder buffer size, network transmission rate, network delay and jitter, and video source characteristics. As an example, the theory is applied to measure and compare the burstiness and delay of video streams coded with MPEG-2, advanced video coding, and high-efficiency video coding standards. The applicability of the theory to IP networks that consist of a specific class of routers is also demonstrated.