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Video traffic is widely expected to account for a large portion of the traffic in future wired and wireless networks. We propose an efficient and accurate approximation method for calculating a given percentile of the end-to-end delay of a video stream depicted by a trace. The queueing delay encountered in the network by the IP packets carrying the video is modeled by a tandem queueing network of infinite capacity queues. The video trace is approximated by a two-stage Markovian Arrival Process (MAP2), which is the arrival process to the tandem network. The proposed method uses only the first queue of the tandem queueing network to construct an upper and lower bound of a given percentile of the end-to-end delay. The percentile value is then approximated by interpolating between the two bounds. We used this method to estimate the 95th percentile of the end-to-end delay of two different types of video traces, Cisco's point-to-point presence and WebEx over a 10-node path. The results obtained were compared against simulation, and have an average relative error of 4.24%. Using this method, we then obtained the minimum amount of bandwidth required to be allocated on each link along the path of the video stream, so that a given 95th percentile of the end-to-end delay is satisfied.