A spectral theory approach for extreme value analysis in a tandem of fluid queues

We consider a model for to evaluate performance of streaming media over an unreliable network. Our model consists of a tandem of two fluid queues. The first fluid queue is a Markov Modulated fluid queue that models the network congestion, and the second queue represents the play-out buffer. For this model the distribution of the total amount of fluid in the congestion and play-out buffer corresponds to the distribution of the maximum attained level of the first buffer. We show that, under proper scaling and when we let time go to infinity, the distribution of the total amount of fluid converges to a Gumbel extreme value distribution. From this result, we derive a simple closed-form expression for the initial play-out buffer level that provides a probabilistic guarantee for undisturbed play-out. 1 Motivation and literature Over the past few years, the tremendous popularity of smart phone end devices and services (like Youtube) has boosted the demand for streaming media applications offered via the Internet. One of the key requirements for the success J.W. Bosman · R. Núñez-Queija Centrum Wiskunde & Informatica (CWI), P.O. Box 94079, 1090 GB Amsterdam Tel.: +31 20 592 9333 E-mail: {J.W.Bosman, sindo}@cwi.nl J.W. Bosman Faculty of Sciences, VU University Amsterdam R. Núñez-Queija Korteweg-de Vries Institute for Mathematics, University of Amsterdam, E-mail: [email protected] 2 J.W. Bosman, R. Núñez-Queija IP Network φ(t) Playout buffer Rplay Streaming video with rate Rplay X(t): process representing packets on flight φ(t): Speed determined by CTMC Y (t): Playout buffer process Playback rate V (t)