New Bounds and Approximations Using Extreme Value Theory for the Queue Length Distribution in High-Speed Networks

In this paper we study P({Q > x}), the tail of the steady state queue length distribution at a high-speed multiplexer. The tail probability distribution P({Q > x}) is a fundamental measure of network congestion and thus important for the efficient design and control of networks. In particular, we focus on the case when the aggregate traffic to the multiplexer can be characterized by a stationary Gaussian process. In our approach, a multiplexer is modeled by a fluid queue serving a large number of input processes. We propose two asymptotic upper bounds for P({Q > x}), and provide several numerical examples to illustrate the tightness of these bounds. We also use these bounds to study important properties of the tail probability. Further, we apply these bounds for a large number of non-Gaussian input sources, and validate their performance via simulations. We have conducted our simulation study using Importance Sampling in order to improve its reliability and to effectively capture rare events. Our analytical study is based on Extreme Value Theory, and therefore different from the approaches using traditional Markovian and Large Deviations techniques.