Jinwoo Choe

A central-limit-theorem-based approach for analyzing queue behavior in high-speed networks

In this paper, we study P(/spl Qscr/>x), the tail of the steady-state queue length distribution at a high-speed multiplexer. In particular, we focus on the case when the aggregate traffic to the multiplexer can be characterized by a stationary Gaussian process. We provide two asymptotic upper bounds for the tail probability and an asymptotic result that emphasizes the importance of the dominant time scale and the maximum variance. One of our bounds is in a single-exponential form and can be used to calculate an upper bound to the asymptotic constant. However, we show that this bound, being of a single-exponential form, may not accurately capture the tail probability. Our asymptotic result on the importance of the maximum variance and our extensive numerical study on a known lower bound motivate the development of our second asymptotic upper bound. This bound is expressed in terms of the maximum variance of a Gaussian process, and enables the accurate estimation of the tail probability over a wide range of queue lengths. We apply our results to Gaussian as well as multiplexed non-Gaussian input sources, and validate their performance via simulations. Wherever possible, 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.

Queueing analysis of high-speed multiplexers including long-range dependent arrival processes

With the advent of high-speed networks, a single link will carry hundreds or even thousands of applications. This results in a very natural application of the central limit theorem, to model the network traffic by a Gaussian stochastic processes. We study the tail probability P({Q>x}) of a queueing system when the input process is assumed to be a very general class of Gaussian processes which includes a large class of self similar or other types of long-range dependent Gaussian processes. For example, past work on fractional Brownian motion, and variations therein, are but a small subset of the work presented in this paper. This study is based on extreme value theory and we show that log P({Q>x})+m/sub x//2 grows at most on the order of logx, where m/sub x/ corresponds to the reciprocal of the maximum (normalized) variance of a Gaussian process directly related to the aggregate input process. The result is considerably stronger than the existing results in the literature based on large deviation theory, and we theoretically show that this improvement can be quite important in characterizing the asymptotic behavior of P({Q>x}). Through numerical examples, we also demonstrate that exp[-m/sub x//2] provides a very accurate estimate for a variety of long-range and short-range dependent input processes over the entire buffer range.

Use of the supremum distribution of Gaussian Processes in queueing analysis with long-range Dependence and self-similarity

In this paper we study the supremum distribution of a general class of Gaussian processes with stationary increments. This distribution is directly related to the steady state queue length distribution of a queueing system. Hence, its study is also important for different queueing applications such as delay analysis in communication networks. Our study is based on Extreme Value Theory and we show that asymptotically grows at most (on the order of) log x, where mx is the reciprocal of the maximum (normalized) variance of Xt This result is considerably stronger than the existing results in the literature based on Large Deviation Theory. We further show that this improvement can be critical in characterizing the asymptotic behavior of . Our results cover a large class of self-similar, short range dependent, and long-range dependent Gaussian processes

A new method to determine the queue length distribution at an ATM multiplexer

In this paper, we develop a simple analytical technique to determine P({Q>q}), the tail of the queue length distribution, at an ATM multiplexer. The ATM multiplexer is modeled as a fluid queue serving a large number of independent sources. Our method is based on the central limit theorem and the maximum variance approximation, and enables us to avoid the state explosion problem. The approach is quite general and not limited by a Markovian framework. We apply our analytical method to study the buffer behavior for various traffic sources such as multiplexed homogeneous and heterogeneous Markov modulated sources, sources that are correlated at multiple time scales, sources whose autocorrelation function exhibits heavy (sub-exponential) tail behavior, and sources generated from real MPEG-encoded video sequences.

New bounds and approximations using extreme value theory for the queue length distribution in high-speed networks

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.