e of exponentials The research was performed while this author was a Ph.D student at the Technion - Israel Institute of Technology This work was supported by the Israel Science Foundation administrated by the Academy of Science and Humanities 2 Starobinski and Sidi / Modeling and Analysis of Power-Tail Distributions 1. Introduction Recent studies have revealed that network trac exhibits burstiness over multiple time scales [15,22]. In many circumstances, power-tail probability distributions have been found appropriate for capturing this salient feature (see [19] and references therein). A random variable X has a power-tail distribution if its complementary cumulative distribution function (ccdf) F (t) satises F (t) = PrfX > tg ct as t !

This paper analyzes how TCP congestion control can propagate self-similarity between distant areas of the Internet. This property of TCP is due to its congestion control algorithm, which adapts to self-similar uctuations on several timescales. The mechanisms and limitations of this propagation are investigated, and it is demonstrated that if a TCP connection shares a bottleneck link with a self-similar background trac ow, it propagates the correlation structure of the background trac ow above a characteristic timescale. The cut-o timescale depends on the end-to-end path properties, e.g., round-trip time and average window size. It is also demonstrated that even short TCP connections can propagate long-range correlations eectively. Our analysis reveals that if congestion periods in a connection's hops are long-range dependent, then the end-user perceived end-to-end trac is also long-range dependent and it is characterized by the largest Hurst exponent. Furthermore, it is shown that...

In this paper we present an overview of the progress made using chaotic maps to model individual and aggregated self-similar traffic streams and in particular their impact on queue performance. Our findings show that the asymptotic behaviour of the queue is a function only of the tail of the ON active periods, and that the Hurst parameter is not a good parameter to achieve traffic control: two different selfsimilar traffic traces can have the same Hurst parameter but have a very different effect on the queue statistics. These results are part of a framework for developing chaotic control of networks.

The thesis proposes models for aggregate data network traffic which incorporate the additional randomness arising from the randomness in the number of data sources. A conditionally-Gaussian scale mixture process is shown to be a limit for the cumulative work from a random superposition of alternating on-off processes. Sub-Fractional Brownian Motion is shown to be the limit in a particular case. Queueing and estimation results for processes which are conditionally Fractional Gaussian Noise are included. A model with a superposition of alternating on-off processes with independent lifetimes is also considered.

We consider an M/M/1 queue in a semi-Markovian environment. The environment is modeled by a two-state semiMarkov process with arbitrary sojourn time distributions F0 (x) and F1 (x). When in state i = 0; 1, customers are generated according to a Poisson process with intensity i and customers are served according to an exponential distribution with rate i . Using the theory of Riemann-Hilbert boundary value problems we compute the z-transform of the queue-length distribution when either F0 (x) or F1 (x) has a rational Laplace-Stieltjes transform and the other may be a general | possibly heavy-tailed | distribution. The arrival process can be used to model bursty trac and/or trac exhibiting long-range dependence, a situation which is commonly encountered in networking. The closed-form results lend themselves for numerical evaluation of performance measures, in particular the mean queue-length.

The impact of bursty traffic on queues is investigated in this paper. We consider a discrete-time single server queue with an infinite storage room, that releases customers at the constant rate of c customers/slot. The queue is fed by an M/G/1 process. The M/G/1 process can be seen as a process resulting from the superposition of infinitely many "sessions": sessions become active according to a Poisson process; a station stays active for a random time, with probability distribution G, after which it becomes inactive. The number of customers entering the queue in the time-interval [t; t + 1) is then defined as the number of active sessions at time t (t = 0, 1, ...) or, equivalently, as the number of busy servers at time t in an M/G/1 queue, thereby explaining the terminology. The M/G/1 process enjoys several attractive features: First, it can display various forms of dependencies, the extent of which being governed by the service time distribution G. The heavier the tail of G, the more bursty the M/G/1 process.

We present a multiplicative multifractal process to model traffic which exhibits long-range dependence. Using traffic trace data captured by Bellcore from operations across local and wide area networks, we examine the interarrival time series and the packet length sequences. We also model the frame size sequences of VBR video traffic process. We prove a number of properties of multiplicative multifractal processes that are most relevant to their use as tra$c models. In particular, we show these processes to characterize e!ectively the long-range dependence properties of the measured processes. Furthermore, we consider a single server queueing system which is loaded, on one hand, by the measured processes, and, on the other hand, by our multifractal processes (the latter forming a MF � /MF � /1 queueing system model). In comparing the performance of both systems, we demonstrate our models to effectively track the behaviour exhibited by the system driven by the actual traffic processes. We show the multiplicative multifractal process to be easy to construct. Through parametric dependence on one or two parameters, this model can be calibrated to fit the measured data. We also show that in simulating the packet loss probability, our multifractal traffic model provides a better fit than that obtained by using a fractional Brownian motion model.