this paper, we analyze a fluid or storage queueing system with LRD input. Fluid systems have been used before (e.g. Bensaou et al. [2], Guibert [7]) to model bursty traffic fed into ATM multiplexer queues, when considering time scales where the granularity of the ATM cells no longer dominates. The input sources are assumed to be of On/Off type, that is, with mutually independent, alternating silence periods (no work arriving) and activity periods (work arriving at a constant rate). We consider a superposition of N identical, independent On/Off sources flowing into an infinite reservoir with fixed output rate C. The object of study is the complementary distribution function Q of the stationary queue content. F. Brichet et al. / Heavy traffic analysis with LRD sources 3 If is the mean arrival rate for a single source, we require C ? N for stability. For N sufficiently large, C then exceeds the peak rate of an individual source. If this rate is proportional to C=N , so that it decreases with N , then we are in the realm of "small" sources. The M=G=1 example above represents the limiting form of the alternative assumption, where instantaneous arrivals idealize the case of a capacity significantly smaller than the individual arrival rate. By selecting "heavy" tails (as defined in section 4) for the silence and/or activity periods, the input process becomes long range dependent and the queueing problem is fundamentally non-Markovian. Despite this, a useful lower bound L to

Modern telecommunications networks are being designed to accomodate a heterogenous mix of traffic classes ranging from traditional telephone calls to video and data services. Thus, traffic models are of crucial importance to the engineering and performance analysis of telecommunications system, notably congestion and overload controls and capacity estimation. This chapter surveys teletraffic models, addressing both theoretical and computational aspects. It first surveys the main classes of teletraffic models commonly used in teletraffic modeling, and then proceeds to survey traffic methods for computing statistics relevant to the engineering a teletraffic network. 1 INTRODUCTION Traffic is the driving force of telecommunications systems, representing customers making phone calls, transferring data files and other electronic information, or more recently, transmitting compressed video frames to a display device. The most common modeling context is queueing; traffic is offered to a qu...

INTRODUCTION In the last decade a number of extensive studies of high resolution traffic measurements from a wide range of packet traffic networks have been reported [1,6,7,10,11,14,22]. The most important finding of these studies is the identified fractal-like behaviour implying the so called long-range dependence and self-similarity properties. As a result of intensive research at Bellcore a series of papers reported these findings in Ethernet LAN [7,9,11,12]. The comprehensive study of Leland's group with the conclusion that this traffic is self-similar was published in detail in [11]. The study of Duffy et al. [6] revealed the self-similarity traffic property in common-channel signalling network. Meier-Hellstern et al. [14] found that the Pareto distribution with infinite variance is applicable for characterizing the D-channel traffic in N-ISDN. Paxson et al. [20,22] reported the self-similar features of TCP traffic. The fractal properties also

A simple construction that will be referred to as an error duration model is shown to generate fractional integration and long memory. An error duration representation also exists for many familiar ARMA models, making error duration an alternative to autoregression for explaining dynamic persistence in economic variables. The results lead to a straightforward procedure for simulating fractional integration and establish a connection between fractional integration and common notions of structural change. Two examples show how the error duration model could account for fractional integration in aggregate employment and in asset price volatility.

Self-similar processes based on fractal point processes (FPPs) provide natural and attractive network tra#c models. We show that the point process formulation yields a wide range of FPPs which in turn yield a diversity of parsimonious, computationally e#cient, and highly practical asymptotic second-order self-similar processes. Using this framework, we show that the relevant second-order fractal characteristics such as long-range dependence (LRD), slowly-decaying variance, and 1/f noise are completely characterized by three fundamental quantities: mean arrival rate, Hurst parameter, and fractal onset time. Four models are proposed, and the relationship between their model parameters and the three fundamental quantities are analyzed. By successfully applying the proposed models to Bellcore's Ethernet traces, we show that the FPP models prove useful in evaluating and predicting the queueing performance of various types of fractal tra#c sources. Keywords: point process, fractal, self-similarity, long-range dependence, tra#c modeling 1 Throughout this paper, self-similarity refers to asymptotic second-order self-similarity [4], [13] unless otherwise defined. 1

This is an overview of a relatively recent application of long-range dependence (LRD) to the area of communication networks, in particular to problems concerned with the dynamic nature of packet flows in high-speed data networks such as the Internet. We demonstrate that this new application area offers unique opportunities for significantly advancing our understanding of LRD and related phenomena. These advances are made possible by moving beyond the conventional approaches associated with the wide-spread "black-box" perspective of traditional time series analysis and exploiting instead the physical mechanisms that exist in the networking context and that are intimately tied to the observed characteristics of measured network traffic. In order to describe this complexity we provide a basic understanding of the design, architecture and operations of data networks, including a description of the TCP/IP protocols used in today's Internet. LRD is observed in the large scale behavior of the data traffic and we provide a physical explanation for its presence. LRD tends to be caused by user and application characteristics and has little to do with the network itself. The network affects mostly small time scales, and this is why a rudimentary understanding of the main protocols is important. We illustrate why multifractals may be relevant for describing some aspects of the highly irregular traffic behavior over small time scales. We distinguish between a time-domain and wavelet-domain approach to analyzing the small time scale dynamics and discuss why the wavelet-domain approach appears to be better suited than the time-domain approach for identifying features in measured traffic (e.g., relatively regular traffic patterns over certain time scales) that have a direct networking interpretation (e....

We study the superposition process of a class of independent renewal processes with long-range dependence. It is known that under two different scalings in time and space either fractional Brownian motion or a stable Levy process may arise in the rescaling asymptotic limit. It is shown here that in a third, intermediate scaling regime a new limit process appears, which is neither Gaussian nor stable. The new limit process is characterized by its cumulant generating function and some of its properties are discussed.

Recent traffic measurement studies suggest that the self-similarity observed in packet traffic arises from aggregating individual sources which behave in an ON/OFF manner with heavy-tailed sojourn times in one or both of the states. In this paper, we investigate the connection between general ON/OFF behavior, self-similarity and queueing performance. We use chaotic maps to model general ON/OFF behavior with combinations of heavy tailed and light tailed sojourn time behavior. We present results which show that chaotic maps which capture the heavytailed sojourn time behavior in the OFF and/or ON states generate traffic that is asymptotically self-similar. However, the resulting queue length distribution decays as a power law with the heavy ON source, and as an exponential with the light ON source, even though both processes exhibit identical 1/f noise behavior. To resolve this apparent paradox, we consider aggregates of ON and OFF sources, and show that the nature of the ON period is le...

We study the effect of investor inertia on stock price fluctuations with a market microstructure model comprising many small investors who are inactive most of the time. It turns out that semi-Markov processes are tailor made for modeling inert investors. With a suitable scaling, we show that when the price is driven by the market imbalance, the log price process is approximated by a process with long range dependence and non-Gaussian returns distributions, driven by a fractional Brownian motion. Consequently, investor inertia may lead to arbitrage opportunities for sophisticated ‘third parties’. The mathematical contributions are a functional central limit theorem for stationary semi-Markov processes, and approximation results for stochastic integrals of continuous semimartingales with respect to fractional Brownian motion.

: Fluctuations of aggregated connectionless traffic are modelled with the fractional Brownian motion. The applicability and interpretation of the self-similar model are studied both by comparison with Ethernet traffic data from Bellcore and conceptually, defining an ideal notion of Free Traffic. Insight into the three model parameters is obtained by finding their counterparts in a burst model with heavy-tailed burst length distribution. A bandwidth allocation formula based on the corresponding queueing system is discussed. Guidelines for short-term traffic prediction are found using an exact formula for the conditional expectation of a future value given the past. Directions for further study are indicated at the end of each section. 1 Introduction: fractional Brownian traffic The principle of a connectionless network is that all data is sent in relatively small datagrams that are labeled with the destination address and routed independently. No bandwidth needs to be reserved. The con...