and Analysis of Self-Similar Traffic-- Part I Model Construction

ferent manifestations and instances of self-similarity yield di erent results. This appears to stem from the nature of self-similar processes themselves. This study focuses on explaining the dynamics of individual arrivals giving rise to (second-order) selfsimilarity, using a point processes formulation. Thus, the fractal point processes (FPPs) introduced in this paper reveals how self-similarity arises from individual packet arrivals in the packet-switching networks. They also enjoy the properties of parsimonious modeling and fast simulation, and have direct applications in modeling Ethernet tra c using TCP/IP (data 18] and video 20]) as well as long-range dependent VBR video tra c over ATM 19]. We begin by introducing the family of fractal point processes (FPPs), whose increments processes are (second-order) self-similar. These models naturally fall into two classes, one based on renewal processes, and the other on doubly stochastic Poisson processes (DSPPs). Next, we construct and analyze four FPP models: the fractal renewal process (FRP), the superposition of several fractal renewal processes (SupFRP), the fractal-shot-noise-driven Poisson process (FSNDP), and the fractal-binomial-noise-driven Poisson process (FBNDP). Finally, we present simulation results for all FPPs, including comparison with theory. We call a point process fractal when a number of the relevant statistics exhibit scaling with related scaling exponents, indicating that the represented phenomenon contains clusters of points over all (or a relatively large set of) time or length scales 14, 15]. Such scaling leads mathematically to power-law dependencies in the scaled quantities 10, 14]. Thus fractals and power-law forms of their statistics are closely related. Each statistic which scales will therefore provide an exponent; for a (mono-) fractal process all are simply related, yielding a single fractal exponent for the process. For a general point process, fractal scaling in one statistic does not necessarily imply fractal scaling in other statistics; if scaling exists in only one statistic, then we do not call this process fractal 10, 14].
Bong K. Ryu Steven B. Lowen Department of Electrical Engineering and Center for Telecommunications Research Columbia University, New York, NY 10027 fryu,steve6g@ We propose four fractal point processes (FPPs) as novel approaches to modeling and analyzing various types of self-similar tra c: the fractal renewal process (FRP), the superposition of several fractal renewal processes (Sup-FRP), the fractal-shot-noise-driven Poisson process (FSNDP), and the fractal-binomialnoise-driven Poisson process (FBNDP). These models fall into two classes depending on their construction. Study of these models provides a thorough understanding of how self-similarity arises in computer network tra c. We nd that (i) all these models are (second-order) self-similar in nature; (ii) the Hurst parameter alone does not fully capture the burstiness of a typical self-similar process; (iii) the heavy-tailed property is not a necessary condition to yield selfsimilarity; and (iv) these models permit parsimonious modeling (using only 2{5 parameters) and fast simulation. Simulation veri es that these models exhibit fractal behavior over a wide range of time scales.
Abstract
1 Introduction
It is expected that future high-speed networks will support a wide variety of services that exhibit extremely diverse tra c characteristics. Characterizing the statistical behavior of a tra c source is crucial to the proper design of high-speed networks, ensuring that they provide the pre-negotiated quality of service (QOS) to users while achieving high network utilization. Traditional tra c models, most of which assume Markov characteristics, have been used extensively; in many cases they prove adequate for evaluating network performance (see, for example, 3, 5] and references therein). However, traditional Markov-based models often prove inadequate to the task of e ciently characterizing the diverse tra c now encountered over current teletra c networks 1, 7, 18, 20, 22]. Recent analyses of high-quality tra c measurements have revealed the prevalence of self-similarity (or long-range dependence) in data 7] and compressed video 1]. Despite the signi cant attention self-similarity has received recently, a clear picture of its nature and importance has not yet emerged 3, 4, 6, 7, 17, 19]. Dif-