Long-range dependence and heavy-tail modeling for teletraffic data

Analysis and modeling of computer network traffic is a daunting task considering the amount of available data. This is quite obvious when considering the spatial dimension of the problem, since the number of interacting computers, gateways and switches can easily reach several thousands, even in a Local Area Network (LAN) setting. This is also true for the time dimension: W. Willinger and V. Paxson in [42] cite the figures of 439 million packets and 89 gigabytes of data for a single week record of the activity of a university gateway in 1995. The complexity of the problem further increases when considering Wide Area Network (WAN) data [28]. In light of the above, it is clear that a notion of importance for modern network engineering is that of invariants, i.e. characteristics that are observed with some reproducibility and independently of the precise settings of the network under consideration. In this tutorial paper, we focus on two such invariants related to the time dimension of the problem, namely, long-range dependence, or selfsimilarity, and heavy-tail marginal distributions. Both characteristics arise in most “scalar signals” that can be extracted from complete network traffic traces [27, 35, 3, 43]. Typical “scalar signals” include continuous-time point processes constructed from recording the arrival times of successive IP (Internet Protocol) packets at some point of the network, or a time series obtained by counting the size of the data transferred during some time intervals. In order to illustrate and motivate the technical part of this tutorial, we begin with a statistical description of the traffic data that will be considered throughout the paper. A striking feature, which corroborates the conjecture that long-range dependence and heavy-tailness are really meaningful traffic invariants, is that they can be observed, to some extent, without using any specific experimental pro