The Infinte Source Model for Internet Traffic: Statistical Analysis and Limit Theorems
Joseph Horowitz, Walter A. Rosenkrantz · Methods and Applications of Analysis · 2002
The discovery by Willinger et.al. that the input process to an Ethernet cable exhibits, at least empirically, long range dependence (LRD) and self-similarity has produced many models of this phenomenon, along with many, sometimes ad-hoc, statistical methods to analyze them ([1],[5],[14], [17]).It has been observed, for example, that detecting long range dependence (LRD) in internet traffic by fitting a straight line to a variance time (VT) plot can give misleading results; in particular, this method appears to be biased towards LRD even when the underlying process is known to be short range dependent (SRD) ([7],[8], [15]).In this paper we consider VT plots in the context of a readily-interpretable model for internet traffic that avoids the use of Gaussian processes, such as fractional Brownian motion (fBm) or fractional Gaussian noise (fGn), which take on negative values, to model a process that does not (Riedi et al. [14]).This model, which has been studied (e.g., in Guerin et al. [5]) under the rubric "infinite source Poisson data traffic model", can capture the essential features of LRD and asymptotic self-similarity, but is simple enough so that closed-form expressions can be found for the variance-covariance function of the process.For our model, we show that the variance of the cumulative input is a nonlinear function of time in log-log scale, which explains the curved appearance of VT plots for simulated and empirical data (see Figure 3.1) .This suggests that nonlinear regression might yield more accurate estimates of the model parameters.These theoretical results are confirmed by simulations, real data, and the use of residual plots, the details of which were first reported in a separate paper [15].Following the referee's suggestion we have incorporated some of the numerical and graphical results presented there into Section 4 and summarized in Tables ((4.1), (4.2)).Our results, which apply to all SRD and LRD M/G/∞ processes, generalize those obtained by Krunz and Matta ([8]) for an important special case.A more detailed analysis of this model, given in Section 3, explains, simply and rigorously, why fitting a straight line to a VT plot is always biased towards LRD, even when the process is known to be SRD.A separate, but closely related question, is obtaining a limit theorem for the cumulative input traffic under "heavy traffic"that is consistent with empirical internet traffic measurements, a topic that has attracted much attention in the current literature ([5],[10],[14], [16],[17]).In some of these papers the limit process is not unique; it depends on how two parameters, the arrival rate (also called the connection rate) and the time scale, go to infinity.In this paper, we derive a central limit type theorem for the cumulative input process that differs significantly from those obtained by others.We show that as the arrival rate of network users tends to infinity the cumulative input process, when suitably normalized, tends (in the sense of weak convergence of processes) to a Gaussian limit which is not fBm and with a covariance function that depends on the service time distribution (Section 5).As a consequence of this result, *