The Mystery of the Missing Scales: Pitfalls in the Use of Fractal Renewal Processes to Simulate LRD Processes

It has now been demonstrated in many studies that network traac exhibits properties consistent with Long Range Dependence (LRD) and self-similarity. While theoretical frameworks are currently being developed to estimate the performance of such systems, simulation will remain a valuable tool for validating these theoretical models, and providing insight into systems which are too complicated to eeectively model. Furthermore, when testing real systems, it is desirable to have traac sources which are realistic, and hence display self-similarity. The Fractal Renewal Process (FRP) and its variants (including On/OO processes and superpositions thereof) have been proposed as models for LRD processes, in particular for network traac. The FRP is a simple renewal point process with heavy-tailed inter-renewal times. The long-range correlations in the process are directly introduced by the heavy tail of the renewal times. The FRP has the great advantage that the number of computations required to generate a time series is linear and the time series can be generated on-line, facilitating generation of real traac. However, there are some problems which arise when using such processes to generate LRD traac. Most notably undersampling of the heavy-tailed random variables used to generate FRPs can lead to a truncation of the sampled autocorrelation that is not consistent with LRD. This problem becomes clear when the processes are investigated using the wavelet based methods of Abry and Veitch which segregate behaviour at diierent scales. This paper will describe the problem of undersampling, and its eeects, and methods for avoiding the problem.