Measuring long-range dependence under changing traffic conditions

Previous measurements of various types of network traffic have shown evidence consistent with long-range dependence and self-similarity. However, an alternative explanation for these measurements is non-stationarity. Standard estimators of LRD parameters such as the Hurst parameter H assume stationarity and are susceptible to bias when this assumption does not hold. Hence LRD may be indicated by these estimators when none is present, or alternatively LRD taken to be non-stationarity. The Abry-Veitch (see IEEE Trans. on on Info. Theory, vol.44, no.1, p.2-15, 1998) joint estimator has much better properties when a time-series is non-stationary. In particular the effect of polynomial trends in data may be intrinsically eliminated from the estimates of LRD parameters. This paper investigates the behavior of the AV estimator when there are non-stationarities in the form of a level shift in the mean and/or the variance of a process. We examine cases where the change occurs both gradually or as a single jump discontinuity, and also examine the effect of the size of the shift. In particular we show that although a jump discontinuity may cause bins in the estimates of the H, the bias is negligible except when the jump is sharp, and large compared with the standard deviation of the process. We explain these effects and suggest how any introduced errors might be minimized. We define a broad class of non-stationary LRD processes so that LRD remains well defined under time varying mean and variance. The results are tested by applying the estimator to a real data set which contains a clear non-stationary event falling within this class