The Concept of Relevant Time Scales

Recent traac analyses from various packet networks have shown the existence of long-range dependence in bursty traf-c. In evaluating its impact on queuing performance, earlier investigations have noted how the presence of long-range dependence , or a high value of the Hurst parameter H, is often associated with surprisingly large queue sizes. As a result, a common impression has been created of expecting queuing performance to be worse as H increases, but this impression can be misleading. In fact, there are examples in which larger values of H are associated with smaller queues. So the question is how can one tell whether queuing performance would improve or degrade as H rises? In this paper, we show that the relative queuing performance can be assessed by identifying a couple of time scales. First, in comparing a high-H process with a low-H process, there is a unique time scale tm at which the variances of the two processes match (assuming exact, second-order self similarity for both processes). Second, there are time scales tqi that are most relevant for queuing the arrivals of process i. If both of the queuing scales tqi exceed the variance-matching scale tm, then the high-H queue is worse; if the queuing scales are smaller, then the low-H queue is worse. However, no rm prediction can be made in the remaining case of tm falling between the two queuing scales. Numerical examples are given to demonstrate our results.