Multiscale Analysis and Control of Networks with Fractal Traffic

A recently-introduced multiscale framework is used to develop efficient analysis and design tech-' niques for networks with self-similar traffic. These allow the interarrival density function for fractal point processes under Bernoulli random erasure to be determined, as well as the counting process distribution for superpositions of these processes. The results suggest that fractal characteristics are preserved under traffic branching and merging, which may, in turn, provide insight into the prevalence of self-similarity in aggregate traffic broadly observed on real networks. Multiscale techniques are also developed for analyzing fractal queueing scenarios. These are used to obtain, as examples, the steady-state customer distribution for a memoryless queue servicing selfsimilar arrivals, and for Poisson customers serviced with self-similar holding times. The persistent memory inherent in the underlying point processes leads to substantially different behavior than is observed in traditional queueing scenarios, and important implications on resource consumption and quality of service are discussed. We show how multiscale methods can be used in conjunction with dynamic programming techniques to develop efficient and practical control policies for these fractal queueing scenarios. In particular, optimal server control is developed for a memoryless queueing system with self-similar traffic input, and optimal flow control is formulated for self-similar service of memoryless traffic. By exploiting past history, these controllers achieve substantially better performance-both in terms of quality of service and resource utilization-than traditionally used queueing control strategies.