Multiple Time Scale Regulation and Worst Case

Recent research on congestion control in high speed ATM networks has sought to extract statistical multiplexing eeciencies while guaranteeing quality of service. To facilitate network control, traac regulators are positioned at network edges, with Leaky Bucket regulation ubiquitous. We consider the composite of conventional Leaky Bucket regulators, one for each time scale. Over longer time scales the enforcement of traac to the mean in the regulation is less stringent, in so far as the bucket is larger, while the mean rate is smaller. The regulated process is required to be compliant with all the Leaky Bucket regulators. In the approach taken here, traac sources are assumed to be mutually independent but otherwise adversarial to the utmost extent permitted by regulators. To mitigate against an overly conservative design, small losses are allowed in accessing the shared unbuuered network resource. We rely on the Chernoo bound, which also gives an asymptotic, large deviations approximation, to estimate the loss probability. This paper resolves a question regarding the constitution of independent, regulated traac processes which maximize the estimate of the loss probability. We show that the estimated loss probability is maximized by certain periodic on-oo processes with random phases. Importantly, this loss probability depends on the composite regulators' parameters only through the ratio of the mean rate in the longest time scale to the peak rate. This is true even in a heterogeneous environment.