Recent experimental studies have shown that α-stable self-similar stochastic processes can accurately characterize various types of aggregate network traffic. Using this traffic modeling approach, we propose some probabilistic envelope processes that can be used to represent the resource demand of a traffic stream in performance evaluation studies. We illustrate the use of the proposed envelopes in resource allocation for data and video traffic and in the design of an admission control mechanism. From our analysis we conclude that the presence of heavy tails in the distribution of a traffic process has a severe impact on the dimensioning of network elements. © 2004 Elsevier B.V. All rights reserved.