Recent experimental studies have shown that several traffic classes exhibit higher variability than the one captured by currently used traffic models. As a result of this, more flexible models based on α-stable stochastic processes have been proposed. They have been satisfactorily justified and proven to capture different levels of long-range dependence and variability. An important application of traffic models is to determine the magnitude of the network resource required to support data transfer with quality of service guarantees. To this end, some probabilistic envelope processes have been defined for earlier models. We generalize this concept to the α-stable case and describe some applications in the context of resource allocation. We also show that envelope processes help to understand the sharp difference in performance between theoretical queuing models with α-stable input and their real-life counterparts.