Weak convergence of high-speed network traffic models

We consider a network traffic model consisting of an infinite number of sources linked to a server. Sources initiate transmissions to the server at Poisson time points. The duration of each transmission has a heavy-tailed distribution. We show that suitable scalings of the traffic process converge to a totally skewed stable Levy motion in Skorohod space, equipped with the Skorohod M-1 topology. This allows us to prove a heavy-traffic theorem for a single-server fluid model.