Several studies of file sizes either being downloaded or stored in the World Wide Web have commented that tails can be so heavy that not only are variances infinite, but so are means. Motivated by this fact, we study the infinite node Poisson model under the assumption that transmission times are heavy tailed with infinite mean. The model is unstable but we are able to provide growth rates. Self-similar but nonstationary Gaussian process approximations are provided for the number of active sources, cumulative input, buffer content and time to buffer overflow.
"Self-similar communication models and very heavy tails." Ann. Appl. Probab. 10 (3) 753 - 778, August 2000. https://doi.org/10.1214/aoap/1019487509