models, a class of models used to describe error distributions for the generalized linear model. These Tweedie distributions are characterized by an inherent scale invariance and thus for any random variable Y that obeys a Tweedie distribution, the variance var(Y) relates to the mean E(Y) by the power law, where a and p are positive constants. The exponent p for the variance to mean power law associated with certain self-similar stochastic processes ranges between 1 and 2 and thus may be modeled in part by a Tweedie compound Poisson–gamma distribution. The additive form of the Tweedie compound Poisson-gamma model has the cumulant generating function (CGF), , where , is the cumulant function, α is the Tweedie exponent , s is the generating function variable, θ is the canonical parameter and λ is the index parameter. The first and second derivatives of the CGF, with s=0, yields the mean and variance, respectively. One can thus confirm that for the additive models the variance relates to the mean by the power law, . Whereas this Tweedie compound Poisson-gamma CGF will represent the probability density function for certain self-similar stochastic processes, it does not return information regarding the long range correlations inherent to the sequence Y. Nonetheless, the Tweedie distributions provide a means understand the possible origins of self-similar stochastic processes for reason of their role as foci for a central limit-like convergence effect known as the Tweedie convergence theorem. In nontechnical terms this theorem tells us that any exponential dispersion model that asymptotically manifests a variance-to-mean power law is required to have a variance function that comes within the domain of attraction of a Tweedie model. The Tweedie convergence theorem can be used to explain the origin of the variance to mean power law, 1/f noise and multifractality, features associated with self-similar processes. Network performance Network performance degrades gradually with increasing self-similarity. The more self-similar the traffic, the longer the queue size. The queue length distribution of self-similar traffic decays more slowly than with Poisson sources. However, long-range dependence implies nothing about its short-term correlations which affect performance in small buffers. Additionally, aggregating streams of self-similar traffic typically intensifies the self-similarity ("burstiness") rather than smoothing it, compounding the problem. Self-similar traffic exhibits the persistence of clustering which has a negative impact on network performance. With Poisson traffic (found in conventional telephony networks), clustering occurs in the short term but smooths out over