The size distribution of waiting times are found to have an exponential distribution in the case of a stationary Poissonian process. In reality, however, the waiting time distributions reveal power law-like distribution functions, which can be modeled in terms of non-stationary Poisson processes by a superposition of Poissonian distribution functions with time-varying event rates. We model the time evolution of such waiting time distributions by polynomial, sinusoidal, and Gaussian functions, which have exact analytical solutions in terms of the incomplete Gamma function, as well as in terms of the Pareto type-II approximation, which has a power law slope of , where represents the linear time evolution, or with representing nonlinear growth rates, which have a power law slope of . Our mathematical modeling confirms the existence of significant deviations from ideal power law size distributions (of waiting times), but no correlation or significant interval–size relationship exists, as would be expected for a simple (linear) energy storage-dissipation model.

The occurrence frequency distributions (size distributions) are the most important diagnostics for self-organized criticality systems. There are at least three formats for size distributions: (i) the differential size distribution function, (ii) the cumulative size distribution function, and (iii) the rank-order plot. Each of the three formats (or methods) has at least three ranges of event sizes: (i) a range with statistically incomplete sampling; (ii) an inertial range or power law fitting range with statistically complete sampling; and (iii) a range bordering finite system sizes. Only the intermediate range with power law behavior should be used to determine the power law slope from fitting the observed size distributions. The establishment of power law functions in a given observed size distribution depends crucially on the choice of the fitting range, which should have a logarithmic range of at least 2–3 decades. Often the fitted distribution functions exhibit significant deviations from an ideal power law and can be fitted better with alternative functions, such as log-normal distributions, Pareto type-II distributions, and Weibull distributions.