  The observed powerlaw distributions of solar flare parameters can be
interpreted in terms of a nonlinear dissipative system in the state of
self-organized criticality (SOC). We present a universal analytical model of a
SOC process that is governed by three conditions: (i) a multiplicative or
exponential growth phase, (ii) a randomly interrupted termination of the growth
phase, and (iii) a linear decay phase. This basic concept approximately
reproduces the observed frequency distributions. We generalize it to a
randomized exponential-growth model, which includes also a (log-normal)
distribution of threshold energies before the instability starts, as well as
randomized decay times, which can reproduce both the observed occurrence
frequency distributions and the scatter of correlated parametyers more
realistically. With this analytical model we can efficiently perform
Monte-Carlo simulations of frequency distributions and parameter correlations
of SOC processes, which are simpler and faster than the iterative simulations
of cellular automaton models. Solar cycle modulations of the powerlaw slopes of
flare frequency distributions can be used to diagnose the thresholds and growth
rates of magnetic instabilities responsible for solar flares.
