  The self-organized criticality in Ehrenfest's historical dog-flea model is
analyzed by simulating the underlying stochastic process. The fluctuations
around the thermal equilibrium in the model are treated as avalanches. We show
that the distributions for the fluctuation length differences at subsequent
time steps are in the shape of a $q$-Gaussian (the distribution which is
obtained naturally in the context of nonextensive statistical mechanics) if one
avoids the finite size effects by increasing the system size. We provide a
clear numerical evidence that the relation between the exponent $\tau$ of
avalanche size distribution obtained by maximum likelihood estimation and the
$q$ value of appropriate q-Gaussian obeys the analytical result recently
introduced by Caruso et al. [Phys. Rev. E \textbf{75}, 055101(R) (2007)]. This
rescues the q parameter to remain as a fitting parameter and allows us to
determine its value a priori from one of the well known exponents of such
dynamical systems.
