  The scaling of the number of Rydberg excitations in a laser-driven cloud of
atoms with the interaction strength is found to be affected by the finite size
of the system. The scaling predicted by a theoretical model is compared with
results extracted from a numerical many-body simulation. We find that the
numerically obtained scaling exponent in general does not agree with the
analytical prediction. By individually testing the assumptions leading to the
theoretical prediction using the results from the numerical analysis, we
identify the origin of the deviations, and explain it as arising from the
finite size of the system. Furthermore, finite-size effects in the pair
correlation function $g^{(2)}$ are predicted. Finally, in larger ensembles, we
find that the theoretical predictions and the numerical results agree, provided
that the system is sufficiently homogeneous.
