Based on microscopic considerations, an analytic statistical model is presented for the study of the temporal variation of up-conversion luminescence produced by the interaction of excited state Ln3+ optical centers in monodoped amorphous materials. The resulting expressions are presented in terms of known mathematical functions, and they account for the detailed dependence on the system's experimental variables. The model allows a correct prediction of the shape of the up-conversion transient curves, and also accounts not only for the dependence of the up-conversion intensity on the square of the optical center concentration and of the external excitation intensity at relatively low pump power, but also allows the prediction of "saturation" effects with increasing values of these factors, a situation comparable to what happens with steady state luminescence. Finally, interpretations of the reproduced observables from a microscopic viewpoint are given.