The photon-avalanche effect, which involves both absorption from a metastable intermediate level and cross-relaxation energy transfer, introduces a new trend in the achievement of efficient up-conversion lasers. In this paper, we report a quite general theoretical treatment of this process which is successfully applied to the case of ${\mathrm{LiYF}}_{4}$:${\mathrm{Nd}}^{3+}$. This model shows clearly that the avalanche effect may occur, above a pumping threshold, only if the cross-relaxation energy-transfer probability is higher than the relaxation rate from the up-converted excited state to the levels located below the metastable intermediate state.