  We present two Monte Carlo algorithms of the Markovian type which solve the
modified QCD evolution equations at the NLO level. The modifications with
respect to the standard DGLAP evolution concern the argument of the strong
coupling constant alpha_S. We analyze the z - dependent argument and then the
k_T - dependent one. The evolution time variable is identified with the
rapidity. The two algorithms are tested to the 0.05% precision level. We find
that the NLO corrections in the evolution of parton momentum distributions with
k_T - dependent coupling constant are of the order of 10 to 20%, and in a small
x region even up to 30%, with respect to the LO contributions.
