Markovian MC simulation of QCD evolution at NLO level with minimum k_T

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.