Currently, simulation is usually used to estimate network degree distribution P(k) and to examine if a network model predicts a scale-free network when an analytical formula does not exist. An alternative Markovian chain-based numerical method was proposed by Shi et al. [Phys. Rev. E 71, 036140(2005)] to compute time-dependent degree distribution P(k,t). Although the numerical results demonstrate a quick convergence of P(k,t) to P(k) for the Barabasi-Albert model, the crucial issue on the rate of convergence has not been addressed formally. In this paper, we propose a simpler Markovian iterative method to compute P(k,t) for a class of growing network models. We also provide an upper bound estimation of the error of using P(k,t) to represent P(k) for sufficiently large t, and we show that with the iterative method, the rate of convergence of P(k,t) is root linear.