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The over-smoothing problem is an obstacle of developing deep graph neural network (GNN). Although many approaches to improve the over-smoothing problem have been proposed, there is still a lack of comprehensive understanding and conclusion of this problem. In this work, we analyze the over-smoothing problem from the Markov chain perspective. We focus on message passing of GNN and first establish a connection between GNNs and Markov chains on the graph. GNNs are divided into two classes of operator-consistent and operator-inconsistent based on whether the corresponding Markov chains are time-homogeneous. Next we attribute the over-smoothing problem to the convergence of an arbitrary initial distribution to a stationary distribution. Based on this, we prove that although the previously proposed methods can alleviate over-smoothing, but these methods cannot avoid the over-smoothing problem. In addition, we give the conclusion of the over-smoothing problem in two types of GNNs in the Markovian sense. On the one hand, operator-consistent GNN cannot avoid over-smoothing at an exponential rate. On the other hand, operator-inconsistent GNN is not always over-smoothing. Further, we investigate the existence of the limiting distribution of the time-inhomogeneous Markov chain, from which we derive a sufficient condition for operator-inconsistent GNN to avoid over-smoothing. Finally, we design experiments to verify our findings. Results show that our proposed sufficient condition can effectively improve over-smoothing problem in operator-inconsistent GNN and enhance the performance of the model.