Results and conditions which quantify the decrease in dependence with lag for a stationary Markov process and enable one to compare the dependence for two stationary Markov processes are obtained. The notions of dependence used in this paper are the supermodular ordering and the concordance ordering. Both discretetime and continuous-time Markov processes are considered. Some applications of the main results are given. In queueing theory, the monotonicity results of the waiting time of the n-th customer as well as the stationary waiting time in an MR/GI/1 queue, and the stationary workload in a Markov-modulated queue are established, thus strengthening previous results while simplifying their derivation. This paper is a continuation of Fang, Hu and Joe [7] and Hu and Joe [10].