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We study multimedia traffic classification into popular applications to assist the quality of service (QoS) support of networking technologies, including but not limited to, WiFi. For this purpose, we propose to model the multimedia traffic flow as a stochastic discrete-time Markov chain in order to take into account the strong sequentiality (i.e. the dependencies across the data instances) in the traffic flow observations. This addresses the shortcoming of the prior techniques that are based on feature extraction which is prone to losing the information of sequentiality. Also, for investigating the best application of our Markov approach to traffic classification, we introduce and test three data driven classification schemes which are all derived from the proposed model and tightly related to each other. Our first classifier has a global perspective of the traffic data via the likelihood function as a mixture of Markov components (MMC). Our second and third classifiers have local perspective based on k-nearest Markov components (kNMC) with the negative loglikelihood as a distance as well as k-nearest Markov parameters (kNMP) with the Euclidean distance. We additionally introduce to the use of researchers a rich multimedia traffic dataset consisting of four application categories, e.g., video on demand, with seven applications, e.g., YouTube. In the presented comprehensive experiments with the introduced dataset, our local Markovian approach kNMC outperforms MMC and kNMP and provides excellent classification performance, 89% accuracy at the category level and 85% accuracy at the application level and particularly over 95% accuracy for live video streaming. Thus, in test time, the nearest Markov components with the largest likelihoods yield the most discrimination power. We also observe that kNMC significantly outperforms the state-of-the-art methods (such as SVM, random forest and autoencoder) on both the introduced dataset and benchmark dataset both at the category and application levels.