Due to the continuous growth of Internet and online applications, network traffic classification is not only becoming one of the most crucial disciplines in network management but is also becoming quintessential for providing advanced tasks such as Quality of Service and network security. Moreover, even though many studies have been undertaken in recent years, real-time encrypted traffic classification continues to be an important challenge in the field of network traffic classification. Therefore, in this paper a real-time network traffic classification system is proposed together with five new models. The real-time classification system classified each incoming real-time packets into appropriate classes of interest and the five new models make use of a cost-sensitive learning strategy to deal with the unbalanced data problem during the training phase. The proposed models, which are called Cost-Sensitive Long-Short Term Memory (CSLSTM), Cost-Sensitive Gated Recurrent Unit (CSGRU), Cost-Sensitive Convolution Neural Network (CSCNN), CSNN with LSTM and CSCNN with GRU, can handle both traffic categorization and application identification. These proposed models were compared with prominent methods in this field and the proposed CSCNN was observed to outperform the researched deep learning models by at least 4% to 16% in correctly classifying packets from the ISCX VPN-nonVPN dataset.