In this paper, we propose a quality-centric congestion control for multimedia streaming over wired IP networks, which we refer to as media-TCP-friendly congestion control (MTCC). Our solution adapts the sending rate to both the network condition and the application characteristics by explicitly considering the distortion impacts, delay deadlines, and interdependencies of different video packet classes. The media-aware solution is able to provide differential services for transmitting various packet classes and thereby, further improves the multimedia streaming quality compared to the conventional network-aware congestion control. We use finite-horizon Markov decision process (FHMDP) to determine the optimal congestion control policy that maximizes the long-term multimedia quality, while adhering to the horizon- <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$K$</tex></formula> TCP-friendliness constraint, which ensures long-term fairness with existing TCP applications. Moreover, the proposed MTCC is able to achieve quality-based fairness among multimedia users. We derive sufficient conditions for multiple multimedia users to achieve quality-based fairness using MTCC congestion control. Note that the proposed solution only modifies the adaptation mechanism of the TCP congestion window size at the sender, without changing the design at the receiver side (i.e., each current TCP receiver can correctly receive and process MTCC streams). Our simulation results show that MTCC achieves more than 3 dB improvement in terms of PSNR over the conventional TCP congestion control approaches, with the largest improvements observed for real-time streaming applications requiring stringent playback delays.