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Data Center Networks represent the convergence of computing and networking, of data and storage networks, and of packet transport mechanisms in Layers 2 and 3. Congestion control algorithms are a key component of data transport in this type of network. Recently, a Layer 2 congestion management algorithm, called QCN (Quantized Congestion Notification), has been adopted for the IEEE 802.1 Data Center Bridging standard: IEEE 802.1Qau. The QCN algorithm has been designed to be stable, responsive, and simple to implement. However, it does not provide weighted fairness, where the weights can be set by the operator on a per-flow or per-class basis. Such a feature can be very useful in multi-tenanted Cloud Computing and Data Center environments. This paper addresses this issue. Specifically, we develop an algorithm, called AF-QCN (for Approximately Fair QCN), which ensures a faster convergence to fairness than QCN, maintains this fairness at fine-grained time scales, and provides programmable weighted fair bandwidth shares to flows/flow-classes. It combines the QCN algorithm developed by some of the authors of this paper, and the AFD algorithm previously developed by Pan et. al. AF-QCN requires no modifications to a QCN source (Reaction Point) and introduces a very light-weight addition to a QCNcapable switch (Congestion Point). The results obtained through simulations and an FPGA implementation on a 1Gbps platform show that AF-QCN retains the good congestion management performance of QCN while achieving rapid and programmable (approximate) weighted fairness.