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Accurate traffic demand estimation is critical for the dynamic evaluation and optimization of signalized intersections. Existing studies based on connected vehicle (CV) data are designed for a single phase only and have not sufficiently studied the real-time traffic demand estimation for oversaturated traffic conditions. Therefore, this study proposes a cycle-by-cycle multi-phase traffic demand joint estimation method at signalized intersections based on CV data that considers both undersaturated and oversaturated traffic conditions. First, a joint weighted likelihood function of traffic demands for multiple phases is derived given real-time observed CV trajectories, which considers the initial queue and relaxes the first-in-first-out assumption by treating each queued CV as an independent observation. Then, the sample size of the historical CVs is used to derive a joint prior distribution of traffic demands. Ultimately, a joint estimation method based on the maximum a posteriori (i.e., the JO-MAP method) is developed for cycle-based multi-phase traffic demand estimation. The proposed method is evaluated using both simulation and empirical data. Simulation results indicate that the proposed method can produce reliable estimates under different penetration rates, arrival patterns, and traffic demands. The feature of joint estimation makes our method less demanding for the penetration rate of CVs and the consideration of prior distribution can significantly improve the estimation accuracy. Empirical results show that the proposed method achieves accurate cycle-based traffic demand estimation with a MAPE of 12.73%, outperforming the other four methods., Comment: 30 pages, 11 figures