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This work is an attempt to elucidate effects that may limit efficiency in magnetrons operated at relativistic voltages (V ~ 500 kV). Three-dimensional (3-D) particle-in-cell (PIC) simulation is used to investigate the behavior of 14- and 22-cavity, cylindrical, rising-sun magnetrons. Power is extracted radially through a single iris located at the end of every other cavity. Numerical results show that in general output power and efficiency increase approximately linearly with increasing iris width (decreasing vacuum Q) until the total Q becomes too low for stable oscillation in the [Pi]-mode to be maintained. Beyond this point, mode competition or switching occur and efficiency decreases. Results reveal that the minimum value of Q (maximum efficiency) that can be achieved prior to the onset of mode competition is significantly affected by the magnitude of the 0-space-harmonic of the [Pi]-mode, a unique characteristic of rising-suns, and by the magnitude of the electron current density (space-charge effects). When these effects are minimized, numerical simulation predicts that gigawatt-level power production at 55% efficiency may be possible at relativistic voltage. Index Terms--High-power microwaves, particle-in-cell simulation, relativistic magnetron, rising-sun magnetron.