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In the ATLAS muon spectrometer, large drift-tube chambers are used for precision tracking. The chambers will be operated at a high neutron and γ background resulting in count rates of up to 500 Hz cm-2 corresponding to 300 kHz per tube. The spatial resolution of the drift tubes is degraded from 82 μm without background to 108 μm at 500 Hz cm-2 background count rate. Due to afterpulsing in the Ar/CO2 gas mixture used in the drift tubes, ionizing radiation causes more than one hit in a tube within the maximum drift time of about 800 ns which is expected for magnetic field strengths around 1.2 T. In order to limit the count rate, the drift tubes are read out with an artificial dead time of 790 ns which causes an efficiency loss of 23% at a rate of 300 kHz per tube. The space-to-drift-time relationship of the tubes varies with background rate, temperature, and magnetic field strength. The mean magnetic field strength in a muon chamber is 0.4 T on the average, but may vary by up to 0.4 T within a chamber. The space-to-drift-time relationship must therefore be determined in short time intervals with an accuracy better than 20 μm using muon tracks and applying corrections for measured magnetic field variations.