Creating stable and efficient compact X-ray sources based on fast capillary discharges that do not incorporate preliminary ionization circuits poses additional restrictions on parameters of voltage pulses and capillary geometry. Applying a voltage pulse with a rise rate of the order of 1 kV/ns results in gradual breakdown of non-ionized gas in the capillary which takes the form of an ionization wave that initiates at the powered electrode and propagates with typical velocities of 1 cm/ns. After the wave reaches the grounded electrode, a plasma channel with gradually increasing conductivity is formed. The current onset therefore appears only after a certain time delay after beginning of the voltage pulse. The ratio between the delay and the applied voltage rise-time will eventually influence the current rise rate that defines plasma heating and compression. It is therefore necessary to have the ability to estimate this delay time for a given capillary geometry and understand its dependence on the properties of a voltage pulse. In this work numerical simulations of fast ionization waves created in an extended Al₂O₃ capillary filled with nitrogen at 2 Torr were performed for cases of voltage pulses of negative polarity with rise-times varying in the range 10-50 ns. The numerical model was based on fluid approach with drift-diffusion approximation for charged particle fluxes. Influence of voltage rise-time on initiation and propagation of a fast ionization wave as well as on consequent rate of current rise is investigated.