The effects of a variation in the mass accretion rate onto a neutron star of 1.41 solar masses in the neighborhood of the critical value is investigated, in order to quantify the differences in the resulting X-ray burst structure and recurrence interval. Two neutron star envelope models are chosen: one for which the hydrogen burning rate equals the accretion rate, and one for which the accretion rate equals the critical rate for which a combined hydrogen-helium shell flash is expected. Numerical results of the evolutionary sequences for each model, up to and including shell flash, are presented. For the steady-state model, pure helium flashes are characterized by long recurrence times, while for the other model the time scale appreciably decreases. The burst profile for the first model shows fast rise time, long duration, and luminosity saturation at the Eddington critical limit; the other model features short rise time, short duration, and no saturation.