  In the single degenerate (SD) scenario for type Ia supernovae (SNe Ia), a
mass-accreting white dwarf is expected to experience a supersoft X-ray source
(SSS) phase. However, some recent observations showed that the expected number
of the mass-accreting WD is much lower than that predicted from theory,
whatever in spiral or elliptical galaxies. In this paper, we did a binary
population synthesis study on the relative duration of the SSS phase to their
whole mass-increasing phase of WDs leading to SNe Ia. We found that for about
40% progenitor systems, the relative duration is shorter than 2% and the
evolution of the mean relative duration shows that it is always smaller than
5%, whatever for young or old SNe Ia. In addition, before SNe Ia explosion,
more than 55% progenitor systems are experiencing a dwarf novae phase, and only
no more than 10% is staying SSS phase. These results are consistent with the
recent observations, and imply that both in early- and late-type galaxies, only
a small fraction of mass-accreting WD resulting in SNe Ia contribute to the
supersoft X-ray flux. So, although our results are not directly related to the
X-ray output of SN Ia progenitor, the low supersoft X-ray luminosity observed
in early type galaxies may have no ability to exclude the validity of SD model.
On the contrary, it is evidence to support the SD scenario.
