  One of the most promising mechanisms for producing radio halos (RHs) in
galaxy clusters is the reacceleration of cosmic-ray electrons by turbulences.
However, the origin of the seed electrons for the reacceleration is still
poorly constrained. In the secondary scenario, most of the seed electrons are
injected via collision of proton cosmic-rays, while non-thermal electrons are
directly injected in the primary scenario. In this paper, we examine the two
scenarios for the seed electrons with the observed statistical properties of
RHs, combining two methods: following the temporal evolutions of the electron
energy and radial distributions in a cluster, and the merger history of
clusters. We find that the RH lifetime largely depends on the seed origin, as
it could be longer than the cosmological timescale in the secondary scenario.
We study the condition for the onset of RHs with the observed RH fraction and
the RH lifetime we obtained, and find that long-lived RHs in the secondary
scenario should be originated from major mergers with a mass ratio of
$\xi\sim0.1$, while the short lifetime in the primary scenario requires more
frequent onsets by minor mergers with $\xi\sim0.01$. Our simple model of the
turbulence acceleration can reproduce the observed radio luminosity-mass
relation. The RH luminosity functions we obtained suggest that the expected RH
number count with the ASKAP survey will detect $\approx10^3$ RHs in both the
scenarios.
