  The High Energy Stereoscopic System (H.E.S.S.) Galactic plane survey (HGPS)
is to date the most comprehensive census of Galactic $\gamma$-ray sources at
very high energies (VHE; $100\,\mathrm{GeV}\leq E\leq 100\,\mathrm{TeV}$). As a
consequence of the limited sensitivity of this survey, the 78 detected
$\gamma$-ray sources comprise only a small and biased subsample of the overall
population. Still, numerical modelling allows us to study the VHE $\gamma$-ray
source population in the Milky Way based on this sample. In this work, several
azimuth-symmetric and spiral-arm models are compared for spatial source
distribution. The luminosity and radius function of the population are derived
from the source properties of the HGPS data set and are corrected for the
sensitivity bias of the HGPS. Based on these models, VHE source populations are
simulated and the subsets of sources detectable according to the HGPS are
compared with HGPS sources. The power-law indices of luminosity and radius
functions are determined to range between $-1.6$ and $-1.9$ for luminosity and
$-1.1$ and $-1.6$ for radius. Azimuth-symmetric distributions and a
distribution following a four-arm spiral structure without Galactic bar
describe the HGPS data reasonably well. The total number of Galactic VHE
sources is predicted to be in the range from 800 to 7000 with a total
luminosity and flux of $(1.6-6.3) \cdot 10^{36}$~ph~s$^{-1}$ and $(3-15) \cdot
10^{-10}$~ph~cm$^{-2}$~s$^{-1}$, respectively. Depending on the model, the HGPS
sample accounts for $(68-87)\%$ of the emission of the population in the
scanned region. This suggests that unresolved sources represent a critical
component of the diffuse emission measurable in the HGPS. With the foreseen
jump in sensitivity of the Cherenkov Telescope Array, the number of detectable
sources is predicted to increase by a factor between 5 - 9.
