  The chemical abundances of Milky Way's satellites reflect their star
formation histories (SFHs), yet, due to the difficulty of determining the ages
of old stars, the SFHs of most satellites are poorly measured. Ongoing and
upcoming surveys will obtain around ten times more medium-resolution spectra
for stars in satellites than are currently available. To correctly extract SFHs
from large samples of chemical abundances, the relationship between chemical
abundances and SFHs needs to be clarified. Here, we perform a high-resolution
cosmological zoom-in simulation of a Milky Way-like galaxy with detailed models
of star formation, supernova feedback, and metal diffusion. We quantify SFHs,
metallicity distribution functions, and the $\alpha$-element (Mg, Ca, and Si)
abundances in satellites of the host galaxy. We find that star formation in
most simulated satellites is quenched before infalling to their host. Star
formation episodes in simulated satellites are separated by a few hundred Myr
owing to supernova feedback; each star formation event produces groups of stars
with similar [$\alpha$/Fe] and [Fe/H]. We then perform a mock observation of
the upcoming Subaru Prime Focus Spectrograph (PFS) observations. We find that
Subaru PFS will be able to detect distinct groups of stars in [$\alpha$/Fe] vs.
[Fe/H] space, produced by episodic star formation. This result means that
episodic SFHs can be estimated from the chemical abundances of $\gtrsim$ 1,000
stars determined with medium-resolution spectroscopy.
