  Radio relics are Mpc-sized synchrotron sources located in the peripheral
regions of galaxy clusters. Models based on the diffuse shock acceleration
(DSA) scenario have been widely accepted to explain the formation of radio
relics. However, a critical challenge to these models is that most observed
shocks seem too weak to generate detectable emission, unless fossil electrons,
a population of mildly energetic electrons that have been accelerated
previously, are included in the models. To address this issue, we present a new
semi-analytical model to describe the formation and evolution of radio relics
by incorporating fossil relativistic electrons into DSA theory, which is
constrained by a sample of 14 observed relics, and employ the Press-Schechter
formalism to simulate the relics in a $20^{\circ} \times 20^{\circ}$ sky field
at 50, 158, and 1400 MHz, respectively. Results show that fossil electrons
contribute significantly to the radio emission, which can generate radiation
four orders of magnitude brighter than that solely produced by thermal
electrons at 158 MHz, and the power distribution of our simulated radio relic
catalog can reconcile the observed $P_{1400}-M_{\mathrm{vir}}$ relation. We
predict that $7.1\%$ clusters with $M_{\mathrm{vir}} > 1.2\times
10^{14}\,\mathrm{M}_{\odot}$ would host relics at 158 MHz, which is consistent
with the result of $10 \pm 6\%$ given by the LoTSS DR2. It is also found that
radio relics are expected to cause severe foreground contamination in future
EoR experiments, similar to that of radio halos. The possibility of AGN
providing seed fossil relativistic electrons is evaluated by calculating the
number of radio-loud AGNs that a shock is expected to encounter during its
propagation.
