  We present Monte-Carlo simulations of a cosmological population of repeating
fast radio burst (FRB) sources whose comoving density follows the cosmic star
formation rate history. We assume a power-law model for the intrinsic energy
distribution for each repeating FRB source located at a randomly chosen
position in the sky and simulate their dispersion measures (DMs) and
propagation effects along the chosen lines-of-sight to various telescopes. In
one scenario, an exponential distribution for the intrinsic wait times between
pulses is chosen, and in a second scenario we model the observed pulse arrival
times to follow a Weibull distribution. For both models we determine whether
the FRB source would be deemed a repeater based on the telescope sensitivity
and time spent on follow-up observations. We are unable to rule out the
existence of a single FRB population based on comparisons of our simulations
with the longest FRB follow-up observations performed. We however rule out the
possibility of FRBs 171020 and 010724 repeating with the same rate statistics
as FRB 121102 and also constrain the slope of a power-law fit to the FRB energy
distribution to be $-2.0 < \gamma <-1.0$. All-sky simulations of repeating FRB
sources imply that the detection of singular events correspond to the bright
tail-end of the adopted energy distribution due to the combination of the
increase in volume probed with distance, and the position of the burst in the
telescope beam.
