  Prompted by various analyses of long (Type II) GRB rates and their
relationship to the cosmic star-formation history, metallicity and luminosity
function evolution, we systematically analyze these effects with a Monte Carlo
code. We test various cosmic star-formation history models including analytical
and empirical models as well as those derived from cosmological simulations. We
also explore expressions for metallicity enhancement of the GRB rate with
redshift, as presented in the literature, and discuss improvements to these
analytic expressions from the point of view of galactic evolution. These are
also compared to cosmological simulations on metal enrichment. Additionally we
explore possible evolutionary effects of the GRB rate and luminosity function
with redshift. The simulated results are tested with the observed Swift sample
including the L, z, and peak flux (log N-log P) distributions. The
observational data imply that an increase in the GRB rate is necessary to
account for the observations at high redshift, although the form of this
enhancement is unclear. A rate increase due to lower metallicity at higher
redshift may not be the singular cause and is subject to a variety of
uncertainties. Alternatively, evolution of the GRB luminosity function break
with redshift shows promise as a possible alternative.
