  A simple physical model for long-duration gamma ray bursts (GRBs) is used to
fit the redshift (z) and the jet opening-angle distributions measured with
earlier GRB missions and with Swift. The effect of different sensitivities for
GRB triggering is sufficient to explain the difference in the z distributions
of the pre-Swift and Swift samples, with mean redshifts of <z> ~1.5 and
<z>~2.7, respectively. Assuming that the emission properties of GRBs do not
change with time, we find that the data can only be fitted if the comoving
rate-density of GRB sources exhibits positive evolution to z >~ 3-5. The mean
intrinsic beaming factor of GRBs is found to range from ~34-42, with the Swift
average opening half-angle <\theta_j> ~10 degree, compared to the pre-Swift
average of <\theta_j> ~7 degree. Within the uniform jet model, the GRB
luminosity function is proportional to L^{-3.25}_*, as inferred from our best
fit to the opening angle distribution. Because of the unlikely detection of
several GRBs with z <~ 0.25, our analysis indicates that low redshift GRBs
represent a different population of GRBs than those detected at higher
redshifts. Neglecting possible metallicity effects on GRB host galaxies, we
find that ~1 GRB occurs every 600,000 yrs in a local L_* spiral galaxy like the
Milky Way. The fraction of high-redshift GRBs is estimated at 8-12% and 2.5-6%
at z >= 5 and z >= 7, respectively, assuming continued positive evolution of
the GRB rate density to high redshifts.
