  We revisit the neutrino and ultra high-energy cosmic ray (UHECR) production
from gamma-ray bursts (GRBs) with time-dependent simulations for the
proton-induced cascades. This method can generate self-consistent photon,
neutrino and escaped neutron spectra. To obtain the integrated background
spectra, we take into account the distributions of the burst luminosity and
pulse duration timescale. A benchmark case with standard GRB luminosity
function, a bulk Lorentz factor $\Gamma=300$ and a proton to gamma-ray
luminosity fraction $f_{\rm p}=10$, is consistent with both the neutrino
upper-limits and the observed UHECR intensity at $\sim 10^{20}$ eV, while
requiring a different type of UHECR source at the ankle. For the benchmark case
the GRBs in the bright end of the luminosity function, which contribute most of
the neutrinos, have their photon spectrum substantially distorted by secondary
photons. Such bright GRBs are few in number, and reducing their $f_p$
eliminates the distortion, while reducing the neutrino production. Even if we
neglect the contribution of the brightest GRBs, the UHECR production rate at
GZK energies is almost unchanged. These nominal GRB models, especially with
$L_{\rm iso} \lesssim 10^{53} ~\mbox{erg} ~\mbox{s}^{-1}$, appear to meet the
current constraints as far as being candidate UHECR sources above the ankle
energy.
