  The reionization of cosmic hydrogen marks a critical juncture in the history
of structure formation in the universe. Here we present a new formulation of
the standard reionization equation for the evolution of the volume-averaged HII
fraction that is more consistent with the accepted conceptual model of
inhomogeneous intergalactic absorption. The revised equation retains the basic
terminology and simplicity of the classic calculation but explicitly accounts
for the presence of the optically thick "Lyman-limit systems" that are known to
determine the mean free path of ionizing radiation after overlap. Integration
of this equation provides a better characterization of the timing of
reionization by smoothly linking the pre-overlap with the post-overlap phases
of such process. We confirm the validity of the quasi-instantaneous
approximation as predictor of reionization completion/maintenance, and discuss
new insights on the sources of cosmic reionization using the improved
formalism. A constant emission rate into the intergalactic medium (IGM) of 3
Lyman continuum (LyC) photons per atom per Gyr leads to a reionization history
that is consistent with a number of observational constraints on the ionization
state of the z=5-9 universe and with the reduced Thomson scattering optical
depth recently reported by the Planck Collaboration. While star-forming
galaxies can dominate the reionization process if the luminosity-weighted
fraction of LyC photons that escape into the IGM, f_esc, exceeds 15% (for a
faint magnitude cut-off of the galaxy UV luminosity function of M_lim=-13 and a
LyC photon yield per unit 1500 AA luminosity of xi_ion=10^{25.3} Hz/erg, simple
models where the product of the two unknowns f_esc xi_ion is not evolving with
redshift fail to reproduce the changing neutrality of the IGM observed at these
epochs.
