  We present semi-analytic models of the epoch of reionization focusing on the
differences between continuous and bursty star formation (SF). Our model
utilizes physically motivated analytic fits to 1D radiative transfer
simulations of HII regions around dark matter halos in a representative cosmic
volume. Constraining our simulations with observed and extrapolated UV
luminosity functions of high redshift galaxies, we find that for a fixed halo
mass, stellar populations forming in bursty models produce larger HII regions
which leave behind long-lived relic HII regions which are able to maintain
partial ionization in the intergalactic medium (IGM) in a manner similar to an
early X-ray background. The overall effect is a significant increase in the
optical depth of the IGM, $\tau_e$, and a milder increase of the redshift of
reionization. To produce $\tau_e =0.066$ observed by Planck and complete
reionization by redshift $z_{\rm re}\sim 6$, models with bursty SF require an
escape fraction $f_{\rm esc}\sim 2\%-10\%$ that is 2-10 times lower than
$f_{\rm esc}\sim 17\%$ found assuming continuous SF and is consistent with
upper limits on $f_{\rm esc}$ from observations at $z=0$ and $z\sim 1.3-6$. The
ionizing photon budget needed to reproduce the observed $\tau_e$ and $z_{\rm
re}$ depends on the period and duty cycle of the bursts of SF and the
temperature of the neutral IGM. These results suggest that the tension between
observed and predicted ionizing photon budget for reionization can be
alleviated if reionization is driven by short bursts of SF, perhaps relating to
the formation of Population~III stars and compact star clusters such as
proto-globular clusters.
