We study the incidence rate of damped Ly-a systems associated with the host galaxies of gamma-ray bursts (GRB-host-DLAs) as functions of neutral hydrogen column density (N_HI) and projected star formation rate (SFR) using cosmological SPH simulations. Assuming that the occurrence of GRBs is correlated with the local SFR, we find that the median N_HI of GRB-host-DLAs progressively shifts to lower N_HI values with increasing redshift, and the incidence rate of GRB-host-DLAs with log N_HI > 21.0 decreases rapidly at z>=6. Our results suggest that the likelihood of observing the signature of IGM attenuation in GRB afterglows increases towards higher redshift, because it will not be blocked by the red damping wing of DLAs in the GRB host galaxies. This enhances the prospects of using high-redshift GRBs to probe the reionization history of the Universe. The overall incidence rate of GRB-host-DLAs decreases monotonically with increasing redshift, whereas that of QSO-DLAs increases up to z=6. A measurement of the difference between the two incidence rates would enable an estimation of the value of eta_grb, which is the mass fraction of stars that become GRBs for a given amount of star formation. Our predictions can be tested by upcoming high-z GRB missions, including JANUS (Joint Astrophysics Nascent Universe Scout) and SVOM (Space multi-band Variable Object Monitor).
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