  The ongoing debate regarding the most accurate accretion model for
supermassive black holes at the center of quasars has remained a contentious
issue in astrophysics. One significant challenge is the variation in calculated
accretion efficiency, with values exceeding the standard range of $0.038 <
\epsilon < 0.42$. This discrepancy is especially pronounced in high redshift
supermassive black holes, necessitating the development of a comprehensive
model that can address the accretion efficiency for supermassive black holes in
both the low and high redshift ranges. In this study, we have focused on low
redshift ($z < 0.5$) PG quasars (79 quasars) and high redshift ($z \geq 3$)
quasars with standard disks from the flux- and volume-limited QUOTAS+QuasarNET
dataset (76 quasars) to establish a model for accretion efficiency. An
interesting trend is revealed where in redshift larger than 3, accretion
efficiency increases as redshift decreases, while in redshift lower than 0.5,
accretion efficiency decreases with reducing redshift. This suggests a peak in
accretion efficiency between the low and high redshift quasars. This peak is
recognized for the flux- and volume-limited QUOTAS+QuasarNET+DL11 dataset,
which is $z \sim 2.675$, and it seems to be related to the peak of the star
formation rate. ($1 < z_{SFR} < 3$). This result can potentially lead to a more
accurate correlation between the star formation rate in quasars and their
relationship with the mass of the central supermassive black holes with a more
comprehensive disk model in future studies.
