  We offer an explanation for the anti-correlation between the minimum
variability timescale ($MTS$) in the prompt emission light curve of gamma-ray
bursts (GRBs) and the estimated bulk Lorentz factor of these GRBs, in the
context of a magnetically arrested disk (MAD) model. In particular, we show
that previously derived limits on the maximum available energy per baryon in a
Blandford-Znajek jet leads to a relationship between the characteristic MAD
timescale, $t_{MAD}$, in GRBs and the maximum bulk Lorentz factor: $t_{MAD}
\propto \Gamma^{-6}$, somewhat steeper than (although within the error bars of)
the fitted relationship found in the GRB data. Similarly, the MAD model also
naturally accounts for the observed anti-correlation between $MTS$ and
gamma-ray luminosity $L$ in the GRB data, and we estimate the accretion rates
of the GRB disks (given these luminosities) in the context of this model. Both
of these correlations ($MTS-\Gamma$ and $MTS-L$) are also observed in the AGN
data, and we discuss the implications of our results in the context of both GRB
and blazar systems.
