  Nonthermal particles can be produced due to incomplete thermalization at
collisionless shocks and further accelerated to very high energies via
diffusive shock acceleration. In a previous study we explored the cosmic ray
(CR) acceleration at cosmic shocks through numerical simulations of CR
modified, quasi-parallel shocks in 1D plane-parallel geometry with the physical
parameters relevant for the shocks emerging in the large scale structure
formation of the universe (Kang & Jones 2002). In order to consider the CR
acceleration at shocks with a broader range of physical properties, in this
contribution we present additional simulations. For a given Mach number, shocks
with higher speeds accelerate CRs faster with a greater number of particles.
However, two shocks with a same Mach number but with different shock speeds
evolve qualitatively similarly when the results are presented in terms of
diffusion length and time scales. Therefore, the time asymptotic value for the
fraction of shock kinetic energy transferred to CRs is mainly controlled by
shock Mach number rather than shock speed. We present the ``CR energy ratio''
for a wide range of shock parameters and for $\epsilon=0.2-0.3$ at terminal
time of our simulations. We suggest that these values can be considered as
time-asymptotic values for the CR acceleration efficiency, since the
time-dependent evolution of CR modified shocks has become approximately
self-similar before the terminal time.
