  A model of homogeneously driven dissipative system, consisting of a
collection of $N$ particles that are characterized by only their velocities, is
considered. Adopting a discrete time dynamics, at each time step, a pair of
velocities is randomly selected. They undergo inelastic collision with
probability $p$. With probability $(1-p)$, energy of the system is changed by
changing the velocities of both the particles independently according to
$v\rightarrow -r_w v +\eta$, where $\eta$ is a Gaussian noise drawn
independently for each particle as well as at each time steps. For the case
$r_w=- 1$, although the energy of the system seems to saturate (indicating a
steady state) after time steps of $O(N)$, it grows linearly with time after
time steps of $O(N^2)$, indicating the absence of a eventual steady state. For
$ -1 <r_w \leq 1$, the system reaches a steady state, where the average energy
per particle and the correlation of velocities are obtained exactly. In the
thermodynamic limit of large $N$, an exact equation is obtained for the moment
generating function. In the limit of nearly elastic collisions and weak energy
injection, the velocity distribution is shown to be a Gaussian. Otherwise, for
$|r_w| < 1$, the high-energy tail of the velocity distribution is Gaussian,
with a different variance, while for $r_w=+1$ the velocity distribution has an
exponential tail.
