  We compute the stationary performance metrics of a single server $M^X/G/1$
queue under a class of generalized processor-sharing scheduling policies that
are proposed by Grishechkin. This class of processor-sharing policies allow
service capacities to be allocated to jobs based on the amount of service they
attained. In [11], Grishechkin derives an integral equation that is satisfied
by the Laplace transform of the stationary performance metrics under these
policies. Our main focus in this paper is to derive the solution to this
integral equation. This is achieved through a series of transforms that convert
the integral equation into a more tractable form, then solve it. Then we derive
approximations to the density functions of the performance metrics through
inverting the Laplace transforms. In the last part of the paper, we apply our
results to some well-known scheduling policies that are either a special case
of the Grishechkin processor-sharing policies or can be treated as the limits
of a sequence Grishechkin processor-sharing policies. These examples include
the egalitarian processor-sharing policy, the discriminatory processor-sharing
policy, shortest residual processing time first rule, foreground-background
policy and the time function scheduling policy.
