Solution methods, both numerical and analytical, are considered for solving the
Liouville master equation associated with discrete-state Markovian initial value
problems. The numerical method, basically a moment (Galerkin) method, is
very general and is validated and shown to converge rapidly by comparison
with an earlier reported analytical result for the ensemble-averaged transmission
of photons through a purely scattering statistical rod. An application of the
numerical method to a simple problem in the extended kinetic theory of gases
is given. It is also shown that for a certain restricted class of problems, the
master equation can be solved analytically using standard Laplace transform
techniques. This solution generalizes the analytical solution for the photon
transmission problem to a wider class of statistical problems.