  We have earlier constructed a generalized entropy concept to show the
direction of time in an evolution following from a Markov generator. In such a
dynamical system, the entity found changes in a monotonic way starting from any
initial state of the system. In this paper, we generalize the treatment to the
case when population is pumped into the system from levels not explicitly
considered. These populations then pass through the coupled levels and exit by
decay to levels outside the system. We derive the form of the equation of
motion and relate it to our earlier treatments. It turns out that the formalism
can be generalized to the new situation. Its physically relevant features are
demonstrated, and the behaviour obtained is illustrated by numerical treatment
of the standard two-level system with pumping and relaxation included.
