We 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 article, 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 behavior obtained is illustrated by a numerical treatment of the standard two-level system with pumping and relaxation included. © 2010 The American Physical Society.