A note on the inverse problem for reducible Markov chains

Suppose a physical process is modelled by a Markov chain with transition probability on S1 ∪ S2, S1 denoting the transient states and S2 a set of absorbing states.If v denotes the output distribution on S2, the question arises as to what input distributions (of raw materials) on S1 produce v. In this note we give an alternative to the formulation of Ray and Margo [2] and reduce the problem to one system of linear inequalities. An application to random walk is given and the equiprobability case examined in detail.