The foundation of the novel stochastic process Backward Non-Homogeneous Markov system ( \(\mathcal {B}\) -NHMS) is provided in the present. This process is closely connected with the problem of tendency to consensus in an information exchanging operation. A problem which has attracted more than a few thousands of citations in the literature and where backward products of stochastic matrices appear. For forward NHMS with chronological order it is known that weak ergodicity does not necessarily imply strong ergodicity. In a basic Theorem it is proved that in \(\mathcal {B}\) -NHMS with chronological order weak ergodicity always imply strong ergodicity. Later sufficient and necessary conditions are provided for strong ergodicity of \(\mathcal {B}\) -NHMS to converge with geometrical rate uniformly. Next ergodicity of \( \mathcal {B}\) -NHMS is studied when the input process is a non-homogeneous Poisson process. Finally an illustration of some of the main results are given for a manpower system with four grades.
