Stability of manpower systems

Members of an organization belong to one of the grades {1,2,...,k}, and movements between grades are governed by a substochastic matrix P. The model can be deterministic, stochastic, or partially stochastic; and we may or may not insist that the total size be restored to a fixed quantity each year. Various notions of the stability of a structure x=(x1,x2,...,xk), where xi represents the number, or proportion, in grade i are considered. Mainly, but not exclusively, the case when P is upper triangular is considered.