  In previous research the importance of both Markov and semi-Markov models in
manpower planning is highlighted. Maintainability of population structures for
different types of personnel strategies (i.e. under control by promotion and
control by recruitment) were extensively investigated for various types of
Markov models (homogeneous as well as non-homogeneous) (Bartholomew, 1967;
Vassiliou and Tsantas, 1984a). Semi-Markov models are extensions of Markov
models that account for duration of stay in the states. Less attention is paid
to the study of maintainability for semi-Markov models. Although, some
interesting maintainability results were obtained for non-homogeneous
semi-Markov models (Vassiliou and Papadopoulou, 1992). The current paper
focuses on discrete-time homogeneous semi-Markov models, and explores the
concept of maintainable population structures in this setting for a system with
constant total size or one with a growth factor. In particular, a new concept
of maintainability is introduced, the so called State Re-union maintainability
(SR-maintainability). Moreover, we show that, under certain conditions, the
seniority-based paths associated with the SR-maintainable structures converge.
This allows to characterize the convex set of SR-maintainable structures.
