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State Re-union Maintainability for Semi-Markov Models in Manpower Planning

Brecht Verbeken, Marie-Anne Guerry

Abstract

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.

State Re-union Maintainability for Semi-Markov Models in Manpower Planning

Abstract

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.
Paper Structure (10 sections, 14 theorems, 57 equations, 2 figures)

This paper contains 10 sections, 14 theorems, 57 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Maintainability under control by recruitment for a Markov chain
  3. SR-maintainability under control by recruitment for a semi-Markov chain
  4. The $SR$-maintainable region
  5. The $SR$-maintainable region for a system with constant size
  6. Extensions to exponential growth and contraction
  7. $SR$-maintainable region and maintainable regions of induced Markov chains
  8. Relationship between $SR$-maintainability and maintainability of induced Markov chains
  9. Illustration: control theory for a semi-Markov chain with constant size
  10. Conclusions and further research avenues

Key Result

Theorem 2.1

bartholomew1967stochastic Suppose that $\mathbf{P}_M^I$ is the internal transition matrix of a Markov system. The maintainable region for a system with constant total size and under control by recruitment is the convex set with vertices given by the normalised rows of $(\mathbf{I}- \mathbf{P}_M^I)^{

Figures (2)

  • Figure 1: The maintainable region $\mathscr{MR}_{SM}$ for the semi-Markov chain with $\mathbf{P}_{SM}$ as in Equation (\ref{['PSM']}).
  • Figure 2: The maintainable regions $\mathscr{MR}_{M1(SM)}$ (blue), $\mathscr{MR}_{M2(SM)}$ (green) and the $SR$-maintainable region (red).

Theorems & Definitions (38)

  • Definition 2.1
  • Theorem 2.1
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.1
  • Proposition 2.2
  • Definition 2.4
  • Example 2.1
  • Proposition 2.3
  • ...and 28 more