In this paper, given personnel distributions that are not attainable, we introduce the grade of attainability in order to measure the degree to which there exists a similar distribution that is attainable. For constant size systems controlled by recruitment, properties of the most similar distribution to a given distribution are formulated.

In a system modelled by a time-discrete deterministic model, predictions of the distribution of the members over the different classes do not result automatically in an integer valued vector. In this paper, for a constant size system, we discuss how to associate with the calculated vector an integer valued vector. Furthermore we examine whether the evolution of the calculated vectors on the one hand, and the evolution of the associated integer valued vectors on the other hand, have the same properties.

A manpower system of constant size, controlled by recruitment, is described by a partially stochastic model in which there are fixed promotion rates, no demotions and stochastic wastage. The geometric-probabilistic relationship is examined for the attainability after one step and after two steps.

Until Guerry's (1990) counterexample to a conjecture of Davies about three-state hierarchical organisations kept at constant size via annual promotion, wastage and recruitment, it was easy to believe that such structures maintainable in t steps would also be maintainable in t + 1 steps. Here we present further counterexamples, which show that t-step maintainability does not imply (t + 1)-step maintainability, for astonishingly large values of t.

In this paper the t-step maintainable regions Mt are examined in a three-graded system under the following conditions: the total size of the system remains constant during each intermediate step, demotions do not occur and recruitment control is considered.

A counterexample, showing that the monotonicity property Mt ⊆ Mt+1 does not exist in general, refutes the conjecture of Davies [3].

This paper presents a unified treatment of the convergence properties of nonhomogeneous Markov systems under different sets of assumptions. First the periodic case is studied and the limiting evolution of the individual cyclically moving subclasses of the state space of the associated Markov replacement chain is completely determined. A special case of the above result is the aperiodic or strongly ergodic convergence. Two numerical examples from the literature on manpower planning highlight the practical aspect of the theoretical results.

In the present paper we study three aspects in the theory of non-homogeneous Markov systems under the continuous-time formulation. Firstly, the relationship between stability and quasi-stationarity is investigated and conditions are provided for a quasi-stationary structure to be stable. Secondly, the concept of asymptotic attainability is studied and the possible regions of asymptotically attainable structures are determined. Finally, the cyclic case is considered, where it is shown that for a system in a periodic environment, the relative structure converges to a periodic vector, independently of the initial distribution. Two numerical examples illustrate the above theoretical results.

We examine a Markov manpower system in continuous time where demotion rates are 0 and promotion rates are time-dependent. The transient and limiting behaviours of the model are discussed, and illustrated with examples.

The question as to what grade structures are maintainable in Markov manpower models, with or without the constraint that the total size be kept constant, is examined. Known results are collected together, examples and further results are given, and an old conjecture is generalised.

The probability of attaining a goal structure in a partially stochastic manpower model is discussed in relation to the position of the structure described in Euclidean space. It is found that, within the regions of structures attainable with non-zero probability, the probabilities decrease on lines emanating from the structure attainable with probability 1.

We study the limiting behaviour of a manpower system where the non-homogeneous Markov chain model proposed by Young and Vassiliou (1974) is applicable. This is done in the cases where the input is a time-homogeneous and time-inhomogeneous Poisson random variable. It is also found that the number in the various grades are asymptotically mutually independent Poisson variates.

A manpower model is examined in which there are fixed promotion rates, no demotions and stochastic wastage. We discuss the probability of attainability of structures and structural paths using recruitment control.

Necessary and sufficient conditions for stability, imposed firstly on the initial structure and the sequence of recruitment, and secondly on the initial structure and the sequence of expansion are provided in forms of two theorems. Also the limiting behaviour of the expected relative grade sizes is studied if we drop the conditions for stability imposed on the initial structure and keep the same sequence of expansion. Finally we examine the limiting behaviour of the expected grade sizes if we drop the assumption of a continuously expanding system.

We examine the geometry of regions of maintainable structures arising in a Markov manpower model. The regions are described in terms of convex hulls, and it is shown that for systems divided into two or three grades these regions form an increasing sequence. It is also shown that the monotonie property fails quite drastically for a four-graded system.

An open problem is discussed for a related model.

In an earlier paper (Bartholomew (1977)) the problem of maintaining a grade (or age) structure by controlling the recruitment flows was discussed. Here we carry out a parallel analysis for the case where control is by promotion. We show how to calculate the probability that a structure can be maintained and investigate the performance of several strategies for keeping a structure as near as possible to that desired. As before, the results demonstrate the limitations of a deterministic analysis of the problem.

This paper presents the general properties of the semi-Markovian manpower model in continuous time. The asymptotic relation for the population numbers in various grades is based on the forces of transition zij(u) from state i to state j at duration u.

Using the expectation recurrence equation of discrete-time Markovian manpower models, an asymptotic relation for the population numbers in various grades is derived.