Measures of concordance have been widely used in insurance and risk management to summarize nonlinear dependence among risks modeled by random variables, which Pearson’s correlation coefficient cannot capture. However, popular measures of concordance, such as Spearman’s rho and Blomqvist’s beta, appear as classical correlations of transformed random variables. We characterize a whole class of such concordance measures arising from correlations of transformed random variables, which includes Spearman’s rho, Blomqvist’s beta and van der Waerden’s coefficient as special cases. Compatibility and attainability of square matrices with entries given by such measures are studied—that is, whether a given square matrix of such measures of concordance can be realized for some random vector and how such a random vector can be constructed. Compatibility and attainability of block matrices and hierarchical matrices are also studied due to their practical importance in insurance and risk management. In particular, a subclass of attainable block Spearman’s rho matrices is proposed to compensate for the drawback that Spearman’s rho matrices are in general not attainable for dimensions larger than three. Another result concerns a novel analytical form of the Cholesky factor of block matrices which allows one, for example, to construct random vectors with given block matrices of van der Waerden’s coefficient.

In the present paper, we study the problem of small-time local attainability (STLA) of a closed set. For doing this, we introduce a new concept of variations of the reachable set well adapted to a given closed set and prove a new attainability result for a general dynamical system. This provide our main result for nonlinear control systems. Some applications to linear and polynomial systems are discussed and STLA necessary and sufficient conditions are obtained when the considered set is a hyperplane.

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.

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.

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.

Our concern is with a particular problem which arises in connection with a discrete-time Markov chain model for a graded manpower system. In this model, the members of an organisation are classified into distinct classes. As time passes, they move from one class to another, or to the outside world, in a random way governed by fixed transition probabilities. In this paper, the emphasis is placed on evaluating exact values of the probabilities of attaining and maintaining a structure.

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.

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.

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.

Grade and age structures in manpower systems are often far from ideal. This fact raises the question of how the flows of people — and particularly the recruitment flow — should be controlled in order to attain and maintain a more desirable structure. The problem has received considerable attention from a deterministic point of view. This paper adopts a stochastic approach to the study of maintainability and shows, among other things, that the problem is more subtle than the deterministic analysis suggests.