Variances and covariances of the grade sizes in manpower systems

Abstract

The asymptotic behaviour of the variances and covariances of the class sizes in closed and open manpower systems is considered. Firstly, the homogeneous case is studied and a theorem is stated which provides the answer to the problem in the most general case for the homogeneous Markov-chain models in manpower systems (open systems) and social mobility models (closed systems). Secondly, the non-homogeneous problem is studied and a theorem is given where under certain conditions it is proved that the vector sequences of means, variances and covariances converge. Finally, we relate our theoretical results to examples from the literature on manpower planning.

The rate of convergence of the vector of variances and covariances in non-homogeneous Markov systems

“…The earliest work on the variability of state sizes in both open and closed manpower systems was that of Pollard (1966Pollard ( ), (1973. These results were extended by Bartholomew «1982), p. 66, 78) and Vassiliou and Gerontidis (1985) for open manpower systems. The concept of non-homogeneous Markov systems (NHMS) was first introduced in Vassiliou (1982b).…”

Periodic strong ergodicity in non-homogeneous Markov systems

“…Sincep o 1' = 1 = PoCt)1', for r = 0, 1,• • ., following the steps ofthe proofofTheorem 3.2 in Vassiliou and Gerontidis (1985) we can show that…”

“…to an irreducible stochastic matrix L with d > 1 eigenvalues of modulus 1 and that the matrices Q(t) have the same incidence matrix as L. This type of problem was raised in Vassiliou (1982) in a work concerned with the relative structure and later on it was considered in Tsaklidis and Vassiliou (1988) in studying the limit of the vector of expectations variances and covariances of the grade sizes. A similar investigation for the homogeneous case is presented in Vassiliou and Gerontidis (1985). In our present approach we provide a more detailed analysis which leads to interesting probabilistic interpretations, by determining the limiting evolution, the inherent variation and the interaction between grade sizes ofthe individual cyclically moving subclasses ofthe state space of the associated Markov replacement chain.…”

“…There are several references dealing with the assumption of time-varying parameters (see Bartholomew and Forbes (1979), Rogoff (1953), Hodge (1966) and Young and Vassiliou (1974». The purpose of this paper is to elaborate further on two recent papers, Vassiliou and Gerontidis (1985) and Tsaklidis and Vassiliou (1988). Our main objective is to introduce a different approach, leading to a detailed analysis and providing interesting probabilistic interpretations under the various modes of convergence.…”

“…In the last few years there has been an increasing interest in the asymptotic behaviour of non-homogeneous Markov systems (NHMS). Important references are the works by Vassiliou (1982), (1984), (1986), Vassiliou and Gerontidis (1985), Tsaklidis and Vassiliou (1988) and Vassiliou and Tsaklidis (1990). The concept of NHMS was first introduced in Vassiliou (1982) with the aim of providing a general framework for a number of Markov chain models in manpower systems, see for example Pollard (1966Pollard ( ), (1973, Bartholomew (1973Bartholomew ( ), (1982, Feichtinger and Mehlmann (1976) and Conlisk (1976).…”