We introduce and define for the first time the concept of a non-homogeneous Markov system in a stochastic environment (S-NHMS). The problem of finding the expected population structure in an S-NHMS is studied, and important properties among the basic parameters of the S-NHMS are established. Moreover, we study the problem of maintaining the relative sizes of the states in a stochastic environment applying control in the input process. Among other things, we provide the probability of maintaining any vector of relative state sizes. Also strategies for attaining in an optimal way a desired relative structure are designed, with the use of a given algorithm. Finally, an illustration is provided of the present results in a manpower system.

In this paper we introduce and define for the first time the concept of a non-homogeneous semi-Markov system (NHSMS). The problem of finding the expected population stucture is studied and a method is provided in order to find it in closed analytic form with the basic parameters of the system. Moreover, the problem of the expected duration structure in the state is studied. It is also proved that all maintainable expected duration structures by recruitment control belong to a convex set the vertices of which are specified. Finally an illustration is provided of the present results in a manpower system.

Let T be a non-empty subset of η X n stochastic matrices. Define The sequence T1, T2, · ·· is called a Markov set-chain. An important problem in this area is to determine when such a set-chain converges. This paper gives a notion of a sequential limiting set and shows how it can be used to obtain a result on set-chain convergence.

The aims of this paper are twofold. The first is to study the rate of convergence of the sequence of vectors of expectations, variances and covariances in NHMS. It is proved that under certain conditions easily met in practice the rate is geometric. The second is to study the rate of convergence of the same sequence when the NHMS is under cyclic behaviour and it is also proved that the rate of convergence of the d subsequences (d being the period of the cycle) in which the sequence splits is geometric.

In this paper we study the asymptotic periodicity of the sequence of means, variances and covariances of the state sizes of non-homogeneous Markov systems. It is proved that under the assumption that the sequence of the extended stochastic transition matrices converge to a matrix which is an irreducible stochastic matrix of period d, and all the matrices in this sequence have the same incidence matrix, then the sequence of means, variances and covariances splits into d subsequences which converge. Finally, we discuss the application of the present results in a manpower system.

We study monotone and associated Markov chains on finite partially ordered state spaces. Both discrete and continuous time, and both time-homogeneous and time-inhomogeneous chains are considered. The results are applied to binary and multistate reliability theory.

The Markov-Dobrushin condition for (weak) ergodicity of non-homogeneous discrete-time Markov chains, and an analogous criterion for continuous chains, are derived by means of coupling techniques.