We generalize a population-size-dependent branching process to a more general branching model called the population-size-dependent branching process in random environments. For the model where {Zn}n≥0 is associated with the stationary environment ξ− = {ξn}n≥0, let B = {ω : Zn(ω) = for some n}, and q(ξ−) = P(B | ξ−, Z0 = 1). The result is that P(q(̅ξ) = 1) is either 1 or 0, and sufficient conditions for certain extinction (i.e. P(q(ξ−) = 1) = 1) and for non-certain extinction (i.e. P(q(ξ−) < 1) = 1) are obtained for the model.

We consider an absorbing semi-Markov chain for which each time absorption occurs there is a resetting of the chain according to some initial (replacement) distribution. The new process is a semi-Markov replacement chain and we study its properties in terms of those of the imbedded Markov replacement chain. A time-dependent version of the model is also defined and analysed asymptotically for two types of environmental behaviour, i.e. either convergent or cyclic. The results contribute to the control theory of semi-Markov chains and extend in a natural manner a wide variety of applied probability models. An application to the modelling of populations with semi-Markovian replacements is also presented.

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.

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.

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.

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.

In this paper we study the cyclic behaviour of non-homogeneous Markov systems, i.e. the behaviour of the system under the assumption of periodic sequences of transition matrices, input probabilities, output probabilities and total numbers in the system. We provide a general theorem for the limiting structure of such a system under the cyclic behaviour. We also study the asymptotic stability of non-homogeneous Markov systems and theorems are given which characterize asymptotic stability. An application of the above results is given for a British firm, BS.

We consider a branching-process model {Zn}, where the law of offspring distribution depends on the population size. We consider the case when the means mn (mn is the mean of offspring distribution when the population size is equal to n) tend to a limit m > 1 as n →∞. For a certain class of processes {Zn} necessary conditions for convergence in L1 and L2 and sufficient conditions for almost sure convergence and convergence in L2 of Wn = Zn/mn are given.

In this paper we study the asymptotic behavior of Markov systems and especially non-homogeneous Markov systems. It is found that the limiting structure and the relative limiting structure exist under certain conditions. The problem of weak ergodicity in the above non-homogeneous systems is studied. Necessary and sufficient conditions are provided for weak ergodicity. Finally, we discuss the application of the present results in manpower systems.

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.

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.

Leslie (1945) models the evolution in discrete time of a closed, single-sex population with discrete age groups by multiplying a vector describing the age structure by a matrix containing the birth and death rates. We suppose that successive matrices are chosen according to a Markov chain from a finite set of matrices. We find exactly the long-run rate of growth and expected age structure. We give two approximations to the variance in age structure and total population size. A numerical example illustrates the ergodic features of the model using Monte Carlo simulation, finds the invariant distribution of age structure from a linear integral equation, and calculates the moments derived here.