The concept of a stable population, which is of basic importance in demography, is generalized to person-flow models as defined by Gani. Stability and asymptotic stability of manpower systems are investigated in terms of the input process and the transition structure. Extending Bartholomew ((1973), Chapter 3) we obtain a result on the limiting behaviour of Gani-type models with asymptotic exponential input processes. The linear population-dynamic model of Leslie (1945) may be considered as a further case.

If Bn is the time of the first birth in the nth generation in a supercritical irreducible multitype Crump–Mode process then when there are people in every generation Bn/n converges to a constant; if Dn is the time of the last birth in the nth generation then Dn/n also converges to a constant on the survival set. Analogous results hold for the extreme members of the nth generation in a branching random walk.

Additional theory important in the design and evaluation of a periodic screening program is presented. The distributions of doubly bounded preclinical state forward and backward recurrence times of the stochastic screening model are derived. Properties of these distributions are used to obtain an expression for estimating the mean lead time gained by screening, an alternative theoretical expression for the mean lead time and an equation for calculating an estimate of the probability of detecting preclinical disease.

At each stage of the packing of a closed interval K, a random number of random open intervals (the packing objects) are placed in that part of K which is as yet unoccupied. No overlapping between the packing objects is allowed. The packing prescription is such that the packing process terminates after at most a finite number of stages. Attention is focused on the final configuration, K = K– + G, where G is a random open subset of K, and is that part of K which is eventually occupied by packing objects, while K–, a random closed subset of K, is that part of K which remains unoccupied.

The paper is concerned with aspects of the behaviour of the products of finite stochastic matrices, the methods used in the proofs being of a probabilistic nature. The main result of the paper (Theorem 1) presents a general picture of the asymptotic behaviour of the transition probabilities between various groups of states. A unified treatment of some results of non-homogeneous Markov chain theory pertaining to weak ergodicity is then given.

Kingman [5] proved a formula that expresses the joint distribution of the processes where b is a regular point in the state space of a Hunt process. We give an extension of this formula, as well as several interesting facts related to it, for the case when Φ is any finely perfect subset of the state space. We also establish some connections between this result and results on last-exit decompositions.

In this paper the limit behaviour of α-recurrent Markov renewal processes and semi-Markov processes is studied by using the recent results on the concept of α-recurrence for Markov renewal processes. Section 1 contains the preliminary results, which are needed later in the paper. In Section 2 we consider the limit behaviour of the transition probabilities Pij(t) of an α-recurrent semi-Markov process. Section 4 deals with quasi-stationarity. Our results extend the results of Cheong (1968), (1970) and of Flaspohler and Holmes (1972) to the case in which the functions to be considered are directly Riemann integrable. We also try to correct the errors we have found in these papers. As a special case from our results we consider continuous-time Markov processes in Sections 3 and 5.

This paper gives a survey of the historical development of the solutions of various fluctuation problems in the theory of queues from the point of view of the mathematical methods used. These methods include Markov chains, Markov processes, integral equations, complex functions, generating functions, Laplace transforms, factorization of functions, operator calculus, Banach algebras and some particular methods, such as calculus of finite differences and combinatorics. In addition, the paper contains several recent results of the author for semi-Markov queuing processes.

In a recent article, Kelly [4] has been able to exhibit interesting equilibrium properties for a wide class of ‘quasi-reversible’ queue networks. The assumption of quasi-reversibility puts restrictions on queue discipline, but not on the distributions of the service requirements of customers: however, because of the method of proof he employed, Kelly was forced to impose the condition that the service requirements were finite mixtures of gamma distributions. The form of the results he obtained led him to conjecture that this condition was in fact unnecessary, as is shown to be the case in this paper. The method used to prove the conjecture is of potentially wide application, in problems where the ‘method of stages' leads to useful simplification.

A technique known as potential cost, used by Faddy [3] for assessing the operation of a dam is seen to be capable of extension to allow for

(i) a very general cost function, as is required for a multipurpose reservoir (the norm nowadays) and

(ii) the use of discounting of future costs, a very widespread accounting procedure.

Numerical results are obtained for an optimal policy based on such an assessment, and demonstrate the need for an accurate specification of the costs associated with the operation of a reservoir. As a by-product a very full description of the steady-state stochastic behaviour of the dam is obtained.