In classical demographic theory, the age structure of a population eventually stabilizes, and the population as a whole grows at a geometric rate. It is possible to prove stochastic analogues of these results if vital rates fluctuate according to a stationary stochastic process. The approach taken here is to study the action of random matrix products on random vectors. This permits the application of Hilbert's projective metric and leads to considerable simplification of the ergodic and central limit theory of population growth.

Two-dimensional bounded birth and death processes are considered. State probabilities are given for the generalized case in which the transition rates are arbitrary time-independent functions of the population sizes. Some applications in epidemic theory and species interaction processes are briefly described.

To evaluate the benefits and risks associated with screening for disease, a model is developed to characterize the changes in incidence and survival distributions effected by a screening program. Screening is presumed to increase survival by resulting in diagnosis of disease at earlier stages. All disease states in the model are observable, thus facilitating application to empirical data. An example of such an application using data from a breast cancer detection project is given for the case of one screening for two-stage disease having Weibull-distributed diagnosis times.

Regenerative stopping problems are stopping problems which recommence from the initial state upon stopping. An algorithm is presented which solves semi-Markov regenerative stopping problem with a finite number of continue actions by solving a sequence of stopping problems. New results for the optimal stopping problem are obtained as well as for the regenerative stopping problem. A model in the literature is used as a detailed example of the algorithm.

Excess life distributions for discrete renewal processes may be computed by using elementary discrete Markov chain concepts involving absorption probabilities. Excess life distributions in general may then be obtained by approximating the renewal process under study by a suitably chosen sequence of discrete renewal processes. The technique is illustrated in the cases of renewal processes with interarrival distributions which are a linear combination of two exponentials and uniform [0, 1]. A related algorithm is described for computer generated approximations of excess life distributions corresponding to continuous interarrival time distributions with an increasing c.d.f. Conditions for convergence of this algorithm are examined.

A strong approximation theorem is proved for some non-stationary complex-valued Gaussian processes and an explicit rate of convergence is achieved. The result answers a problem raised by S. Csörgő.

A one-dimensional random walk is studied in which, at each stage, the probabilities of continuing in the same direction or of changing direction are p and 1 – p, respectively. Exact expressions are derived for the n-step transition probabilities, and various limiting distributions are investigated.

This paper explores best choice problems which allow both recall of applicants and uncertainty of a current applicant accepting an offer of employment. Properties of optimal selection procedures are derived for the general case. Optimal procedures and the associated probabilities of obtaining the best applicant are found in two special cases. The results unify and extend those of Yang (1974) and Smith (1975).

The problem of Monte Carlo estimation of M(t) = EN(t), the expected number of renewals in [0, t] for a renewal process with known interarrival time distribution F, is considered. Several unbiased estimators which compete favorably with the naive estimator, N(t), are presented and studied. An approach to reduce the variance of the Monte Carlo estimator is developed and illustrated.

Consider n cells into which balls are thrown at random until k cells contain at least l + 1 balls each. Let Yl, · ··, Yn be the number of balls in the cells when stopping. In this paper two representations are given for the characteristic functions of random variables of the form The usefulness of these representations are illustrated by two examples. In the first the number of cells with exactly one ball when each cell contains at least one ball is considered. In the second the waiting time until the ball-throwing process stops is discussed.

Time series models for non-linear random vibrations are discussed from the viewpoint of the specification of the dynamics of the damping and restoring force of vibrations, and a non-linear threshold autoregressive model is introduced. Typical non-linear phenomena of vibrations are demonstrated using the models. Stationarity conditions and some structural aspects of the model are briefly discussed. Applications of the model in the statistical analysis of real data are also shown with numerical results.

This paper considers certain stochastic models for token and type counts in literary texts. Elaborating on some models of Gani, it is shown that reasonable fits can be obtained to some data of Yule on Macaulay's Essays.

The steady-state distribution of the inventory position for a continuous-review (s, S) inventory system is derived in a computationally tractable form. Demands for items in inventory are assumed to form an N-process which is the ‘versatile Markovian point process' introduced by Neuts (1979). The N-process includes the phase-type renewal process, Markov-modulated Poisson process etc., as special cases and is especially useful in modelling a wide variety of qualitative phenomena such as peaked arrivals, interruptions, inhibition or stimulation of arrivals by certain events etc.

This paper considers a semi-infinite storage model, of the type studied by Senturia and Puri [13] and Balagopal [2], defined on a Markov renewal process, {(Xn, Tn), n = 0, 1, ·· ·}, with 0 = T0 < T1 < · ··, almost surely, where Xn takes values in the set {1, 2, ·· ·}. If at Tn, Xn = j, then there is a random ‘input' Vn (j) (a negative input implying a demand) of ‘type' j, having distribution function Fj(·). We assume that {Vn (j)} is an i.i.d. sequence of random variables, taken to be independent of {(Xn, Tn)} and of {Vn (k)}, for k ≠ j, and that Vn (j) has first and second moments. Here the random variables Vn (j) represent instantaneous ‘inputs' (a negative value implying a demand) of type j for our storage model. Under these assumptions, we establish certain limit distributions for the joint process (Z(t), L(t)), where Z(t) (defined in (2)) is the level of storage at time t and L(t) (defined in (3)) is the demand lost due to shortage of supply during [0, t]. Different limit distributions are obtained for the cases when the ‘average stationary input' ρ, as defined in (5), is positive, zero or negative.

In most contemporary queueing network analysis, the assumption is made that a network is in a state of equilibrium. That is, the network's state space probabilities are assumed to be time independent. It is therefore important to be able to quantify precisely when this assumption is valid. Furthermore there are also situations in which it is desirable to model the transient behaviour of networks which occur in practice, such as computer and communication systems. For example, the immediate effects of component failure or instantaneous alteration of system status may be predicted.

In this paper an iterative solution is derived to the time-dependent Kolmogorov equations of queueing networks, and is shown to be convergent. From the solution, modelling of transient situations becomes possible and the time periods during which the equilibrium assumption can and should not be made may be identified; for example in terms of a time constant which is easily computed to a first-order approximation.

A non-linear stochastic model for the single-server machine interference problem is investigated. The service rate is a rational function of the number of failed machines, the numerator as well as the denominator of the function being linear in n. Depending on the relative values of the parameters involved, the service rate may increase or decrease with n, exhibiting the incentive or disincentive effect. The analysis is carried out by the usual diffusion approximation technique. A notable feature of this model is the threshold effect exhibited in the mean queue size.

Exact expressions are derived for the transition probabilities of the birth-death process with parameters and which serves as a queueing model where potential customers are discouraged by queue length.

The possibility that interspecific contest tactics might be able to stabilize two species predator-prey populations is investigated using a matrix model of the population. It is found that if the interspecific tactics have no influence on intraspecific encounters, the non-trivial equilibrium population composition is necessarily unstable. The result provides some insight into a reported instability of a predator-prey population with ‘stable' strategies.

Conditions for the non-sure extinction of Galton-Watson processes with generation dependence are obtained. Also a condition is given for such processes to have a strictly increasing probability generating function.