Asymptotic formulas and Laplace–Stieltjes transforms are derived for the first two moments of a renewal process with a random number of delays. These are simplified when all the delays follow the same distribution. An asymptotic occupancy result is also derived for two-stage renewal processes with random numbers of delays. As an example, a demographic model of conception and birth is discussed. This model represents the sequence of live births to a woman as a renewal process. If the woman practises birth control after achieving her desired family composition, the renewal process has a random number of delays.

This article deals with the generalized Smoluchowski process, {n(t), t ≧ 0}, defined by the temporal fluctuating of the numbers of randomly moving particles contained in m < ∞ disjoint regions of space. The relationship of the Smoluchowski process {n(t), t ≧ 0} to the emigration–immigration process is discussed and conditions for their equivalence are presented.

A stochastic model for the competition between two species is considered and a limit theorem is obtained for the size of the surviving species. A limit theorem is also obtained for the probability that a given species should survive the other.

A model for the variation in time of the fitness distribution in a large haploid population is shown to have simple limiting properties which can be elucidated in fairly explicit terms. The novel feature is that mutation is not assumed to cause a small perturbation in fitness but to bring down the evolutionary ‘house of cards’. A threshold phenomenon appears: if a certain inequality holds the limiting distribution is a skewed version of the mutant fitness distribution, but otherwise an atom of probability builds up at the upper limit of fitness.

The vast majority of reliability analyses assume that components and system are in either of two states: functioning or failed. The present paper develops basic theory for the study of systems of components in which any of a finite number of states may occur, representing at one extreme perfect functioning and at the other extreme complete failure. We lay down axioms extending the standard notion of a coherent system to the new notion of a multistate coherent system. For such systems we obtain deterministic and probabilistic properties for system performance which are analogous to well-known results for coherent system reliability.

It is known that for a Bienaymé– Galton–Watson process {Zn} whose mean m satisfies 1 < m < ∞, the limiting random variable in the strong limit theorem can be represented as a random sum of i.i.d. random variables and hence that convergence rate results follow from a random sum central limit theorem.

This paper develops an analogous theory for the case m = ∞ which replaces ‘sum' by ‘maximum'. In particular we obtain convergence rate results involving a limiting extreme value distribution. An associated estimation problem is considered.

Let denote the simple branching process with Z0 = 1 and let G denote the distribution function of Z1. Suppose G satisfies x−α−γ(x)≦1 − G(x) ≦ x−α+γ(x) for large x, where (i) 0 < α < 1, (ii) γ (x) is non-negative and non-increasing, (iii) xγ(x) is non-decreasing and (iv) Then limn→∞α n log (Zn + 1) converges almost surely to a non-degenerate finite random variable W satisfying P(W = 0) = q = probability of extinction of the process.

Linear and non-linear secondary Poisson process models are developed for the level of carboxyhemoglobin in cigarette smoking. The moments of the time of first exit above a critical threshold are expanded as power series in the initial state of the process. Recurrence relations for the coefficients in the moment series are solved numerically. An explicit formula involving Poisson probability functions is obtained for the mean exit time of shot noise.

The optimal strategy for a class of non-Markov decision processes is characterised and has the property that changes of action may occur between successive transitions of the process. Results are given which enable the optimal strategy to be computed iteratively.

A recursion formula is obtained by rearranging Lawrance's (1973) result concerning the superposition of independent stationary point processes for which there exist joint probability density functions for the intervals between successive points. When these component point processes are identically distributed, the formula can in principle be inverted to describe their probability structure given that of the superposition process.

Let Z(t) ··· (Z1(t), …, Zk (t)) be an indecomposable critical k-type age-dependent branching process with generating function F(s, t). Denote the right and left eigenvalues of the mean matrix M by u and v respectively and suppose μ is the vector of mean lifetimes, i.e. Mu = u, vM = v.

It is shown that, under second moment assumptions, uniformly for s ∈ ([0, 1]k of the form s = 1 – cu, c a constant. Here vμ is the componentwise product of the vectors and Q[u] is a constant.

This result is then used to give a new proof of the exponential limit law.

The continuous-time Markov model of a multigrade organization is extended in several ways. Firstly the internal transitions and the leaving process are generalized to a semi-Markov formulation which allows for the inclusion of well-authenticated leaving distributions such as the mixed exponential distribution. The previous assumption of Poisson recruitment is then generalized to allow for a time-dependent Poisson arrival distribution in which the instantaneous probability of an arrival is a mixture of exponential terms. Finally we extend the capital-related manpower model to describe a multigrade organization.

Random processes of convex plates and line segments imbedded in R3 are considered in this paper, and the expected values of certain random variables associated with such processes are computed under a mean stationarity assumption, by resorting to some general formulas of integral geometry.

The concept of strong ergodicity for discrete-time homogeneous Markov chains has been characterized in several ways (Dobrushin (1956), Lin (1975), Isaacson and Tweedie (1978)). In this paper the characterization using mean visit times (Huang and Isaacson (1977)) is extended to continuous-time Markov chains. From this it follows that for a certain subclass of continuous-time Markov chains, X(t), is strongly ergodic if and only if the associated embedded chain is Cesaro strongly ergodic.

Many species of birds have a characteristic clutch size which is either fixed at k or is of the form k or k + 1 for some appropriate integer k. In this paper we show, using a multitype Galton–Watson process to model a bird population, that such behaviour can correspond to maximization of the probability of survival of the species to time t for each finite t. This is also a conclusion which might be drawn from the theory of natural selection and hence provides some mathematical evidence of the force of evolution. The results of the paper rest on a bounding of probability generating functions.

A model is given to describe the concurrent evolution of vocabularies of different languages belonging to the same family. Lexical change involves the replacement of old words by new words acquired by inheritance, borrowing or some other form of innovation. This process is described by a set of simultaneous differential equations involving the probabilities that, at any time, languages share similar words for any particular meaning. Various special cases of the model are compared using data for some Indo-European languages.

Consider a finite irreducible aperiodic Markov chain with nearly-completely decomposable stochastic matrix: i.e. a Markov chain for which the states can be grouped into disjoint aggregates, in such a way that the probabilities of transition between states of the same aggregate are large compared to the probabilities of transition between states belonging to different aggregates. Let Ω be a subset of one of the aggregates. Second-order approximations are determined for the first and second moments of the time to reach Ω and the return time to Ω.

We derive explicit formulas for the second moments of the absorption time matrices in the Markov renewal branching process. These formulas may easily be computationally implemented and are useful in the iterative computation of the semi-Markov matrices, which give the distributions of the duration and of the number of customers served in a busy period of a great variety of complex queueing models.

For homogeneous, two-dimensional random field ξ(t), t ∈ R2 we develop the ‘half' spectral theory sufficient to rigorously define its envelope η (t). We then specialise to the case of ξ Gaussian, which implies η is Rayleigh, and consider the mean value of a certain characteristic of the sets {t:η(t) ≧ u} (u ≧ 0). From this we deduce some qualitative information about the sample path behaviour of the Rayleigh field η .

Let Z(t) denote the number of particles alive at time t in a critical age-dependent branching process. It is proved that, for k ≧ 1, there exists a constant Ak > 0 such that t2P(Z(t) = k)→Ak as t→∞.