In design stereology, and in the context of geometric sampling in general, the problem often arises of estimating the integral of a bounded non-random function over a bounded manifold D ⊂ ℝn by systematic sampling with geometric probes. Variance predictors, often based on Matheron's theory of regionalized variables, are available when the relevant function is sampled at the points of a grid intersecting D, but not when the dimension of the probes is greater than zero. For instance, the volume of a bounded object may be estimated using parallel systematic planes, which amounts to sampling on ℝ1 with systematic points, or using parallel systematic slabs of thickness t > 0, which amounts to sampling on ℝ1 with non-overlapping systematic segments of length t > 0. Useful variance predictors exist for the former case, but not for the latter. In this paper we set out a general scheme to predict estimation variances when the dimension of either D, or of the probes, is n. We make some progress when both dimensions are equal to n, and obtain explicit results for n = 1 (e.g. for systematic slice sampling). We check and illustrate our results for the volume estimators of ellipsoids and of rat lung.

An unbiased stereological estimator for surface area density is derived for gradient surface processes which form a particular class of non-stationary spatial surface processes. Vertical planar sections are used for the estimation. The variance of the estimator is studied and found to be infinite for certain types of surface processes. A modification of the estimator is presented which exhibits finite variance.

Mixings of stationary Poisson hyperplane tessellations in d-dimensional Euclidean space are considered. The intention of the paper is to show that the 0-cell of a mixed stationary Poisson hyperplane tessellation Y is in some sense larger than that of stationary Poisson hyperplane tessellations Y' with the same intensity and directional distribution as Y. Related results concerning the moments for the volume of the 0-cell are derived. In special cases, similar statements with respect to the typical cell are proved.

In this paper we study finite clusters in a high density Boolean model with balls of two distinct sizes. Alexander (1993) studied the geometric structures of finite clusters in a high density Boolean model with balls of fixed size and showed that the only possible structure admitted by such events is that all Poisson points comprising the cluster are packed tightly inside a small sphere. When the balls are of varying sizes, the event that the cluster consists of k1 big balls and k2 small balls (both k1, k2 ≥ 1) occurs only when the centres of all big balls are compressed in a small sphere and the centres of the small balls are distributed uniformly inside the region formed by the big balls in such a way that the small balls are totally contained inside the big balls. We also show that it is most likely that a finite cluster in a high density Boolean model with varying ball sizes is made up only of small balls.

This paper considers a branching process generated by an offspring distribution F with mean m < ∞ and variance σ2 < ∞ and such that, at each generation n, there is an observed δ-migration, according to a binomial law Bpvn*Nnbef which depends on the total population size Nnbef. The δ-migration is defined as an emigration, an immigration or a null migration, depending on the value of δ, which is assumed constant throughout the different generations. The process with δ-migration is a generation-dependent Galton-Watson process, whereas the observed process is not in general a martingale. Under the assumption that the process with δ-migration is supercritical, we generalize for the observed migrating process the results relative to the Galton-Watson supercritical case that concern the asymptotic behaviour of the process and the estimation of m and σ2, as n → ∞. Moreover, an asymptotic confidence interval of the initial population size is given.

The paper considers stability and instability properties of the Markov chain generated by the composition of an i.i.d. sequence of random transformations. The transformations are assumed to be either linear mappings or else mappings which can be well approximated near 0 by linear mappings. The main results concern the risk probabilities that the Markov chain enters or exits certain balls centered at 0. An application is given to the probability of extinction in a model from population dynamics.

A number of stationary stochastic processes are presented with properties pertinent to modelling time series from turbulence and finance. Specifically, the one-dimensional marginal distributions have log-linear tails and the autocorrelation may have two or more time scales. Discrete time models with a given marginal distribution are constructed as sums of independent autoregressions. A similar construction is made in continuous time by considering sums of Ornstein-Uhlenbeck-type processes. To prepare for this, a new property of self-decomposable distributions is presented. Also another, rather different, construction of stationary processes with generalized logistic marginal distributions as an infinite sum of Gaussian processes is proposed. In this way processes with continuous sample paths can be constructed. Multivariate versions of the various constructions are also given.

Let {Yn | n = 1, 2,…} be a stochastic process and M a positive real number. Define the time of ruin by T = inf{n | Yn > M} (T = +∞ if Yn ≤ M for n = 1, 2,…). Using the techniques of large deviations theory we obtain rough exponential estimates for ruin probabilities for a general class of processes. Special attention is given to the probability that ruin occurs up to a certain time point. We also generalize the concept of the safety loading and consider its importance to ruin probabilities.

Modelling malaria with consistency necessitates the introduction of at least two families of interconnected processes. Even in a Markovian context the simplest fully stochastic model is intractable and is usually transformed into a hybrid model, by supposing that these two families are stochastically independent and linked only through two deterministic connections. A model closer to the fully stochastic model is presented here, where one of the two families is subordinated to the other and just a unique deterministic connection is required. For this model a threshold theorem can be proved but the threshold level is not the one obtained in a hybrid model. The difference disappears only when the human population size approaches infinity.

Much of the literature in reliability and survival analysis considers failure models indexed by a single scale. There are situations which require that failure be described by several scales. An example from reliability is items under warranty whose failure is recorded by time and amount of use. An example from survival analysis is the death of a mine worker which is noted by age and the duration of exposure to dust.

This paper proposes an approach for developing probabilistic models indexed by two scales: time, and usage, a quantity that is related to time. The relationship between the scales is described by an additive hazards model. The evolution of usage is described by stochastic processes like the Poisson, the gamma and the Markov additive. The paper concludes with an application involving the setting of warranties. Two features differentiate this work from related efforts: a use of specific processes for describing usage, and a use of Monte Carlo techniques for generating the models.

A set of jobs is to be processed on a machine which is subject to breakdown and repair. When the processing of a job is interrupted by a machine breakdown, the processing later resumes at the point at which the breakdown occurred. We assume that the machine uptime is Erlang distributed and that processing and repair times follow general distributions. Simple permutation policies on both machine parameters and the processing distributions are given which minimize the weighted number of tardy jobs, weighted flow times and the weighted sum of the job delays.

The paper studies a model of repairable systems which is flexible enough to incorporate the standard imperfect repair and many other models from the literature. Palm stationarity of virtual ages, inter-failure times and degrees of repair is studied. A Loynes-type scheme and Harris recurrent Markov chains combined with coupling methods are used. Results on the weak total variation and moment convergences are obtained and illustrated by examples with IFR, DFR, heavy-tailed and light-tailed lifetime distributions. Some convergences obtained are monotone and/or at a geometric rate.

Queueing networks have been rather restricted in order to have product form distributions for network states. Recently, several new models have appeared and enlarged this class of product form networks. In this paper, we consider another new type of queueing network with concurrent batch movements in terms of such product form results. A joint distribution of the requested batch sizes for departures and the batch sizes of the corresponding arrivals may be arbitrary. Under a certain modification of the network and mild regularity conditions, we give necessary and sufficient conditions for the network state to have the product form distribution, which is shown to provide an upper bound for the one in the original network. It is shown that two special settings satisfy these conditions. Algorithms to calculate their stationary distributions are considered, with numerical examples.

In this paper we develop policies for scheduling dynamically arriving jobs to a broad class of parallel-processing queueing systems. We show that in heavy traffic the policies asymptotically minimize a measure of the expected system backlog, which we call system work. Our results yield succinct, closed-form expressions for optimal system work in heavy traffic.