A biphase image, representing the normal and degenerated fibres in a vertical cross-section of a nerve, is considered. A random set model based on a Gibbs point process is proposed for the union of the two phases. A kind of independence between the degeneration process and the original fibres is defined and tested.

We use the fact that the Palm measure of a stationary random measure is invariant to phase space change to generalize the light traffic formula initially obtained for stationary processes on a line to general spaces. This formula gives a first-order expansion for the expectation of a functional of the random measure when its intensity vanishes. This generalization leads to new algorithms for estimating gradients of functionals of geometrical random processes.

Formulas for anisotropic stereology of fibre and surface processes are presented. They concern the relation between second-order quantities of the original process and its projections and sections. Various mathematical tools for handling these formulas are presented, including stochastic optimization. Finally applications in stereology are discussed, relating to intensity estimators using anisotropic sampling designs. Variances of these estimators are expressed and evaluated for processes with the Poisson property.

Quite often in the statistical analysis of medical and biological problems, data are images corresponding to entire objects that include smaller objects within them. In these cases, we need models of random closed sets (RACS) confined to compact subsets of the plane. There is no room for stationarity hypotheses and the increase of statistical information comes from independent replicates of the same phenomena rather than increasing our sample window. We investigate practical methods of modelling RACS by means of circumscribed balls, leading to a natural definition of location, size and shape. We discuss the possibilities of using these random variables in order to define statistical spaces of RACS that will allow us to use maximum likelihood methods.

The corpuscle problem of Wicksell is discussed. We give a numerical quadrature of Gauss–Chebyshev type for Wicksell's integral equation which combines a size distribution of discs on a sectional plane with that of spheres. We also give an estimation procedure of three-dimensional size distributions based on this quadrature and examine its theoretical properties. In practice, we need a smoothing technique for empirical distribution functions before applying this estimator. Simulation results are given. Our idea also is applied to the thick section case and an analysis of microscopic data is given.

Simulation procedures for typical Johnson-Mehl crystals generated under various models for random nucleation are proposed. These procedures include algorithms for simulating spatio-time-inhomogeneous Poisson processes. Empirical results for a particular class of Johnson-Mehl tessellations in two and three dimensions show remarkably different crystals.

A general method for solving stereological problems for particle systems is applied to polyhedron structures. We suggested computing the kernel function of the respective stereological integral equation by means of computer simulation. Two models of random polyhedrons are investigated. First, regular prisms are considered which are described by their size and shape. The size-shape distribution of a stationary and isotropic spatial ensemble of regular prisms can be estimated from the size-shape distribution of the polygons observed in a section plane. Secondly, random polyhedrons are constructed as the convex hull of points which are uniformly distributed on surfaces of spheres. It is assumed that the size of the polyhedrons and the number of points (i.e. the number of vertices) are random variables. Then the distribution of a spatially distributed ensemble of polyhedrons is determined by its size-number distribution. The corresponding numerical density of this bivariate size-number distribution can be stereologically determined from the estimated numerical density of the bivariate size-number distribution of the intersection profiles.

In the early 1940s David Kendall conjectured that the shapes of the ‘large' (i.e. large area A) convex polygons determined by a standard Poisson line process in the plane tend to circularity (as A increases). Subject only to one heuristic argument, this conjecture and the corresponding two results with A replaced in turn by number of sides N and perimeter S, are proved. Two further similar limiting distributions are considered and, finally, corresponding limiting non-deterministic shape distributions for the small polygons are determined.

We obtain a weak approximation for the reduced family tree in a near-critical Markov branching process when the time interval considered is long; we also extend Yaglom's theorem and the exponential law to this case. These results are then applied to the problem of estimating the age of our most recent common female ancestor, using mitochondrial DNA sequences taken from a sample of contemporary humans.

We present a stochastic model for the movement of a white blood cell both in uniform concentration of chemoattractant and in the presence of a chemoattractant gradient. It is assumed that the rotational velocity is proportional to the weighted difference of the occupied receptors in the two halves of the cell and that each of the receptors stays free or occupied for an exponential length of time. We define processes corresponding to a cell with 2nß + 1 receptors (receptor sites). In the case of constant concentration, we show that the limiting process for the rotational velocity is an Ornstein–Uhlenbeck process. Its drift coefficient depends on the parameters of the exponential waiting times and its diffusion coefficient depends in addition also on the weight function. In the inhomogeneous case, the velocity process has a diffusion limit with drift coefficient depending on the concentration gradient and diffusion coefficient depending on the concentration and the weight function.

Each feasible transition between two distinct states i and j of a continuous-time, uniform, ergodic, countable-state Markov process gives a counting process counting the number of such transitions executed by the process. Traffic processes in Markovian queueing networks can, for instance, be represented as sums of such counting processes. We prove joint functional central limit theorems for the family of counting processes generated by all feasible transitions. We characterize which weighted sums of counts have zero covariance in the limit in terms of balance equations in the transition diagram of the process. Finally, we apply our results to traffic processes in a Jackson network. In particular, we derive simple formulas for the asymptotic covariances between the processes counting the number of customers moving between pairs of nodes in such a network.

We consider M/G/1 queues with exhaustive service and generalized vacations, where at the end of every busy period the server either follows a mixed vacation policy from a given vacation policy set or stays idle. A simple recursive formula for the moments of the stationary waiting time is provided. This formula results in the decomposition property for our model immediately. It also enables us to derive many existing results for the M/G/1 queues with various vacation policies.

We consider a multiserver queuing process specified by i.i.d. interarrival time, batch size and service time sequences. In the case that different servers have different service time distributions we say the system is heterogeneous. In this paper we establish conditions for the queuing process to be characterized as a geometrically Harris recurrent Markov chain, and we characterize the stationary probabilities of large queue lengths and waiting times. The queue length is asymptotically geometric and the waiting time is asymptotically exponential. Our analysis is a generalization of the well-known characterization of the GI/G/1 queue obtained using classical probabilistic techniques of exponential change of measure and renewal theory.

We continue our investigation of the batch arrival-heterogeneous multiserver queue begun in Part I. In a general setting we prove the positive Harris recurrence of the system, and with no additional conditions we prove logarithmic tail limits for the stationary queue length and waiting time distributions.