A basic class of spatial stochastic processes is that in which the phenomenon spreads between static individuals with probability of ‘infection’ falling off with distance according to a contact distribution (Mollison (1972)). The following questions then arise:

(1) How does the velocity of spread depend on the contact distribution? In particular, for which contact distributions is it finite?

(2) Does the phenomenon spread at a steady velocity, or does it spread in jumps?

This second meeting in the series was organised principally by Denis Mollison and Eric Renshaw with the help of their respective departments and a grant from the Royal Society. The fifty or so participants were drawn from various backgrounds, both theoretical and applied, and a diverse programme of twelve papers was presented, embracing both the development of mathematical theory and the collection and analysis of field data. Many of the non-speakers contributed to the lively discussion which prevailed. Short abstracts of the papers are appended, and the authors would be happy to answer requests for further information.

In a previous paper (Dagsvik (1975)) the waiting time process of the single server bulk queue is considered and a corresponding waiting time equation is established. In this paper the waiting time equation is solved when the inter-arrival or service time distribution is a linear combination of Erlang distributions. The analysis is essentially based on algebraic arguments.

A point process is obtained by considering a square lattice of side p½, 0 < p ≦ 1, and supposing that each lattice point contains a point of the process or does not independently, with probabilities p and 1 – p. The family of processes thus includes the deterministic lattice for p = 1, and has the Poisson as a limit as p → 0.

Estimation of the properties of particles embedded in a solid medium via observations on random plane sections is studied. The particular properties considered are those associated with steel inclusions, for example, projected area fraction, total caliper length per unit volume and elongation factor. The approach is general, and with the minimum of assumptions about the particle shape, formulae are obtained for suitable estimators and their variances. Generalised moment relations are derived for the case of given, arbitrary shapes. Estimations of the density of particle centres based upon reciprocal observations are shown to lack robustness to small changes in shape and a more reliable density estimator is discussed. Consideration is given to the case of particles with elliptical profiles and to the use of aggregated results which arise when using automatic scanning equipment. Results are given for the case when there is length-biassing of the intercept distribution due to edge-effects for finite sample areas. Finally, the implication of a positive lower resolution point is discussed.

Competition models for spatially distributed individuals may be developed from the basic assumption that each individual will exert a competitive influence within a circle, centered on the individual's location and with radius determined by a random variable, R (Gerrard (1969)).

Variability in space of plankton can arise from stochastic effects on growth and predation, and should be smoothed by horizontal diffusion of the water. The interactions of these factors within a non-linear system should permit small-scale perturbations to appear as larger-scale patches. A simple simulation illustrates this possibility.

This meeting was envisaged by the organisers as the first of a continuing series of short, informal conferences within the general area of applied stochastic processes. The twenty-seven participants were drawn mostly from statistics and mathematics departments of British Universities, but also included a small number of biologists and geographers. The meeting was notable for its pleasantly informal atmosphere, and the consequent lively interchange of ideas, although for future meetings in the series a better balance between statisticians and applied scientists should, we feel, be effected.

For every N ≧ 1, let be a real-valued stationary process, and let . Suppose that and var, where ∊N and τN are positive null sequences. Limiting distributions of as N → ∞ are obtained for the cases τN = ∊N and τN = o(∊N). These results are established by an extension of a method due to Moran. The theory is illustrated by a variety of applications to genetic models.

A shot process with bursts of events is constructed using an occurrence time structure similar to that of clustering point processes. The characteristic function and a simple functional form for the power spectral density of the process are found. It is shown that the burst property might not be observable. Applications of a specific example of this process are given in modeling a 1/f noise commonly found in some electronic components.

A continuous-time Markov branching process is modified to allow some dependence of offspring generating function on population size. The model involves a given population size M, below which the offspring generating function is supercritical and above which it is subcritical. Immigration is allowed when the population size is 0. The process has a stationary measure, and an expression for its generating function is found. A limit theorem for the stationary measure as M tends to ∞ is then obtained.

The steady-state input and output processes are considered for a birth-and-death queueing model with N waiting positions (0 ≦ N ≦ ∞), s servers (1 ≦ s ≦ ∞) and an arbitrary queueing discipline. Let an index n indicate that the quantity in question depends on the system state but not on time t. The instantaneous arrival rate is λ, the probability of balking (i.e., not trying to obtain service) being ξn. The instantaneous departure rate, μn, of customers having joined the system is the sum of the rate of service completions and the rate of defections before service completion. Three cases are considered. We start by ignoring balking customers; in the first case treating a lost customer neither as an input nor as an output, then secondly as both. Finally, balking and lost customers are considered both as inputs and outputs.

The migration and death model (Bartlett (1949)) is useful in describing animal movement among distinct areas.

Cell kinetics deals with experiments and theories designed to determine the proportion of cells in a tissue which are engaged in replication, and the times they take to pass through the various stages of their reproductive cycle.

Some years ago Hammersley and Clifford considered the problem of assigning probabilities to colourings of a graph according to certain local conditions. Subsequent expositions such as those of Grimmett, Besag and Preston, whilst providing more elegant derivations, have shown that the assignment of probabilities coincides with the limiting distribution of spatial-temporal processes arising from theoretical physics. There has been relatively little work on spatial-temporal processes outside this class. In this paper the approximate behaviour of an alternative class of processes is investigated. Topics will include statistical estimation of parameters and existence of unique equilibria.

This paper (Cohen (1975)) establishes a new class of ergodicity theorems for the age structure of populations, and proposes applications.

The paper explores three different aspects of spatial modelling for systems in statistical equilibrium; that is, systems which are both mean and covariance stationary. Attention is primarily directed at models relating to observations for a single time period.

The moment-generating function of the traffic noise from a stream of vehicles with identical noise emissions cannot be readily inverted. If the emissions are not equal, this generating function can be inverted to obtain the exact form of the distribution function in some particular cases. Noise intensity has a maximally skew stable distribution with exponent 1/2 for observers on the highway, whatever the distribution of emissions. The distribution at any distance from the highway is an exponentially modified stable law with exponent 1/2 for an improper exponential distribution of emissions, and an infinite series involving this stable law and iterated error functions when emissions have an exponential distribution. A doubly stochastic process for emissions produces distributions of traffic noise intensity in the domain of attraction of skew stable laws with exponent α, 1/2 < α < 2. The inverse Gaussian (exponentially modified skew stable law with exponent 1/2) is recommended as the best choice of a two-parameter family for fitting traffic noise intensity distributions.