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)).

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.

For many years animal behaviourists, sociologists and the like have been studying how animals and humans form groups, and how the groups change in time. A brief review was given of some of the models devised for such group changing situations, and some new models were described.

We examine birth and death processes where the birth/death rates depend on the particular configuration of the population. The easiest case is when the configurations form the subsets of some finite lattice (thus the state space is finite). In this case we look at the relationships between time-reversibility, nearest neighbour interactions, and the equilibrium state being a Markov random field. A more interesting case is when the entities which can be born or die can do so at any point in some bounded region of space (or the plane). This gives us a pure jump process, whose general properties are well-known, (see for example, Feller (1966) Vol. II Chapter X.3). We examine some particular examples and compute equilibrium distributions.

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

We construct an immigration-branching process from an inhomogeneous Poisson process, a parameter-dependent probability distribution of populations and a Markov branching process with homogeneous transition function. The set of types is arbitrary, and the parameter is allowed to be discrete or continuous. Assuming a supercritical branching part with primitive first moments and finite second moments, we prove propositions on the mean square convergence and the almost sure convergence of normalized averaging processes associated with the immigration-branching process.

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.

A simple path transformation is described which connects one-dimensional Brownian motion with the radial part of three-dimensional Brownian motion. This provides simple proofs of various path decompositions for these processes described by David Williams.

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.

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.

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.

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 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.

This paper is a survey of the solutions of various first-passage time problems in the theory of queues.

The relationship between the Wiener-Hopf factorisation of matrices and the solution of systems of certain operator equations is discussed in an algebraic setting. It is shown that the study of the waiting time process of the nth arriving group of the general single server bulk queue leads to equations of that type. This system of equations may be considered as an extension of Lindley's waiting-time equation.

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.

This paper studies a controlled queueing system in which the decisionmaker may change servers according to rules which depend only on the queue length. It is proved that for a given control policy a properly normalised sequence of these controlled queue length processes converges weakly to a controlled diffusion process as the queueing systems approach a state of heavy traffic.