In the present paper a spatially homogeneous distance measure of Edwards and Cavalli-Sforza type is derived for multiple allele random genetic drift with a (possibly vanishing) symmetric mutation field, using the technique of random time substitution. Since the mutation field is of gradient type in gene frequency space, the transformed process is proved to be Brownian motion relative to a new Riemannian geometry and a new time measure. The new geometry is conformally related to spherical geometry of the original process but is not of constant curvature, generally. A formula relating the stationary density of the old and new process is derived and the edge length formula for the new geometry on the n-simplex frequency space is given and analysed.

As an approach to modelling the ‘matching’ of optical receptors in animals to the objects they are designed to see, we study the problem of locating regions of high intensity in a point process on the real line, using the counts of points in a movable interval of fixed length. We define performance measures analogous to statistical size and power for this procedure and, for points forming a renewal process, give conditions on the quantiles of the convolutions of the interpoint distribution which ensure that the optimal length for the ‘detector’ is close to that of the ‘object’ to be detected. We show that these conditions are satisfied for a Poisson process. Similar conditions ensure that the optimal length is close to zero, and we give a class of distributions satisfying these conditions. Finally we show that the results can be extended to simple two-dimensional models.

We consider a Markov chain on the d-dimensional (d-allelî) non-negative lattice points with the sum of components being N, for which one-step transition consists of two stages—independent reproduction and random sampling. Convergence to a degenerate diffusion process when N → ∞ is proved. We show how difference among alleles in means and variances of offspring numbers affects the limit diffusion, giving a rigorous multi-allelic version of a result of Gillespie.

The distribution of a fixed sum of independent and identically distributed random variables with the modified geometric distribution is the same as the distribution obtained by the compounding by a binomial distribution of either a negative binomial distribution or a Pascal distribution. This result can be used to obtain three summations for the game score probabilities of a two-person game, and leads to the consideration of various ways of dividing up the trials of the game. The game score probabilities are then used to consider the ‘fairness’ of four games and to analyse various methods of ‘setting’ (or ‘tie-breaking’) the games.

We consider an M/G/1 queue in which the service rate is subject to control. The control is exercised continuously and is based on the observations of the residual workload process. For both the discounted cost and the average cost criteria we obtain conditions which are sufficient for a stationary policy to be optimal. When the service cost rate and the holding cost rates are non-decreasing and convex it is shown that these sufficient conditions are satisfied by a monotonic policy, thus showing its optimality.

A plantation is modelled in terms of discs of random size centred on the points of a two-dimensional lattice, together with rules for partitioning the union of these discs. The resulting sets are functions of random sequences, and some properties of these functions are deduced. Results are given which relate various properties of measurements on a plantation to measurements of individual plants and of plants in isolation. These include limit theorems for complete plantations and for samples. They also lead to a rigorous demonstration of certain well-known empirical relations between typical measurements and survival frequencies and to some new relations which are amenable to test. The model is a formulation of computer simulations which have had success in describing competition in plantations, but are too costly for routine forest management.