The problem discussed is that of controlling optimally the release of water from a finite dam, when the release rate may vary continuously and the optimality is defined in terms of a cost structure imposed on the operation of the dam. A diffusion model is suggested and by considering a family of plausible output policies, the control problem is reduced to the solution of a free boundary problem associated with a certain partial differential equation. A set of necessary conditions for the optimal choice of these boundaries is established and a method of solution is suggested. By using this method, together with well-established computational techniques, numerical solutions are obtained. These numerical solutions indicate that this optimal policy does not result in very much improvement over a much simpler policy where the output rate is constrained to take only two values.

In a paper of the same title, Ewens has considered the problem of inferring whether the genotypic frequencies observed in a small sample are consistent with a particular population model, in which all types are selectively neutral. The present paper gives the general theory of such sampling schemes, when the sampling is from either a deterministic, or a stochastically varying, population.

The theory is illustrated on a population model which was introduced by Karlin and McGregor. Various results are proved for both the population, and its sample. It is shown that Ewens' sampling theory applies to this population model, and conjectures as to why different population models may have the same sampling theory are made.

The imago of the peppered moth exists in two sharply distinguished forms, speckled and black. Clarke and Sheppard (1966) have shown that the frequency of the black moth decreases from 99% in Liverpool to 10% in North Wales. If these frequencies are in equilibrium, albeit dynamic equilibrium, then such a gradient is called a cline. Haldane (1948) briefly describes several clines, and then defines a model that may represent the life-cycle of certain species. The assumptions defining his model are as follows.

In 1958 Kimura introduced his Maximum Principle in the Genetical Theory of Natural Selection, which has since been described in several books (Ewens (1969), Crow and Kimura (1970), Kojima (1970)). I shall use Ewens' notation and treatment which, although it uses the discrete-generation model, is formally equivalent to Kimura's continuous-time treatment.

We obtain results connecting the distributions of the random variables Z1 and W in the supercritical Galton-Watson process. For example, if a > 1, and converge or diverge together, and regular variation of the tail of one of Z1, W with non-integer exponent α > 1 is equivalent to regular variation of the tail of the other.

It is shown that two different population models are equivalent to each other when a certain marginal stationary distribution is considered.

For a Markov process in discrete time, both (a) the time-dependent behaviour of the process at time t, and (b) the behaviour of moments (if finite) of the process, whether or not stationary, as a function of time-lag t, are determined by solutions to a single equation, the key equation. The state space of the process is required only to be both a measure space, and a locally convex topological vector space, which may have infinite dimension. The asymptotic behaviour of both (a) and (b), for large times t, is shown to be (nearly) the same, being proportional to tξe−βt, where the exponent β is calculable from an eigenvalue of a linear operator, describing the Markov process. For a large class of Markov processes, the spectrum of this operator is shown to consist of discrete eigenvalues only. Queueing applications are discussed.

Policies of immunising the susceptibles in the general stochastic epidemic are considered. With costs assigned to the infection and immunisation of susceptibles, an optimal policy is found which at any stage minimises the expected future cost. Optimal policies are also found for the general deterministic epidemic and for an epidemic with carriers model. The three policies are compared and numerical examples are provided. An optimal policy is described in the situation when both isolation of infectives and immunisation of susceptibles are allowed.

There is now a large literature on linkage disequilibrium between pairs of loci, both for selection in infinite populations and for neutral genes in finite populations, but there have been few studies with more loci. Bennett (1954) showed how the frequencies of chromosomes with any number of neutral genes would change in an infinite population, and the author (unpublished) has extended Bennett's results to find expected changes in chromosome frequencies with up to six loci in finite populations. For two linked neutral genes in finite populations the expected disequilibrium is zero, but the variance of the disequilibrium or the correlation of gene frequencies in segregating populations has been found. This has been done by Monte Carlo simulation (Hill and Robertson (1968)), but an approximation can be obtained by diffusion methods (Ohta and Kimura (1969)) and the asymptotic values using inbreeding theory (Sved (1971)). In this note we discuss the case of disequilibrium between three neutral loci and show how it relates to that between two loci.

To treat the transient behavior of a system modeled by a stationary Markov process in continuous time, the state space is partitioned into good and bad states. The distribution of sojourn times on the good set and that of exit times from this set have a simple renewal theoretic relationship. The latter permits useful bounds on the exit time survival function obtainable from the ergodic distribution of the process. Applications to reliability theory and communication nets are given.

Many physical problems are described by stochastic processes with stationary increments. We present a general description of such processes. In particular we give an expression of a process in terms of its increments and we show that there are two classes of processes: diffusion and asymptotically stationary. Moreover, we show that the nth increments are given by a linear filtering of an arbitrary stationary process.

The paper discussed the value of the ‘Optimum Model” in describing the inter-relationship between metric characters and natural fitness. This model represents one extreme of a whole spectrum of models. It can be used to predict changes in the mean of the character under natural selection, but the basic premises underlying such use are not well understood.

As an alternative to conventional discrete time models for stochastic processes that fluctuate within the sampling interval, we propose difference equations containing non-integral lags. We discuss the problems of stability, identification and estimation, for which an approximate model is needed. Least squares applied to an approximate Fourier-transformed model yields estimators of the coefficients that are consistent with respect to the true model under some conditions. The conditions are weak when the model contains predetermined variables that obey an “aliasing condition”; estimators of the lags as well as coefficients can then be found that are consistent, efficient and satisfy a central limit theorem. Optimal estimators for stochastic difference-differential equations are also available.

We illustrate a technique for computing certain integrals that arise in probability theory by giving a new derivation of a formula of Kingman. This formula contains the joint distribution of the processes F(t) = inf {s: X(t + s) = b} and B(t) = inf{s: X(t - s) = b} where X is a time homogeneous, continuous parameter, Markov process and b is a fixed point in its state space. We then extend this formula to the situation in which b is replaced by a finite set {b1, …, bn}.