We obtain results connecting the distribution of the random variables Y and W in the supercritical generalized branching processes introduced by Crump and Mode. For example, if β > 1, EYβ and EWβ converge or diverge together and regular variation of the tail of one of Y, W with non-integer exponent β > 1 is equivalent to regular variation of the other. We also prove analogous results for the continuous-time continuous state-space branching processes introduced by Jirina.

A criterion for ergodicity of lattice interactions has been given by Dobrushin [2], and improved by Harris [3] and Holley [5]. In this paper we present a simplified derivation of these results, and obtain stronger conditions for interactions on the one- and two-dimensional integer lattices, and the 3–Bethe lattice.

For the Ek/G/1 queue with finite waiting room the phase technique is used to analyse the Markov chain imbedded in the queueing process at successive instants at which customers complete service, and the distribution of queue length is obtained. The limit as the size of the waiting room becomes infinite is found.

Maynard Smith and Price (1973) have shown that the methods and terminology of Game Theory can be used to study the evolution of various strategies in animal species. They write aij as the mean ‘value’ to the user of strategy i when his opponent uses strategy j; these values aij are related to the fitness of the user of i, in the sense that, if aij is large, it is advantageous to use strategy i when one's opponent uses strategy j, and this advantage will be reflected in an increase in one's reproductive potential. For example, these may be a contest for territory, or dominance rights.

Effects of sampling procedures on the probability distributions of pedigrees are illustrated. Elimination of these effects by suitable conditioning is demonstrated.

Balanced, autosomal chromosome mutations do not change the genotype of animals that are hetero- or homozygotes for the mutation. However, in the gametes produced by a heterozygote animal, there normally exists a fraction with unbalanced chromosome complements. Such gametes are functionally effective in mammals and produce embryos with unbalanced genotypes, which leads to zygotic loss. The fertility of chromosomally heterozygote animals is thus lower than the fertility of homozygote animals. This fertility decrease of heterozygotes has been shown in a number of studies on translocations in mice, cattle and humans. The problem is, how such mutations can ever increase in a population until all animals are homozygotes for the chromosome change.

Following the success of the first meeting in Sheffield in April 1973, a second symposium was held in the School of Biological Sciences, University of Sussex, on 26th and 27th March 1974. (See Adv. Appl. Prob., 6, 1–20, for abstracts of Sheffield meeting.) The objective was, as before, to provide an opportunity for the presentation of papers on mathematical aspects of genetics.

In populations of species such as cattle and sheep which are being selected for commercially important quantitative traits like milk yield and growth rate, the breeder cannot avoid overlapping generations since the female reproductive rate is so low. There is a classical theory for predicting the rate of response to selection in continuing programmes in which the selection differentials and age structure of the population do not change (Rendel and Robertson, (1950)). In a new programme, however, these rates of response are approached only asymptotically and may not be a useful approximation for several years. Recurrence relations have been developed by several authors to enable predictions of short term responses, but these can be simplified with Leslie's (1945) matrix formulation. Some basic principles will be given here; for details see Hill (1974).

The structure of the probability space associated with a general point process, when regarded as a counting process, is reviewed using the coincidence formalism. The rest of the paper is devoted to the class of regular point processes for which all coincidence probabilities admit densities. It is shown that their distribution is completely specified by the system of coincidence densities. The specification formalism is stressed for ‘completely’ regular point processes. A construction theorem gives a characterization of the system of coincidence densities of such a process. It permits the study of most models of point processes. New results on the photon process, a particular type of conditioned Poisson process, are derived. New examples are exhibited, including the Gauss-Poisson process and the ‘fermion’ process that is suitable whenever the points are repulsive.

The paper examines those continuous time Markov processes Z(·) on the positive integers which have the ‘skip free upwards’ property, with regard to their asymptotic behaviour in the event of Z(t) tending to infinity. The behaviour is characterised in terms of the convergence or divergence of an appropriate function of Z(t), and the description is improved by central limit and iterated logarithm theorems. The conditions of the theorems are expressed entirely in terms of the matrix Q of instantaneous transition rates for Z(·). The method is applied, by way of example, to the super-critical linear birth and death process.

A finite number of colonies, each subject to a simple birth-death and immigration process is studied under the condition of migration between the colonies.

Kolmogorov's backward equations for the process are solved for some special cases, and a sequence of functions uniformly converging to the p.g.f. of the process is given for the general case. Further, a set of algebraic equations for the extinction probabilities are studied for the process without immigration, and a necessary and sufficient condition that the extinction probability be one is obtained.

An approximation is found to the distribution of the maximum number of infectives present at any time during the course of a closed epidemic. The technique used is applicable to a commonly occurring type of random walk problem where there is a curved absorbing boundary which is far from the mean path except over a narrow range.

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.

Particles enter a state space at random times. Each particle travels in the space independent of the other particles until its death. Functionals of the particle system are studied with strong laws and central limit theorems being obtained.

The theory of the linear model is incomplete in that it fails to deal with variables possessing infinite variance. To fill an important part of this gap, we give an unbiased estimate, the “screened ratio estimate”, for λ in the regression E(X|Z) = λX; X and Z are linear combinations of independent, identically distributed symmetric random variables that are either stable or asymptotically Pareto distributed of index α ≤ 2. By way of comparison, the usual least squares estimate of λ is shown not to converge in general to any constant when α < 2. However, in the autoregression Xn = a1Xn-1 + … + akXn-k + Un, the least squares estimates are shown to be consistent as long as the roots of 1 - a1x2 - a2x2 - … - akxk = 0 are outside the complex unit circle, Xn is independent of Un+j,j ≥ 1, and the Un are independent and identically distributed and in the domain of attraction of a stable law of index a ≤ 2. Finally, the consistency of least squares estimates for finite moving averages is established.

A control scheme for the immunisation of susceptibles in the Kermack-McKendrick epidemic model for a closed population is proposed. The bounded control appears linearly in both dynamics and integral cost functionals and any optimal policies are of the “bang-bang” type. The approach uses Dynamic Programming and Pontryagin's Maximum Principle and allows one, for certain values of the cost and removal rates, to apply necessary and sufficient conditions for optimality and show that a one-switch candidate is the optimal control. In the remaining cases we are still able to show that an optimal control, if it exists, has at most one switch.

A single server faces a renewal stream of low and another one of high priority customers, one of them being a Poisson stream. For the preemptive resume and head-of-the line disciplines the transient behaviour of the queue is studied. Various limiting distributions are also obtained.