In an irreducible Markov renewal process either all states are periodic or none are. In the former case they all have the same period. Periodicity and the period can be determined by direct inspection from the semi-Markov kernel defining the process. The periodicity considerably increases the complexity of the limits in Markov renewal theory especially for transient initial states. Two Markov renewal limit theorems will be given with particular attention to the roles of periodicity and transient states. The results are applied to semi-Markov and semi-regenerative processes.

Wright (1931) introduced a simple model for a bisexual, diploid population of fixed size and with non-overlapping generations. With N1 males and N2 females, the rate of drift was λ0 = 1 – (N/8N1N2) where N = N1 + N2.

For the generalized single server queueing system described herein weak convergence results are obtained for the processes {Wa, n ≧ 0}, {W(t), t ≧ 0}, and {Q (t), t ≧ 0}, where Wn is the waiting time of customer n, W(t) is the workload of the server at time t, and Q(t) is the number of customers present in the system at time t. We also provide a functional strong law, a functional central limit theorem, and a functional law of the iterated logarithm for various cumulative processes in the system.

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.

When two closely-related species have a partially-overlapping geographical range, it sometimes happens that populations of the two species are very similar in the non-overlapping parts of the range where only one species is present, but that they diverge and are easily distinguishable in the area of overlap where both species occur together. This phenomenon is known as character displacement. One of the best known examples occurs in two species of Rock Nuthatches (Sitta neumayer and S. tephronota) in their area of overlap in Iran (Vaurie (1951)). A plausible explanation of this phenomenon is that it is due to competition between the two species. The purpose of this paper is to construct a model for character displacement based on this idea. A full account of this work will be published elsewhere (Bulmer (1973)).

The model considered here consists of n operating units which are subject to stochastic failure according to an exponential failure time distribution. These operating units are backed up by mn spare units. Failures can be of two types. With probability p (q) a failure is of type 1(2) and is sent to repair facility 1(2) for repair. Repair facility 1(2) operates as a -server queue with exponential repair times having parameter μ1 (μ2). The number of units waiting for or undergoing repair at each of the two facilities is a continuous parameter Markov chain with finite state space. This paper derives limit theorems for the stationary distribution of this Markov chain as n becomes large under the assumption that and mn grow linearly with n. These limit theorems give very useful approximations, in terms of the seven parameters characterizing the model, to a distribution that would be difficult to calculate in practice.

Kennedy (1972) showed that the standard single-server queueing model is continuous. These results are extended to the standard multi-server model here. Even when there is only one server, an additional condition is needed for the queue length 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.

Information is available from a diallel cross consisting of all possible crosses among n parental lines, excluding the parental lines themselves and assuming that reciprocal crosses are identical. The lines are assumed to be a random sample from an infinite population of such lines. The general combining abilities of the lines may be more important than the specific combining abilities of the individual crosses. If so, we may decide to rank the lines on the basis of their estimated general combining abilities and make crosses for further testing, or for commercial production, using only lines with high estimated general combining abilities. Existing statistical selection theory can be used to predict the expected genetic superiority of these crosses. For n = 6, 10, and 20, the best five crosses would be expected to result from crossing the lines as follows, where the line number is its rank order on estimated general combining ability: 1 × 2, 1 × 3, 1 × 4, 2 × 3, 1 × 5. Crosses 2×3 and 1 × 5 may not necessarily be expected to be in the stated order, depending on the value of n. In a particular trial, the estimated general combining abilities may suggest that other crosses should be made, but the practical importance of this is doubtful. Note that if only three crosses are to be selected, 1 × 4 is expected to be better than 2 × 3, i.e., the best three crosses are not obtained by crossing the best three lines. This method of selection based on general combining abilities will be referred to as line selection.

The Hardy-Weinberg law is generally regarded as one of the most important results of population genetics. It was originally proved for the case of populations with distinct generations (Hardy (1908), Weinberg (1908)); a general proof for populations with overlapping generations has apparently not been given before. The case of a single autosomal locus with an arbitrary number of alleles is considered here. Births and deaths are assumed to occur in continuous time. The weak ergodicity property of the birth rate and age structure of such a population, first derived by Norton (1928), is used to establish the fact that allele frequencies tend to constant limits in the absence of mutation, migration, selection and genetic drift.

We consider immigration-branching processes constructable from an inhomogeneous Poisson process, a sequence of population probability distributions, and a homogeneous branching transition function. The set of types is arbitrary, and the process parameter is allowed to be discrete or continuous. For the branching part a weak form of positive regularity, criticality, and the existence of second moments are assumed. Varying the conditions on the immigration law, we obtain several results concerning asymptotic extinction, the rate of extinction, and limiting distribution functions of properly normalized, vector-valued counting processes associated with the immigration branching process. The proofs are based on the generating functional method.

The intrinsic random functions (IRF) are a particular case of the Guelfand generalized processes with stationary increments. They constitute a much wider class than the stationary RF, and are used in practical applications for representing non-stationary phenomena. The most important topics are: existence of a generalized covariance (GC) for which statistical inference is possible from a unique realization; theory of the best linear intrinsic estimator (BLIE) used for contouring and estimating problems; the turning bands method for simulating IRF; and the models with polynomial GC, for which statistical inference may be performed by automatic procedures.

This paper is concerned with the general problem of finding an optimal transition matrix for a finite Markov chain, where the probabilities for each transition must be chosen from a given convex family of distributions. The immediate cost is determined by this choice, but it is required to minimise the average expected cost in the long run. The problem is investigated by classifying the states according to the accessibility relations between them. If an optimal policy exists, it can be found by considering the convex subsystems associated with the states at different levels in the classification scheme.

A Markov process in discrete time with a finite state space is controlled by choosing the transition probabilities from a given convex family of distributions depending on the present state. The immediate cost is prescribed for each choice and it is required to minimise the average expected cost over an infinite future. The paper considers a special case of this general problem and provides the foundation for a general solution. The main result is that an optimal policy exists if each state of the system can be reached with positive probability from any other state by choosing a suitable policy.

In the last ten years the theory of weak convergence of probability measures has been used extensively in studying the models of applied probability. By far the greatest consumer of weak convergence has been the area of queueing theory. This survey paper represents an attempt to summarize the experience in queueing theory with the hope that it will prove helpful in other areas of applied probability. The paper is organized into the following sections: queues in light traffic, queues in heavy traffic, queues with a large number of servers, continuity of queues, rates of convergence, and special queueing models.

The joint distributions of the exit time and exit value of a spectrally positive process, from semi-infinite and finite intervals, are derived in the form of Fourier-Laplace transforms. Also the probability that such a process makes its first exit from a finite interval via the lower end point is obtained explicitly.