Population genetics is that branch of genetics, whose object is the study of the genetical make-up of natural populations. By investigating the laws which govern the genetic structure of natural populations, we intend to clarify the mechanism of evolution.

The probability theory of storage systems formulated by P. A. P. Moran in 1954 has now developed into an active branch of applied probability. An excellent account of the theory, describing results obtained up to 1958 is contained in Moran's (1959) monograph, Considerable progress has since been made in several directions-the study ofthe time-dependent behaviour ofstochastic processes underlying Moran's original model, modifications of this model, as well as the formulation and solution of new models. The aim of this paper is to give an expository account of these developments; a comprehensive treatment will be found in the author's forthcoming book [Prabhu (1964)].

Smoluchowski's classical analysis of the temporal fluctuation, under diffusion equilibrium, of the number of particles in a fixed region R of space is generalised to a set of disjoint regions; specifically, the single Smoluchowski region is divided into a finite number of non-intersecting subregions. The generalisation allows a more rigorous test of some of the consequences of the Einstein-Smoluchowski theory of Brownian motion to be carried out, and at the same time enables the Avogadro constant to be estimated with greater precision than is possible with the single region. In particular, the reversibility paradox of Loschmidt and the recurrence paradox of Zermelo are reexamined from the point of view of the fluctuation of configurations (a configuration being defined as the set of occupation numbers for the various subregions) rather than that of total concentration for the single region.

Moran (1958a) (1958b) introduced two population genetic models of a general kind to investigate the combined effects on evolution of four factors,

In this paper we shall be concerned with two mathematical models of infinite dams. In the first model independent random inputs occur at regular time intervals and in the second model independent random inputs occur in accordance with a Poisson process. The first model has already been studied by Gani, Yeo and others, and the second model by Gani and Prabhu, Gani and Pyke, Kendall, and others. For both models we shall find explicit formulas for the distribution of the content of the dam and that of the lengths of the wet periods and dry periods. The proofs are elementary and based on two generalizations of the classical ballot theorem.

Our concern is with passage times and rate of approach to ergodicity for two types of temporally homogeneous processes, doubly bounded diffusion processes in one dimension, and birth-death processes on the finite lattice. The passage time problems associated with these processes are of considerable practical interest, but for many important cases, e.g., the Uhlenbeck-Ornstein process, only formal solutions such as Laplace transforms (cf. Darling and Siegert, 1953) have been given, with limited numerical potential.

The theory of multiplicative (or branching) population processes where the states of each individual in the population range over a fixed finite set has been studied by a number of authors: see for example, Everett and Ulam, Kolmogorov and Sevastyanov, Harris; the continuous parameter case is treated in Arley; further references will be found in the books by Bartlett and Bharucha-Reid. The purpose of the present paper is to study the theory of such processes in the case where the individual state space is arbitrary (i.e., an abstract space).

Let us suppose that telephone calls arrive at a telephone exchange according to a recurrent process. We shall assume that there are infinitely many lines available and that, therefore, no call is ever lost. Ordinarily, it is assumed that the holding times (the durations of the connections) are identically distributed, independent, random variables with an exponential distribution. The American folklore, however, assumes that women talk more than men. In this paper, we shall consider the possibility of calls requiring one of several (s) different exponential holding times.

Homogeneous additive processes on a finite or semi-infinite interval have been studied in many forms. Wald's identity for the first passage process on the finite interval (see for example Miller, 1961), the waiting time process of Lindley (1952), and a variety of problems in the theory of queues, dams, and inventories come to mind. These processes have been treated by and large by methods in the complex plane. Lindley's discrete parameter process on the continuum, for example, described by where the ξn are independent identically distributed random variables, has been discussed by Wiener-Hopf methods in recent years by Lindley (1952), Smith (1953), Kemperman (1961), Keilson (1961), and many others. A review of earlier studies is given by Kemperman.

1. In 1923 Eggenberger and Pólya introduced the following ‘urn scheme’ as a model for the development of a contagious phenomenon. A box contains b black and r red balls, and a ball is drawn from it at random with ‘double replacement’ (i.e. whatever ball is drawn, it is returned to the box together with a fresh ball of the same colour); the procedure is then continued indefinitely. A slightly more complicated version with m-fold replacement is sometimes discussed, but it will be sufficient for our purposes to keep m = 2 and it will be convenient further to simplify the scheme by taking b = r = 1 as the initial condition. We shall however generalise the scheme in another direction by allowing an arbitrary number k(≧2) of colours. Thus initially the box will contain k differently coloured balls and successive random drawings will be followed by double replacement as before. We write sn (a k-vector with jth component ) for the numerical composition of the box immediately after the nth replacement, so that and we observe that is a Markov process for which the state-space consists of all ordered k-ads of positive integers, the (constant) transition-probability matrix having elements determined by where Sn is the sum of the components of sn and (e(i))j = δij. We shall calculate the Martin boundary for this Markov process, and point out some applications to stochastic models for population growth.

The purpose of this paper is to construct a theory of the amount of information provided by an experiment which does not rely on what Good (1962) has termed the modern Bayesian principle that it is legitimate to use the axioms of probability even when this involves the use of probabilities of hypotheses. In this respect the theory of this paper differs from the Lindley (1956), Mallows (1959) and Good (1960) each of which is written from a Bayesian viewpoint. Lindley (1956) expresses the opinion that Bayesian ideas would seem to be necessary to the development of a theory of the amount of information provided by an experiment and it is of interest therefore to determine how far such a theory may be developed without Bayesian ideas. There is further a need to use such a theory to examine how prior knowledge can be expressed quantitatively, and used in accordance with that theory.

A model for road traffic delays at intersections is considered where vehicles arriving, possibly in bunches, in a Poisson process in a one way minor road yield right of way to traffic, which forms alternate bunches and gaps, in a major road. The gap acceptance times are random variables, and depend on whether or not a minor road vehicle is immediately following another minor road vehicle into the intersection or not.

The transforms of the stationary waiting time and queue size distributions, and the mean stationary delay, for minor road vehicles are obtained by substitution of determined service time distributions into results for a generalisation of the M/G/1 queueing system. Some numerical results are given to illustrate the increase in the mean delay for variable gap acceptance times for a Borel-Tanner distribution of major road traffic, and a partial solution is given for a two way major road.

In the present paper we deal with two models of the traffic on a divided highway. In both models traffic is flowing in one direction only, in two lanes, of which one (the left hand lane) is used only for overtaking. We suppose that vehicles enter the highway at the same entrance so that the instants at which a vehicle enters the highway form a homogeneous Poisson process, with density λ. Thus λ is the rate at which vehicles enter the highway per unit time. We suppose that there are no junctions, or exits (i.e. the highway extends in one direction to infinity).

In genetics one frequently encounters variates which behave, to a close approximation, as diffusion variates with drift and diffusion coefficients m(x),v(x), respectively, where after a suitable rescaling of the time axis, m(x) and v(x) are of the form

In a separate paper (Davis [2]) the characteristic functional for a replacement process due to Gani (see Gani and Yeo [1]) was obtained explicitly. The purpose of the present note is to consider the product densities for this process.

The classical two-player gambler's ruin problem is well known and has attracted a great deal of attention in the literature in various forms. Extensions of these two-player problems to three or more players do not appear to have been discussed however, and this paper is concerned with one particular extension of the problem to the case of three players.

An asymptotic expansion is derived for Sr(n)/Tr(n), where r and n are integers, and . Bounds for the remainder after kterms of the expansion are obtained and it is shown how the expansion may be applied in hypothesis testing and the study of the transient behaviour of a queue with batch arrivals.

The problem of determining infinitely dilute resonance integrals is formulated in renewal theoretical terms. The mean value of the integral for a single resonance is determined in simple closed form. On the assumption that Wigner's hypothesis holds, the resonance density is determined, and a usable approximation to it is derived. An expression for the infinitely dilute resonance integral in the statistical region is then given and its value calculated in special cases and compared with the results of a previous computation.

In a previous paper a branching Poisson process model was derived to explain deviations from a Poisson process in computer failure patterns. Physically the deviations arise because an attempt to repair a computer is not always successful and the failure recurs a relatively short time later. In this paper we discuss the implications of this model for the use and maintenance of computers.

In a ballot, candidate A scores a votes and candidate B scores b votes. Suppose the ballots are drawn out one at a time, and denote αr and βr the number of votes registered for A and B, respectively, among the first r votes recorded. Further, let Δa,b be the number of subscripts r satisfying the strict lead condition , let be the number of subscripts r satisfying the weak lead condition ; and suppose all possible () voting records are equally probable. The probability distributions of the number of strict and weak lead positions corresponding to and , respectively, have been determined in [4] for a≧b.