In this paper the theory of genetic algebras introduced by Etherington [3], [5] is developed to cover the case where the population is divided into a number of partially isolated subpopulations, between which a limited amount of migration is permitted. It is known [3], [6], [7], [8], [10], that in the absence of selective factors most genetic segregation systems can be represented by algebras belonging to the class of special train algebras, discussed in detail in [4].

Wright (1921) computed various correlations of relatives by a rather cumbersome procedure called the “method of path coefficients”. Wright's method is basically a disguised form of the use of Bayes' rule and the law of total probabilities. Malecot (1948) reorganized Wright's calculations by introducing the fundamental concept of identity by descent and exploiting its properties. The method of identity by descent has been perfected and developed by Malécot and his students, especially Gillois, Jauquard and Bouffette. Kempthorne (1957) has applied the concept of identity by descent to the study of quantitative inheritance. Kimura (1963) elegantly employed the ideas of identity by descent in determining rates of approach to homozygosity in certain mating situations with finite population size. Later in this chapter we will extend and refine the results of Kimura (1963) to give a more complete study of rates of approach to homozygosity. Ellison (1966) established several important limit theorems corresponding to polyploid, multi-locus random mating infinite populations by judicious enlargement of the concepts of identity by descent. Kesten (unpublished)) has recently refined the technique of Ellison.

We consider a system of colonies in which migration, immigration and emigration take place according to a fairly general rule. Necessary and sufficient conditions for equilibrium are given in the theorem of Section (3). The equilibrium distribution of colony numbers is determined by Equations (7) and (9); the colony sizes are thus statistically independent in equilibrium.

Attention is also paid to the case where one colony becomes a “bottleneck” to circulation, when equilibrium changes character

Generations are assumed to be non-overlapping. We consider a haploid population divided into K parts, each of which contain N adults in any generation. These are obtained by a random sampling of the offspring of the previous generation. We assume that the probability of an adult offspring of an individual in one subpopulation being in some other subpopulation is the same small positive number, no matter what two subpopulations are considered.

If the population initially has individuals of two types, A and a, it is of interest to study approximations, if n is large, to

1. (1) the rate at which A or a is lost between generations n-1 and n,

2. (2) the probability that A and a are still present in generation n,

3. (3) the joint distribution of frequencies of A in the subpopulations.

A solution is given for the first problem. It is found that if the mean number of migrants per generation from one subpopulation to another is at least as large as 1, the population behaves almost as if it were not subdivided. But if this number is considerably less that 1, then the rate at which one or the other gene is lost is slower than in an undivided population. The other two problems are discussed for K = 2.

This paper treats the well known model of sib mating of individuals with autosomal genes; a single locus is studied. The general derivations are valid for any number of alleles. At the end of the paper, some simple examples are included; here several familiar results concerning models for two alleles are obtained as special cases of the general theory.

The well-known Wright's genetic model is extended to a population whose size varies cyclically. Explicit expressions are obtained for the absorption probability and the rate of approach to homozygosity in the case where there is no selection. When selection operates, diffusion approximations are obtained for the various quantities of interest.

Two cases of multiple linearly interconnected linear birth and death processes are considered. It is found that in general the solution of the Kolmogorov differential equations for the probability generating function (p.g.f) g of the random variables involved is not obtainable by standard methods, although one can obtain moments of the random variables from these equations. A method is considered for obtaining an approximate solution for g. This is based on the introduction of a sequence of stochastic processes such that the sequence {f(n)} of their p.g.f.'s tends to g as n → ∞ in an appropriate manner. The method is applied to the simple case of two birth and death processes with birth and death rates λi and μi, i = 1,2, interconnected linearly with transition rates v and δ (see Figure 2). For this case some limit theorems are established and the probability of ultimate extinction of both the processes is considered. In addition, for the special cases (i) λ1 = δ = 0, with the remaining rates time dependent and (ii) λ2 = δ = 0, with the remaining rates constant, explicit solutions for g are obtained and studied.

In recent papers Yassky (1962) and Gani (1965) have considered respectively deterministic and stochastic models for the attachment of phages to bacteria. Following Brenner's (1965) conjecture they assumed that there is a maximum, S (say), to the number of phages that can be attached to a bacterium. In this note, Gani's (1965) results will be obtained starting from a different set of assumptions. This modification enables us to consider the case when S is a random variable, which probably is a more realistic assumption. Some remarks on the problem of estimation for the latter model are given in Section 3.

Consider a finite Markov process {Xn} described by the one-step transition probabilities In describing the transition probabilities in the above manner we are adopting the convention that (0)0 = 1 so that the states 0 and M are absorbing, and the states 1,2,···,M-1 are transient.

Much of the theory of epidemics (see Bailey, 1957) has been concerned with models for their behaviour in closed populations. In such models the epidemic ultimately dies out, and interest has been concentrated on, for example, the ultimate size of the epidemic and its duration in time. In practice a population is rarely completely closed, and for many diseases an endemic model rather than an epidemic model is appropriate. To create models for endemic diseases it is necessary to introduce both new persons susceptible to the disease into the population and new sources of infection. For the so-called general stochastic epidemic, Ridler-Rowe (1967) has obtained certain limiting properties of the population where immigration of both susceptibles and infectives into the population takes place, but much work remains to be done to obtain, for example, the general equilibrium behaviour of this model.

During a rush hour, the arrival rate λ(t) of customers to a service facility is assumed to increase to a maximum value exceeding the service rate μ, and then decrease again. In Part I it was shown that, after λ(t) has passed μ, the expected queue E{X(t)} exceeds that given by the deterministic theory by a fixed amount, (0.95)L, which is proportional to the (–1/3) power of a(t) = dλ(t)/dt evaluated at time t = 0 when λ(t) = μ. The maximum of E{X(t)}, therefore, occurs when λ(t) again is equal to μ at time t1 as predicted by deterministic queueing theory, but is larger than given by the deterministic theory by this same constant (0.95)L (provided t1 is sufficiently large). It is shown here that the maximum queue, suptX(t) is, approximately normally distributed with a mean (0.95) (L + L1) larger than predicted by deterministic theory where L, is proportional to the (–1/3) power of a(t1). We also investigate the distribution of X(t) at the end of the rush hour when the queue distribution returns to equilibrium. During the transition, the queue distribution is approximately a mixture of a truncated normal and the equilibrium distributions. These results are applied to a case where λ(t) is quadratic in t.

We will consider a problem which arises in the theories of neutron transport and radiative (photon) transfer. We will consider, in a particular case, the random motion of a particle in a scattering medium and determine asymptotics for the probability distribution of the random variable for the number of collisions before exit from the medium.

The arrival rate of customers to a service facility is assumed to have the form λ(t) = λ(0) — βt2 for some constant β. Diffusion approximations show that for λ(0) sufficiently close to the service rate μ, the mean queue length at time 0 is proportional to β–1/5. A dimensionless form of the diffusion equation is evaluated numerically from which queue lengths can be evaluated as a function of time for all λ(0) and β. Particular attention is given to those situations in which neither deterministic queueing theory nor equilibrium stochastic queueing theory apply.

The article first reviews briefly the field of statistical process control and its relation to automatic process control. Some recent results obtained by various workers in Czechoslovakia in the field of statistical aspects of automatic process control are then presented.

A previous paper [2] was concerned with the determination of optimal policies for restocking an inventory which is continuously depleted by a random process of demands. The purpose of the present paper is to develop a similar model for controlling the output of a dam whose random input depends on a homogeneous Wiener process. This reversal of the roles of input and output does not, by itself, change the character of the problem. But the consideration of set-up costs for ordering replacements, which leads to inventory policies of the (s, S) type, has no counterpart here. It is natural to regard the dam as a device for smoothing out random fluctuations in a flow of water and, under utility assumptions which reflect this attitude, it follows that the optimal output rate is a continuous function of the level of water in the reservoir. Our main object is to determine this function.

The relation between “forward” and “backward” definitions of quantum velocity is discussed. A connection between the Feynman Integral and Wigner's phase space distributions which was noted in an earlier paper is explored further and some new and useful formulae are derived.

In a variety of the congestion or queueing problems that arise in practice, for example, in studies of the crossing and entry problems of road traffic, (see Evans, Herman, and Weiss [2]), and recently of the service afforded by large centralized and shared computer facilities, (see Scherr [12]), the understanding of system performance furnished by the present mathematical theory is inadequate. The reason is that while the consideration of simple problems typically yields elegant mathematical results, the form of these results—often expressed in terms of integral transforms—is not immediately comprehensible nor useful for simple comparisons. This fact has been remarked upon by Newell, who in [9] has suggested certain more comprehensible but approximate approaches based on diffusion theory; further promising developments and elaborations will be found in [10]. The latter approach is related to the “heavy traffic theory” of J. F. C. Kingman [8], and to some recent work of Iglehart [6]. Of course, the idea of approximating complex discrete-state processes by diffusion processes with continuous paths is not new. It has long been used in genetics, see Feller [3], and the review paper by Kimura [7]. Nonetheless, applications to congestion theory are apparently still rather rare.

In [3], Kendall proved a solidarity theorem for irreducible denumerable discrete time Markov chains. Vere-Jones refined Kendall's theorem by obtaining uniform estimates [14], while Kingman proved analogous results for an irreducible continuous time Markov chain [4], [5].

We derive similar solidarity theorems for an irreducible Markov renewal process. The transient case is discussed in Section 3, and Section 4 deals with the positive recurrent case. Recently Cheong also proved solidarity theorems for Semi-Markov processes [1]. His theorems use the Markovian structure, while our emphasis is on the renewal aspects of Markov renewal processes.

An application to the M/G/1 queue is included in the last section.

Moran's [1] early investigations into the theory of storage systems began in 1954 with a paper on finite dams; the inputs flowing into these during consecutive annual time-intervals were assumed to form a sequence of independent and identically distributed random variables. Until 1963, storage theory concentrated essentially on an examination of dams, both finite and infinite, fed by inputs (discrete or continuous) which were additive. For reviews of the literature in this field up to 1963, the reader is referred to Gani [2] and Prabhu [3].