The more one studies the vast and ever-growing literature of the theory of queues, the more one is bewildered by the wide variety of mathematical techniques which different authors have used to analyse the same or similar problems. And yet it seems that, beneath the superficial diversity, there is a deeper unity obscured by the special devices and notations characteristic of the different approaches. It is the thesis of this paper that there is indeed such a unity, and that it is best appreciated by observing that the central results of queueing theory, despite the analytical and combinatorial accretions which, by historical accident, they have acquired, are essentially algebraic in character.

Information theory, in the strict sense, is a rapidly developing branch of probability theory originating from a paper by Claude E. Shannon in the Bell System Technical Journal in 1948,in which anew mathematical model ofcommunications systems was proposed and investigated.

One of the central innovations of this model was in regarding the prime components of a communications system (the source of messages and the communication channel) as probabilistic entities. Shannon also proposed a quantitative measure of the amount of information based on his notion of entropy and proved the basic theorem of this theory concerning the possi bility of reliable transmission of information over a particular class of noisy channels.

In this paper a stochastic calculus, based on limits in quadratic mean and in probability, is introduced and applied to some fundamental theorems of natural selection. The paper is divided into five principal sections. In Section 2 some elements of stochastic calculus are given, and in Section 3 the stochastic calculus is applied to the haploid case. Section 4 is devoted to the development of an analysis of variance structure applicable to a haploid population, and in Section 5 the results of Sections 3 and 4 are generalized to cover the diploid case. Finally, Section 6 is devoted to two specific examples in which the results of previous sections apply.

This paper is concerned with the distribution of “types” of individuals in an infinite population after indefinitely many nonoverlapping generations of random mating. The absence of selection and mutation is assumed. The probabilistic law which governs the production of an offspring may be asymmetrical with respect to the “sexes” of the two parents, but the law is assumed to apply independently of the “sex” of the offspring. The question of the existence of a limit distribution of types, the rate at which a limit distribution is approached, and properties of limit distributions are treated.

Some comparisons are made between various characteristics of the genetic structures of populations of the same size and age, which have (i) evolved from a small founder population, and (ii) evolved from a population which has been of constant size throughout the period considered.

The utility of the moments of the binomial distribution in finding the n-stage transition probabilities was indicated by Khazanie and McKean (1966). In this paper, maintaining the same breeding structure as in the two-allele case, it is proposed to extend the technique to cover the multiallelic case in general, with specific emphasis on the triallelic case. The existing literature on the multiple allelic case is at best scanty (see for example Kimura (1955)). Section 6 will briefly touch on a further extension to multiple loci which segregate independently.

The random walk between an absorbing and a reflecting or semi-reflecting barrier has been discussed in the literature. Indeed, a biological example of such a random walk, arising from a system of countercurrent dialysis, was described by this author. In this earlier paper a technique was suggested for obtaining the probability of being in a specific state after a given number of transitions, if the probability is unity of being in the reflecting state at time zero. The present paper extends this technique to yield the full matrix of higher transition probabilities.

We consider a dioecious diploid population of N individuals, N1 males and N2 = N – N1 females. The alleles will be represented by a and A, and the population reproduces according to the Wright scheme, that is, by random mating with non-overlapping generations.

1. Recently Weiss [7] constructed a stochastic model for the spread of diseases e.g. typhoid, by carriers. He assumed that a carrier is eliminated in (t, t + dt) with probability βrdt, where r is the number of carriers extant at time t and that a susceptible is infected in (t, t + dt) with probability arsdt where s is the number of susceptibles at time t. Thus only the carriers are responsible for the spread of the disease.

A classical type of problem in population genetics is that of calculating the probability that a line descended from a particular gene will become extinct. In one problem of this sort, dealt with by Fisher (1922) and Haldane (1927), it is assumed that the population being studied is very large and that initially the number of genes of a particular type, say type A, is small. These authors obtained the solution by the use of the theory of branching processes.

This paper deals with asymptotic properties of various models of age dependent branching processes, relying heavily on Harris [3].

One of the earliest known distributions is that of the Binomial distribution which arises from a Bernoulli sequence defined on two symbols (or generators) a and b. The corresponding limit theorem is that of Demoivre and Laplace and states (in an obvious notation) that (r – np)/√npq converges to the standard Normal distribution N(0,1). If the generators do not commute the situation is a good deal more complicated and in order to say very much about the sequences generated we must be able to put them in some simple canonical form. One case in which this can certainly be done is when the two symbols generate a semi-group of nilpotency class two. This means that although ba ≠ ab, we do have ba = ab (b,a) where (b, a) is a symbol which commutes with both a and b. Each sequence can then be expressed in the form aαbβ(b,a)γ. In this paper we examine first the conditional distribution of γ given α and β for Bernoulli sequences in the symbols a and b and obtain central limit theorems when γ is appropriately normed. We then consider the more general problem of the m-generator semi-group of nilpotency class two and obtain the corresponding multi-dimensional central limit theorem in the case where the probability measure is discrete and is distributed over the generators.

Distributions appropriate to the description of long-term behaviour within an irreducible class of discrete-time denumerably infinite Markov chains are considered. The first four sections are concerned with general reslts, extending recent work on this subject. In Section 5 these are applied to the branching process, and give refinements of several well-known results. The last section deals with the semi-infinite random walk with an absorbing barrier at the origin.

Suppose that we have a non-negative, real valued random variable x, whose distribution is governed by some unknown moment generating function M(t). Suppose further that we are given certain moments of x, then the question to be discussed in this paper is : can we find a sharp upper bounding function for the m.g.f.? It will be shown that this is usually possible both in the single variate case and in its natural extension to the multivariate case.

This paper considers asymptotic properties of increasing population age dependent branching processes which have a limiting age distribution. In Section I, the Bellman-Harris model [2] is altered in accord with a suggestion by Kendall [3]. The effect of this modification on the applicable results of [3] is indicated. An extension to the case where each cell passes through a sequence of states to mitosis or proceeds from state to state in accord with a semi-Markov process to mitosis is considered. Section 1.1 considers asymptotic moments for the total number of cells. Section 1.2 treats moments in a sequence-of-states model [4]. Section 1.3 extends the results of 1.2 to a semi-Markov model. Section 1.4 treats various asymptotic lifetime distributions and fraction of cells in a given state for both the sequence-of-states and semi-Markov models.

Let x be a non-negative, non-decreasing, right-continuous function of t defined on the interval [O, T) and let A = x(T–). Define the function y on [O, A] by Then y is non-decreasing and left-continuous and If t is a time variable it is natural to call y the passage time function.

We assume that the successive service times in a single server queue with Poisson arrivals form an m-state semi-Markov process.

The results for the M/G/1 queue are extended to this case. Both the time-dependence and the stationary solutions are discussed.

An expression is found (Theorem 1) for the probability that a sample function of a Poisson process never exceeds some given function and some examples are given.

Throughout this paper we shall be interested in a certain integer-valued process n(t) (t ≧ 0) which is encountered in queueing theory and textile research.

The subject of geometrical probability has recently shown signs of renewed vigour which is partly due to the attractive nature of the many unsolved problems which it contains, and partly the ever widening field of applications. These applications are so diverse and so scattered in the literatures of different branches of science that when Professor M.G. Kendall and I published a monograph on the subject in 1963 some interesting papers were ignored. Others have since appeared and the purpose of the present paper is to survey, in a very brief manner, work not described in that book. In fact the present bibliography, which contains 94 items, is disjoint from the bibliography in our joint monograph.