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.

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.

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.

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.

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

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.

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.

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.

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.

The problem discussed in this paper arose from the study of the effects of unresolved resonances on neutron cross-sections, but it is considered here in more general terms.

Events in a modified renewal process occur at successive intervals x1, x2, …, and the ith event has associated with it a parameter Γi. The random variables xi and Γi are all independent, and their probability distribution functions are known. Each event contributes to two quantities F(u) and G(v) measured at u and v respectively. The value of the total contribution of all events to G(v) is assumed to be known from observation: this paper gives formulae for the mean value of F(u) conditional on this known value of G(v).

Throughout this paper, we shall write ℒ(W) = ℒ(Z) to mean the random variables W and Z have the same distribution. The relation “ℒ(W) = ℒ(;Z)” reads “the law of W equals the law of Z”.

The optimal strategies of any finite matrix game can be characterized by means of the Snow-Shapley Theorem [1]. However, in order to use this theorem to compute the optimal strategies, it may be necessary to invert a large number of matrices, most of which are not related to the solutions of the game. The present paper will show that when the columns of the pay-off matrix satisfy some special relations, it is possible to enumerate a much smaller class of matrices from which the optimal strategies may be obtained. Furthermore, the maximizing strategies that are determined by these matrices can be written down by inspection as soon as the matrices are specified.

Probability generating functions are used to relate the joint distribution of the numbers of customers left behind by two successive departing customers to the marginal distribution of the number left behind by each departing customer. A probability generating function is then found for the joint distribution of the numbers of customers arriving in two successive departure intervals using the joint distribution of the numbers of customers left behind by three successive departing customers. The results could be obtained from general Markov chain theory but the method used in this paper is quicker.

When a complex system such as the Apollo vehicle arrives on the Moon, certain parts may no longer function. To restore the system to a working state, these parts must be replaced by new ones; but the vehicle can carry only a limited number of spare parts and must also be able to leave at a specified time after arrival. This paper investigates an abstract (and simplified) version of the problem of the optimal maintenance procedure for such a system. As a bonus of the abstract formulation, the results obtained are applicable to a variety of specific systems.

This paper discusses an optimization problem arising in the theory of inventory control. Much of the previous work in this field has been focused on the Arrow-Harris-Marschak model, [1], [2], in which the inventory level can be modified only at the instants of discrete time. Here, we shall be concerned with a continuous time analogue of the model, in an attempt to avoid the difficulties experienced in solving the basic integral equations. The approach was suggested by recent investigations of a statistical decision problem, [3], [5], which exploited the advantages of a continuous treatment. Although the ideas discussed here are relatively straightforward and involve strong assumptions as to the behavior of the inventory, the explicit character of the optimal policy is encouraging and particular solutions might nevertheless provide useful restocking procedures.

Moments of the measure of the intersection of a given set with the union of random sets may be called moments of coverage. These moments specified for the case of independent random sets in the Euclidean space, were treated by several authors. Kolmogorov (1933) and Robbins (1944), (1945), (1947) derived transformation formulas for the moments of coverage under some specified conditions. Robbins' formula was applied by Robbins (1945), Santaló (1947) and Garwood (1947) to computations of expectations and variances of coverage in the cases of spheres and intervals in 2-dimensional and in n-dimensional Euclidean space. The computations were carried out under the assumption of a simple distribution function for the centers of the covering figures (a homogeneous distribution in most of the cases and a normal one in some of them). Neyman and Bronowski (1945) computed by a different method the variance of coverage for some specific cases in the Euclidean plane.

In this note, we refine an upper bound, established by P. Whittle, for the probability that the maximum of a number of random variables x1, …, xn is less than a given value C when and are given probabilities in terms of which the bound will be expressed.

Consider a plane containing a set of curves. Suppose a separate set of curves is constrained to lie at random on a particular region of this plane. In this note an expression is obtained for the expected number of intersections between the two sets of curves, and it is shown that the angles between the two sets at their points of intersection are distributed as ½ sin θ.