This paper applies the constant population approximation to the study of epidemics which involve more than a single type of infective. An example of this would be a situation in which both clinically infected individuals and subclinically infected individuals or carriers are present.

We derive equations for the expected numbers of clinically infected individuals and carriers at any time t for the model with zero latent period and infectious periods having negative exponential distributions. From these equations we derive conditions under which a unimodal incidence curve can result, and expressions for the expected total epidemic size.

In a recent paper Weiss (1965) has suggested a simple model for a carrier-borne epidemic such as typhoid. He considers a population (of size m) of susceptibles into which a number (k) of carriers is introduced. These carriers exhibit no overt symptoms and are only detectable by the discovery of infected persons. He supposed that after the initial introduction of the carriers, the population remains entirely closed and no new carriers arise. The epidemic then progresses until either all the carriers have been traced and isolated or until the entire population has succumbed to the disease.

We present two theorems that provide simple iterative solutions of special systems of differential-difference equations. We show as examples of the theorems the simple stochastic epidemic (cf. Bailey, 1957, p. 39, and Bailey, 1963) and the general stochastic epidemic (cf. Bailey, 1957; Gani, 1965; and Siskind, 1965), in each of which we let the initial distribution of the number of uninfected susceptibles and the number of infectives be arbitrary but assume the total population size bounded. In all of the references cited above the methods of solution involve solving a corresponding partial differential equation, whereas we deal directly with the original system of ordinary differential-difference equations. Furthermore in the cited references the authors begin at time t = 0 with a population having a fixed number of uninfected susceptibles and a fixed number of infectives. For the simple stochastic epidemic with arbitrary initial distribution we provide solutions not obtainable by the results given by Bailey (1957 or 1963). For the general stochastic epidemic, if we use the results of Gani or Siskind, then the solution of the problem having an arbitrary initial distribution would involve additional steps that would sum proportionally-weighted conditional results.

A number of years ago, R. A. Fisher discussed the problem of the propagation of a virile mutant in a population. At about the same time, Kolmogorov, Petrovsky, and Piscounoff, whom we shall refer to as KPP, investigated a general class of partial differential equations which describe simultaneous growth and diffusion processes.

In a previous paper (Marshall and Olkin (1966)) the authors have derived a multivariate exponential distribution from points of view designed to indicate the applicability of the distribution. Two of these derivations are based on “shock models” and one is based on the requirement that residual life is independent of age.

The practical importance of the univariate exponential distribution is partially due to the fact that it governs waiting times in a Poisson process. In this paper, the distribution of joint waiting times in a bivariate Poisson process is investigated. There are several ways to define “joint waiting time”. Some of these lead to the bivariate exponential distribution previously obtained by the authors, but others lead to a generalization of it. This generalized bivariate exponential distribution is also derived from shock models. The moment generating function and other properties of the distribution are investigated.

If X1, …, Xn obey a multivariate normal distribution with zero means, then the probability that Xi > a for all i = 1, …, n is often called a multivariate normal integral. Such integrals have been considered by various investigators, particularly when a = 0. (For a bibliography, see Gupta.)

Let U denote the set of all integers, and suppose that Y = {Yu; u ∈ U} is a process of standardized, independent and identically distributed random variables with finite third moment and with a common absolutely continuous distribution function (d.f.) G (·). Let a = {au; u ∈ U} be a sequence of real numbers with Σuau2 = 1. Then Xu = ΣwawYu–w defines a stationary linear process X = {Xu; u ɛ U} with E(Xu) = 0, E(Xu2) = 1 for u ∊ U. Let F(·) be the d.f. of X0. We prove that if maxu |au| is small, then (i) for each w, Xw is close to Gaussian in the sense that ∫∞−∞(F(y) − Φ(y))2dy ≦ g maxu |au | where Φ(·) is the standard Gaussian d.f., and g depends only on G(·); (ii) for each finite set (w1, … wn), (Xw1, … Xwn) is close to Gaussian in a similar sense; (iii) the process X is close to Gaussian in a somewhat restricted sense. Several properties of the measures of distance from Gaussianity employed are investigated, and the relation of maxu|au| to the bandwidth of the filter a is studied.

This paper deals with the problem of predicting or extracting a certain type of non-stationary sequence or signal from another sequence composed of the signal and a stationary noise. The noise is assumed to be uncorrelated with the signal. Although all sequences in the body of this paper are multivariate, for purposes of discussion, this introduction will deal with only the univariate case.

In the present paper the solutions of two integral equations are derived. One of the integral equations dominates the mathematical description of the stochastic process {vn, n = 1,2, …}, recursively defined by K is a positive constant, τ1, τ2, …; Σ1, Σ2, …; are independent, non-negative variables, with τ1, τ2,…, identically distributed, similarly, the variables Σ1, Σ2, …, are identically distributed.

A single server queueing system with Poisson input is considered. There are a finite number of types of customers and the service time of the nth customers depends on the types of the nth and the (n – l)th customers. The time dependence of the queue size process will be studied, (it will be clear how the methods of the paper can be applied to other processes of interest,) and limiting as well as transient results will be given.

A queueing system with a single server is considered. There are a finite number of types of customers, and the types of successive arrivals form a Markov chain. Further, the nth interarrival time has a distribution function which may depend on the types of the nth and the n–1th arrivals. The queue size, waiting time, and busy period processes are investigated. Both transient and limiting results are given.

In a previous paper (Moran (1956)) the theory of a dam with a continuous release was developed for a situation in which the input was defined by an additive homogeneous random process of gamma distribution type, and the release was defined to occur continuously at a fixed rate when the content of the dam was non-zero. In the present paper we modify these conditions by assuming first that the input is defined by a general additive homogeneous process with non-negative increments with finite second moment, and secondly that the rate of release is proportional to the content of the dam. This modification to the release rule ensures that the content of the dam is never zero and the theory is then so simplified that the distributional properties of the contents of a sequence of dams in series can be easily found.

This paper arises from the observation by Dirac [1] that an exact measurement of the velocity of an electron should give one of the instantaneous values ± c. The velocity v < c with which the electron moves bodily is in some statistical sense an average. This aspect of the quantum states of motion of a relativistic particle has frequently been mentioned [2], [3] and, as is wellknown, is connected with the self energy problem. Such deductions from quantum mechanics have led the author to consider the possibility of constructing from purely statistical ideas an entity having the physical properties of a relativistic elementary particle. One might reasonably hope to build up characteristics such as rest mass and spin in such a scheme. Properties such as energy and momentum will arise as averages over fluctuating random quantities. In this paper we shall concentrate on rest mass, energy and momentum. We shall show that a zero rest mass object which is regarded as being imbedded in a statistical structure emulates the characteristic of rest mass. The average energy and momentum of this statistical system are then shown to satisfy the usual classical relativistic relation between energy and momentum. The statistical structure chosen is that of an object with a randomly orientated velocity vector, c, which has a fixed magnitude c … |c| where c is the velocity of light.

Let G = (V, E) be a graph with N vertices and A a set of N real numbers. Then a one-to-one mapping ϕ: V → A is called a numbering of G. The elements of A will always be assumed ordered a1 ≦ a2 ≦ … ≦ aN. In [3] and [7] it was shown how to construct numberings of the n-cube, in fact all numberings, which minimize Σe ∈ E Δe where ∆e = |ϕ(v) – ϕ(w)| and e is the edge between vertices v and w. A variant problem of considerable interest is to do the same for Σe ∈ E (Δe)2. It is conjectured that when A = {1,2, …,2n} the natural numbering is the unique minimizer of Σe ∈ E (Δe)2 for every n, but this has so far resisted all efforts. Theorem 1 in the following is the result of attempts to get weaker results, namely to thin out the ranks of those numberings which could possibly minimize Σe ∈ E (Δe)2. The second problem posed and solved in this paper is a generalization of the results in [3], where the n-cube becomes the n-torus.

Let X(t) be the position at time t of a particle undergoing a simple symmetrical random walk in continuous time, i.e. the particle starts at the origin at time t = 0 and at times T1, T1 + T2, … it undergoes jumps ξ1, ξ2, …, where the time intervals T1, T2, … between successive jumps are mutually independent random variables each following the exponential density e–t while the jumps, which are independent of the τi, are mutually independent random variables with the distribution . The process X(t) is clearly a Markov process whose state space is the set of all integers.

Consider the following service system.

The deleting or delay effect which a recurrent point process exerts upon another recurrent point process (or process with stationary independent increments) is treated. The concept arises in the statistical analysis of neuronal firing patterns, but may also be useful in applications to storage systems in operations research.