This note is prompted by the papers of Weiss (this Symposium) and Bailey (1968). Weiss develops a technique for approximation to the moments of an epidemic process by regarding them as expandable in powers of N-1 where N is the size of the population, assumed constant. He first considers the simple stochastic epidemic with no removals and obtains explicit formulae for the terms of order N-1, the zero order terms being the deterministic values. Bailey is concerned with a similar type of approximation and he derives explicit results to the same order. Bailey uses an eigenfunction approach whereas Weiss's method is more direct and perhaps easier to generalise. However, in attempting to extend the method to the case of a closed epidemic with removals Weiss is led to intractable difference equations.

The behaviour of the Galton-Watson process in near critical conditions is discussed, both with and without immigration. Limit theorems are obtained which show that, suitably normalized, and conditional on non-extinction when there is no immigration, the number of individuals remaining in the population after a large number of generations has approximately a gamma distribution. The error estimates are uniform within a specified class of offspring distributions, and are independent of whether the critical situation is approached from above or below. These results parallel those given for continuous time branching processes by Sevast'yanov (1959), and extend recent work by Nagaev and Mohammedhanova (1966), Quineand Seneta (1969), and Seneta (1970).

The theory of epidemics has been formulated as a deterministic theory and as a stochastic theory. There has been little work, except for a recent paper by Bailey [1], on the transition between the two formulations. In this paper we present an alternate formulation to that of Bailey that leads to a perturbation method more easily extendible to higher orders.

Gaver (1963) and Antelman and Savage (1965) have proposed models for the distribution of the time to failure of a simple device exposed to a randomly varying environment. Each model represents cumulative wear as a specified function of a non-negative stochastic process with independent increments, and assumes that the reliability of the device is conditioned upon realizations of this process. From these models are derived the corresponding unconditional joint distributions for the random failure time vector of n independent, identical devices exposed to the same realization of the wear process. It is shown that the identical failure time distribution for one component can arise from each model. In the Gaver model simultaneous failure times occur with positive probability. The probabilities of specific tie configurations are developed.

For an interesting class of Gaver models involving a time scale parameter, the maximum likelihood estimates from several devices in one environment are examined. In that case the tie configuration probability does not depend on the parameter. For the corresponding Antelman-Savage models a consistent sequence of estimators is obtained; the maximum likelihood theory did not appear tractable.

Consider the well known Galton-Watson branching process (Harris (1963)) in which individuals reproduce independently of each other and have probability aj (j = 0, 1, · · ·) of giving rise to j individuals in the next generation. In recent years some attention has been given to the branching process in which there is an independent immigration component at each generation, bj (j = 0, 1, · · ·) being the probability of j immigrants entering each generation. For a review of this work see Seneta (1969), and Pakes (1971 a, b and c) for further results.

Let v be a positive integer and for each k = 1, · · ·, v let mk and Nk be a positive and a non-negative integer, respectively. Denote by S'Nk,mk the set of (mk + 1) -tuples rk = (rk,mk, · · ·, rk,1, rk,0) having non-negative components summing to Nk, and by Xk(t) = (Xk,mk(t), · · ·, Xk,1(t), Xk,0(t)) an (mk + 1)-tuple random variable taking on values only from the set S′Nk,mk.

In modern medicine it becomes more and more urgent every year to solve new epidemiological problems. One of these is the evolution of resistance to antibiotics of a large spectrum of microorganisms; this resistance is a consequence of the mass administration of antibiotics. In this problem it becomes necessary to apply the mathematical methods of epidemic theory as here the key lies in a purely epidemiological factor, namely the mass character of the use of antibiotics. A first model of the phenomenon encountered is presented.

Three simulation models for stochastic discrete-time epidemics involving two competing and interfering viruses in highly structured communities were presented.

We consider the usual stochastic model of a general epidemic, consisting of n + 1 homogeneously mixing individuals, assuming that initially when t = 0 there is one infective and n susceptibles (Bailey (1957)). At any time t we suppose that there are r susceptibles still uninfected, s infectives in circulation, and u individuals who have been removed (and are dead, isolated, or recovered and immune), where

Some obscure, yet fundamental, formulae of integral geometry are re-considered. They are applied to determine all the moments of the random volume of various isotropic random r-dimensional simplices in En (r = 1, …, n).

The effect of an epidemic of chickenpox on the course of an epidemic of yaws in a village of New Guinea was studied by Gart and deVries (1966) and Gart (1968). A deterministic continuous time model was used to confirm that the yaws epidemic accelerated dramatically in the month following the chickenpox epidemic. However, by the nature of the model, no statistical tests are possible.

Investigations in the theory of heavy traffic were initiated by Kingman ([5], [6] and [7]) in an effort to obtain approximations for stable queues. He considered the Markov chains {Wni} of a sequence {Qi} of stable GI/G/1 queues, where Wni is the waiting time of the nth customer in the ith queueing system, and by making use of Spitzer's identity obtained limit theorems as first n → ∞ and then ρi ↑ 1 as i → ∞. Here &rHi is the traffic intensity of the ith queueing system. After Kingman the theory of heavy traffic was developed by a number of Russians mainly. Prohorov [10] considered the double sequence of waiting times {Wni} and obtained limit theorems in the three cases when n1/2(ρi-1) approaches (i) - ∞, (ii) -δ and (iii) 0 as n → ∞ and i → ∞ simultaneously. The case (i) includes the result of Kingman. Viskov [12] also studied the double sequence {Wni} and obtained limits in the two cases when n1/2(ρi − 1) approaches + δ and + ∞ as n → ∞ and i → ∞ simultaneously.

For the linear growth birth-death process with parameters λn = nλ, μn = nμ, Puri ((1966), (1968)) has investigated the joint distribution of the number X(t) of survivors in the process and the associated integral Y(t) = ∫0tX(τ)dτ. In particular, he has obtained limiting results as t → ∞. Recently one of us (McNeil (1970)) has derived the distribution of the integral functional Wx = ∫0Txg{X(τ)}dτ, where Tx is the first passage time to the origin in a general birth-death process with X(0) = x and g(·) is an arbitrary function. Functionals of the form Wx arise naturally in traffic and storage theory; for example Wx may represent the total cost of a traffic jam, or the cost of storing a commodity until expiration of the stock. Moments of such functionals were found in the case of M/G/1 and GI/M/1 queues by Gaver (1969) and Daley (1969).

Many service systems exhibit variations of a random nature in the intensity of the arrival process or of the speed of service or of both. Changes in work shifts, rush hours, interruptions in the arrival process, server breakdowns, etc. all fall into this category.

The present study deals with a generalization of the classical M/G/1 queue by considering an extraneous process of phases which can be in one of the states {1, …, m}. During any interval spent in phase i, the arrivals are according to a homogeneous Poisson process of rate λi and any service initiated during such an interval has a duration distributed according to Hi(·). The process of phases is assumed to be an irreducible Markov chain in continuous time and is fully characterized by its initial conditions, by an irreducible stochastic matrix P and by the mean sojourn times σ1-1, …, σm-1 in each phase.

Independently of the queueing aspects, this arrival process is a generalization of the classical Poisson process which can be of interest in modelling simple point processes with randomly fluctuating “arrival” rate.

Two approaches to the time dependent study of this queue are presented; one generalizes the imbedded semi-Markov process obtained by considering the queue immediately following departure points; the other approach exploits the relationship between this queue and branching processes. The latter is more elegant from a purely theoretical viewpoint and involves iterates of a general type of matrix function introduced by the author. By making extensive use of the Perron-Frobenius theory of positive matrices the equilibrium condition of the queue is obtained. While retaining a similar intuitive interpretation the equilibrium condition is substantially more complicated than for the M/G/1 model.

The recurrence relations which yield the joint distribution of the phase state at time t, the queue length, the total number served and the virtual waiting time at t are exhibited in detail. Via transform techniques a number of limiting and marginal distributions are discussed. The discussion relies heavily on the theory of Markov renewal processes.

Throughout the paper and in a final section the author advocates the use of the structural properties of the queue and the resulting recurrence relations to organize the numerical analysis of complex queueing models such as the present one.

More explicit results for the case of two phases are given and are compared to results obtained by Yechiali and Naor for a closely related two-phase generalization of the M/M/1 queue.

The problem of transmitting information in a specified direction over a communication channel with three terminals is considered. Examples are given of the various ways of sending information. Basic inequalities for average mutual information rates are obtained. A coding theorem and weak converse are proved and a necessary and sufficient condition for a positive capacity is derived. Upper and lower bounds on the capacity are obtained, which coincide for channels with symmetric structure.

The set of rooted trees, generated by random bifurcation at the terminal nodes, is considered with the aims of enumerating it and of determining its probability distribution. The account of enumeration collates much previous work and attempts a complete perspective of the problems and their solutions. Asymptotic and numerical results are given, and some unsolved problems are pointed out. The problem of ascertaining the probability distribution is solved by obtaining its governing recurrence equation, and numerical results are given. The difficult problem of determining the most probable tree-shape of given size is considered, and for labelled trees a conjecture at its solution is offered. For unlabelled shapes the problem remains open. These mathematical problems arise in attempting to reconstruct evolutionary trees by the statistical approach of Cavalli-Sforza and Edwards.

We consider a branching process in which each individual reproduces independently of all others and has probability aj(j = 0, 1, · · ·) of giving rise to j progeny in the following generation. It is assumed, without further comment, that 0 < a0, a0 + a1 < 1.