An exposition is given of the properties of the iterates of complex functions near a fixed point, with explicit expressions for their power series in certain cases. The relevance to problems in genetics and statistics is pointed out.

Let {Z(t)}t0 be an age-dependent branching process with immigration. For a general class of functions Φ(x), a necessary and sufficient condition is given for when E{Φ (Z(t))} <∞. This result is a direct generalization of a theorem proven for the branching process without immigration.

In this article we examine the behavior of Mm(τ), the number of susceptibles remaining uninfected at time τ in a carrier-borne epidemic as the initial number of susceptibles m tends to infinity for fixed initial number of carriers n. We show that as m → ∞ the proportion of susceptibles process {Mm(τ)/m : ≧ 0} converges weakly to a limiting process. The limiting process has sample functions almost surely of the form exp (– h(τ)), where the form of h(τ) is determined by the removal times of the carriers.

Paz (1963), (1971) has shown that for an n × n stochastic matrix P the property that some power of P has a positive ergodic coefficient is decidable. In particular he shows that α (Pk) > 0 for some k, only if it is positive for k = ½n(n – 1). However he states that it is not known whether this bound is sharp. In this paper a sharp bound is given, namely k =½(n – 1)2 + ½ or ½(n − 1)2 + 1 depending on whether n is even or odd. The proof of this is based on some results from number theory.

In this paper a dam with autoregressive inputs and unit outputs is considered. Approximations for the distributions of the times to first emptiness and overflow are obtained using a Wald-type identity. The limiting behaviour of the dam content is also investigated

The behaviour in equilibrium of networks of queues in which customers may be of different types is studied. The type of a customer is allowed to influence his choice of path through the network and, under certain conditions, his service time distribution at each queue. The model assumed will usually cause each service time distribution to be of a form related to the negative exponential distribution.

Theorems 1 and 2 establish the equilibrium distribution for the basic model in the closed and open cases; in the open case the individual queues are independent in equilibrium. In Section 4 similar results are obtained for other models, models which include processes better described as networks of colonies or as networks of stacks. In Section 5 the effect of time reversal upon certain processes is used to obtain further information about the equilibrium behaviour of those processes.

The class of fractional linear generating functions is used to illustrate various aspects of the theory of branching processes in varying and random environments. In particular, it is shown that Church's theorem on convergence of the varying environments process admits of an elementary proof in this particular case. For random environments, examples are given on the asymptotic behavior of extinction probabilities in the supercritical case and conditional expectation given non-extinction in the subcritical case.

Let Zk denote the number in the kth generation of a Galton-Watson process initiated by one individual and let N be the total progeny, i.e., As n → ∞ the limiting behaviour of the process {Zk, 0 ≦ k ≦ n} conditioned on the event {N =n} is studied. The results obtained are of exactly the same form for the subcritical, critical and supercritical cases. This is in marked contrast to the analogous situation got by conditioning on non-extinction by the nth generation and letting n → ∞. In the latter case the limiting results differ in form for the critical and non-critical cases.

A sufficient condition is given for a supercritical generalized age-dependent branching process, properly normalized, to converge to zero. This condition is the strongest one possible, for if it does not hold, then it is known that the normalized process does indeed converge to a legitimate random variable.

In the article below, we consider sets of non-random functions of time t admitting certain asymptotic distributions. Purely temporal and deterministic considerations lead us to associate to a set , say, of functions H(t) of this type, a space Ω of samples ω.

To each function H(t) ⊂ , there corresponds a random variable h (ω). To the set of translated functions H(t + λ) of a function H(t) ⊂ , there corresponds a stationary random function of the translation parameter λ, say, h(λ, ω). We study the transposition to the set of non-random functions H(t) of such properties as moments, gaussian character, independence, harmonic analysis, and others, of the random variables h (ω) and of the random functions h (λ, ω).

Some remarks are made concerning the links between ergodicity and the above problems.

A single server is fed by a renewal stream of individual customers. These are of type k with probability πk, k = 1, …, N, and are all served individually. Upon completion of a service the server proceeds immediately with a customer of the lowest type (= highest priority) present, if any. Service times for type k are drawn from a general distribution function Bk(t) concentrated on (0, ∞).

We lay the foundations for a broad analysis of the model.

A class of multivariate exponential distributions is suggested which includes those presented by Marshall and Olkin, Downton, and Hawkes. It is based on a concept of hierarchical successive damage. Recursive expressions for the Laplace transforms and for first and second moments are derived in the bivariate case. Related multivariate geometric distributions are described.

Individuals arrive at a geographical area that is initially empty, in accordance with a pure birth process with birth parameters λj = jλ + θ, j ≧ 0. Due to contamination, this geographical area is unsafe for its population and at some fixed time T in the future everyone in the area will be evacuated and no further immigration will be allowed.

Suppose now than an intermediate evacuation time τ, 0 ≦ τ ≦ T, at which time everyone present in the area would be evacuated, is to be chosen. The area would then again fill up with individuals between times τ and T, and, at T, the final evacuation would be made and the area would be permanently sealed off. The problem is to choose τ so as to minimise the total expected cost incurred by time T, where a cost g(x) is incurred for each individual that spends a time x in the area.

This note deals with a q-dimensional stochastic vector process x(t) = {x1 (t), …, xq (t)}, satisfying certain stated general conditions. For such a process, there is a representation (1) in terms of stochastic innovations acting throughout the past of the process. The number N of terms in this representation is called the multiplicity of the x(t) process, and is uniquely determined by the process. For a one-dimensional process (q = 1) it is known that under certain conditions we have N = 1. For an arbitrary value of q, this note gives conditions under which we have N ≦ q.

Results are given which relate the tail behaviour of the service and limiting waiting time distributions of a GI/G/1 queue. A limit theorem for the maxima of waiting times is given.

We discuss here an extension of a queueing model studied by Neuts and also by Çinlar. We obtain a matrix form of Takács' equations and exhibit the equilibrium conditions. We also show that the conditions imposed by Neuts and by Çinlar in order to obtain their results concerning the busy period are not necessary.

When passing from two to more dimensions, the study of non-isotropic scattering transport processes, and chain molecules, which are both covered by the same mathematical model, becomes affected by the non-commutativity of rotations. The techniques developed in [2], together with results on radially symmetric direction changes, are used in this paper to obtain a functional central limit theorem for those random processes, with a suitable normalization, the limit being a Brownian motion process which is completely identified.

It is shown in this note that if the population increases geometrically, then the asymptotic distribution for the inter-record times is also geometric. The records in Olympic games are used as an example. Also, it is noted that the rapid breaking of Olympic records is not due mainly to the increase in population.

A model is proposed for the control of a deterministic carrier-borne epidemic by immunising susceptibles. An optimal control is of bang-bang type and it is proved that it can have at most one switch.

Recent work by Moussouris [10] has clarified our present knowledge of finite Markov fields. The present note examines, in a loose and general fashion, whether one can extend the treatment to infinite fields.