In this paper the theory of genetic algebras introduced by Etherington [3], [5] is developed to cover the case where the population is divided into a number of partially isolated subpopulations, between which a limited amount of migration is permitted. It is known [3], [6], [7], [8], [10], that in the absence of selective factors most genetic segregation systems can be represented by algebras belonging to the class of special train algebras, discussed in detail in [4].

This paper treats the well known model of sib mating of individuals with autosomal genes; a single locus is studied. The general derivations are valid for any number of alleles. At the end of the paper, some simple examples are included; here several familiar results concerning models for two alleles are obtained as special cases of the general theory.

The well-known Wright's genetic model is extended to a population whose size varies cyclically. Explicit expressions are obtained for the absorption probability and the rate of approach to homozygosity in the case where there is no selection. When selection operates, diffusion approximations are obtained for the various quantities of interest.

Much of the theory of epidemics (see Bailey, 1957) has been concerned with models for their behaviour in closed populations. In such models the epidemic ultimately dies out, and interest has been concentrated on, for example, the ultimate size of the epidemic and its duration in time. In practice a population is rarely completely closed, and for many diseases an endemic model rather than an epidemic model is appropriate. To create models for endemic diseases it is necessary to introduce both new persons susceptible to the disease into the population and new sources of infection. For the so-called general stochastic epidemic, Ridler-Rowe (1967) has obtained certain limiting properties of the population where immigration of both susceptibles and infectives into the population takes place, but much work remains to be done to obtain, for example, the general equilibrium behaviour of this model.

The article first reviews briefly the field of statistical process control and its relation to automatic process control. Some recent results obtained by various workers in Czechoslovakia in the field of statistical aspects of automatic process control are then presented.

A previous paper [2] was concerned with the determination of optimal policies for restocking an inventory which is continuously depleted by a random process of demands. The purpose of the present paper is to develop a similar model for controlling the output of a dam whose random input depends on a homogeneous Wiener process. This reversal of the roles of input and output does not, by itself, change the character of the problem. But the consideration of set-up costs for ordering replacements, which leads to inventory policies of the (s, S) type, has no counterpart here. It is natural to regard the dam as a device for smoothing out random fluctuations in a flow of water and, under utility assumptions which reflect this attitude, it follows that the optimal output rate is a continuous function of the level of water in the reservoir. Our main object is to determine this function.

Moran's [1] early investigations into the theory of storage systems began in 1954 with a paper on finite dams; the inputs flowing into these during consecutive annual time-intervals were assumed to form a sequence of independent and identically distributed random variables. Until 1963, storage theory concentrated essentially on an examination of dams, both finite and infinite, fed by inputs (discrete or continuous) which were additive. For reviews of the literature in this field up to 1963, the reader is referred to Gani [2] and Prabhu [3].

We find the distribution of delay to minor road vehicles waiting to merge or cross a single stream of major road traffic. The decision to cross is taken on the basis of a gap-acceptance function. The model turns out to be a simple queueing problem in which a customer finding an empty queue has a different service time distribution from queueing customers. The key to this representation is given in Section 3. Some numerical results in Section 6 indicate that in most circumstances a simple model will give adequate results.

In a previous paper [4] the author studied the stochastic process {wn, n = 1,2, …}, recursively defined by with K a positive constant, τ1, τ2, … σ1, σ2, …, independent, nonnegative stochastic variables. τ1,τ2…, are identically distributed, and σ1,σ2,…, are also identically distributed variables. For this process the generating function of the Laplace-Stieltjes transforms of the joint distribution of Wn, σ2 + … + σn and τ1 + … + τn−1 has been obtained. Closely related to the process {wn, n = 1, 2,…} is the process {un, n = 1, 2,…} with {un = K + [wn + τn − K]−, n = 1,2,…; these are dual processes.

In the present paper we study the stationary distributions of the processes {wn, n= 1,2, …} and {un, n = 1,2, …}, and the distributions ot the entrance times and return times of the events “wn, n = 0” and “un = K” for some n, for discrete as well as for continuous time. For these events various taboo probabilities are also investigated. The mathematical descri ption of the processes {wn, n = 1,2, …} and {un, n= 1,2, …} gives all the necessary information about the time-dependent behaviour for the general dam model with finite capacity K, since the process {wn, n= 1,2, …} is the basic process for such dam models. In Sections 5, 6 and 7 the general theory is applied to the models M/G/1 and G/M/1. Complete explicit solutions are obtained for these models.

The present theory also leads to new and important results for the queueing system or dam model G/G/1 with infinite capacity. For instance the joint distribution of the busy period (or wet period) and of the supremum of the dam content dunng this period is obtained.

A service unit I, with Poisson input and general service times is in series with a unit II, with negative-exponential service times. The intermediate waitingroom can accomodate at most k persons and a customer cannot leave unit I when the waitingroom is full.

The paper shows that this system of queues can be studied in terms of an imbedded semi-Markov process. Equations for the time dependent distributions are given, but the main emphasis of the paper is on the equilibrium conditions and on asymptotic results.

We follow the terminology of Khintchine's monograph [4].

Let us consider a random stream of calls entering a service system which consists of n lines. The service durations are assumed to be identically distributed random variables which are independent both of each other and of the course of the incoming stream.

Customers arrive in a recurrent process and get served by one of the s (≧1) servers wit han exponential service time distribution. The equilibrium behavior of the queue length process has been studied by earlier authors. In this paper the transient behavior of this process is investigated.

Consider n independent vector valued point processes. Superposition is defined component by component as a natural extension of the definition for the one-dimensional case. Under proper conditions as n → ∞, it is shown that the superposed process is a many-dimensional Poisson process with independent components. The results are applied to the superposition of Markov renewal processes.

This paper applies the invariance principle to the problem of testing a distribution on a compact homogeneous space for uniformity. The notion of using a reduction by invariance in such a situation is due to Ajne[1], who considers tests invariant under rotation on a circle. In his paper, he derives the distribution of the maximal invariant and gives the general form of the most powerful invariant test for uniformity on the circle.

Probability properties of the measure of the union of random sets have theoretical as well as practical importance (David (1950), Garwood (1947), Hemmer (1959)). In the present paper we derive asymptotic properties of the distributions of these measures and apply the derived properties to the investigation of the asymptotic behavior of empirical distribution functions. Thus, an asymptotic distribution function for the relative lengths of steps in the empirical distribution function is obtained.

In the study of dynamical systems perturbed by noise, it is important to know whether the stochastic process of interest has a stationary distribution. Four necessary and sufficient conditions are formulated for the existence of a finite invariant measure for a Feller process on a σ-compact metric (state) space. These conditions link together stability notions from several fields. The first uses a Lyapunov function reminiscent of Lagrange stability in differential equations; the second depends on Prokhorov's condition for sequential compactness of measures; the third is a recurrence condition on the ergodic averages of the transition operator; and the fourth is analogous to a condition of Ulam and Oxtoby for the nonstochastic case.

Let Xi, i = 1, 2, 3,… be a sequence of independent and identically distributed random variables with law ℓ(X) and write. if EX= 0 and EX2=σ2< ∞, the law of the iterated logarithm (Hartman and Wintner [1]) tells us that

Let F(z) = σ fjzj be the generating function for the offspring distribution {fj} from a single ancestor in the usual Galton-Watson process. It is well-known (see Harris [1]) that if Π(z) is the generating function of the distribution of ancestors in the 0th generation, the distribution of offspring at the nth generation has generating function where Fn(z), the nth functional iterate of F(z), gives the distribution of offspring at the nth generation from a single ancestor.

Let {X(n): n = 0, 1, 2, …} be a Markov chain with state space {0, 1, 2, …, N} and transition probability matrix P = (pij).

In the M/G/1 queue, the joint distribution of the virtual waitingtime and the residual duration of the busy period at any time t, is given by a formula analogous to the one for the virtual waitingtime by itself. In the limit, as t → ∞, the limiting joint distribution is given by a formula of the Pollaczek-Khintchine type.