The threshold behaviour of the stochastic spatial general epidemic model on a discrete location space is investigated by making use of the general percolation theory of McDiarmid.

The general stochastic epidemic model is used as a model for the spread of rumours. Recursive expressions are found for the mean of the final size of each generation of hearers. Simple expressions are found for the generation size and the asymptotic form of its final size in the deterministic model. An approximating process is presented.

The kin number problem involves the relationship between sibship sizes and offspring numbers, and also numbers of relatives of other degrees of affinity of a random member of a population, to be called Ego. The problem has been well known to demographers for some time, but results obtained only gave expected numbers. Recently a study of it, based on the Galton-Watson process, was made, with a view to obtaining joint distributions (Waugh (1981)). In the latter study it was assumed that the population was large, and thus some of the results obtained were approximations.

In the present paper exact distributions are obtained, for any size of population. This can be of use in applications, where the population considered may be a small, isolated tribe or other special group. As a theoretical investigation, it replaces some heuristic arguments with limiting properties that are intrinsic to the process and it makes it possible to evaluate the previous approximations.

In this paper, we consider a Bienaymé– Galton–Watson process {Xn ; n ≧ 0; Xn = 1} and develop a recurrence formula for P(Xn = k), k = 1, 2, ···. The problem of obtaining the maximum likelihood estimate of the age of the process when p 0 = 0 is discussed. Furthermore the maximum likelihood estimate of the age of the process when the offspring distribution is negative binomial (p 0 ≠ 0) is obtained, and a comparison with Stigler's estimator (1970) of the age of the process is made.

The species-area problem in biology and the type-token problem in literary studies are analogues of one another but have nearly disjoint literatures. Here their relationship is treated, a critique of models used in each is made and Markovian models are provided for two well-known empirical models from the species-area literature. Finally, some of the assumptions underlying each of these models are considered in the light of literary data.

Formulae are presented for the variance and higher moments of the present value of single-stage rewards in a finite Markov decision process. Similar formulae are exhibited for a semi-Markov decision process. There is a short discussion of the obstacles to using the variance formula in algorithms to maximize the mean minus a multiple of the standard deviation.

The decision-maker drives a car along a straight highway towards his destination and looks for a parking place. When he finds a parking place, he can either park there and walk the distance to his destination or continue driving. Parking places are assumed to occur in accordance with a Poisson process along the highway. The decision-maker does not know the distance Y to his destination exactly in advance. Only an a priori distribution is assumed for Y and cases of typically important distribution are examined. When we take as loss the distance the decision-maker must walk and wish to minimize the expected loss, the optimal stopping rule and the minimum expected loss are obtained. In Section 3 a generalization to the cases of a non-homogeneous Poisson process and a renewal process is considered.

The long-run average cost per unit time of operating a finite dam controlled by a PlM policy (Faddy (1974), Zuckerman (1977)) is determined when the cumulative input process is (a) a Wiener process with drift and (b) the integral of a Markov chain. It is shown how the cost for (a) can be obtained as the limit of the costs associated with a sequence of input processes of the type (b).

This paper considers the equilibrium behaviour of the M/G/k group-arrival group-departure loss system. Such a system has k servers whose customers arrive in groups, the arrival epochs of groups being points of a Poisson process. The duration of a service can be characteristic of the group size; however, customers who belong to the same group have equal service times. The customers of a group start being served immediately upon their arrival, unless their number is greater than the number of idle servers. In this case the whole group leaves and does not return later (i.e. is lost). Among other things, a generalization of the Erlang B-formula is given and it is shown that the arrival and departure processes are statistically indistinguishable.

The probability distributions of the size and the duration of simple stochastic epidemic models are well known. However, in most instances, the solutions are too complicated to be of practical use. In this note, interarrival times of the infectives are utilized to study asymptotic distributions of the duration of the epidemic for a class of simple epidemic models. A brief summary of the results on simple epidemic models in terms of interarrival times is included.

Estimation of the offspring distribution from a single realization of a supercritical Galton-Watson process is studied. It is shown that, based on population totals, a parameter of the offspring distribution cannot be estimated unless it is determined by the mean, variance, lattice size and lattice offset of the offspring distribution.

Mating functions considered by Asmussen (1980) and Daley (1968) for the class of bisexual Galton-Watson branching processes (GWBP) are shown to be superadditive. Consideration of a process that allows only sibling mating leads to a necessary condition for almost sure extinction in bisexual GWBP governed by superadditive mating functions. A simple example shows that the condition is not sufficient.

In this paper we study the asymptotic behavior of Markov systems and especially non-homogeneous Markov systems. It is found that the limiting structure and the relative limiting structure exist under certain conditions. The problem of weak ergodicity in the above non-homogeneous systems is studied. Necessary and sufficient conditions are provided for weak ergodicity. Finally, we discuss the application of the present results in manpower systems.

Seneta, in a recent paper, presented a general treatment of the concept of ‘coefficient of ergodicity', τ (P), for a finite stochastic matrix P. In this paper, a functional form for τ∞(P) in terms of the attributes of P is determined. It is shown that, by increasing the dimension of P, τ∞(Ρ) can assume any large value. In view of this, τ∞ will be practically useful only in the case τ∞(P) ≦ τ1(P), where τ1(P) is the well-known Dobrushin or delta coefficient.

Some recent results of Oxley and Welsh (1979) and McDiarmid (1981) concerning Bernoulli percolation on clutters are generalized. Our results allow to consider quantitative aspects of percolation on graphs and clutters.

The paper contains a simple proof that every finite discrete distribution is the equiprobable mixture of r distributions, each of which has at most a mass points.

It is shown how a triangle can be recognised from the odd moments of the length of random chords and how a quadrilateral can be recognised from the derivative of its chord length density.

We consider a random field {Xij, i, j = 1, ···, n} where the random variables Xij takes on values 1 or 0. The collection {Xij } can be viewed as a random graph with nodes {1, ···, n} by interpreting X ij = 1 as the existence of an arc emanating from the node i to the node j. Such a representation will enable us to study ordered and unordered graphs, being also the general representation of a random graph. In this note the probability that the graph is connected is computed under the condition that ΣiXki=l for k = 1, · ··, n. This result extends Ross's recent theorems on connectivity of random graphs.

The extended block replacement policy with used items as advocated by Tango (1978) uses used items of age T only although the policy creates used items of lower age. In this note we modify Tango's policy so as to make use of all used items created by the policy.

In this note we consider the marginal effect of adding an additional spare to a system containing operating units and their spares. We shall see that under the circumstance that the number of spares is greater than the number of repair facilities, the marginal effect will assume a simple interesting structure.