We introduce a class of infinite-dimensional diffusion processes which contains a limiting version of the Ohta–Kimura model in population genetics. For this a necessary and sufficient condition for existence of stationary distributions is obtained. We are especially interested in the case where there is no stationary distribution. Then it is shown that an individual ergodic theorem holds for a suitably centralized process. As a corollary the wandering distribution exists.

This paper deals with the parity of individuals in an age-dependent birth and death process. A more general model with parity and age-dependent birth rates is also considered. The mean number of individuals with parity 0, 1, 2, ·· ·is obtained for the two models. The first moments of the total number of births in the population up to time t and the sum of the parities of the individuals existing at time t are obtained. A brief discussion on the parity of individuals in a population including ‘twins' is also given.

Bounds of exponential type are derived for the first-hitting time and occupation times of a real-valued random sequence which has a uniform negative drift whenever the sequence is above a fixed level. The only other assumption on the random sequence is that the increments satisfy a uniform exponential decay condition. The bounds provide a flexible technique for proving stability of processes frequently encountered in the control of queues.

Two applications are given. First, exponential-type bounds are derived for a GI/G/1 queue when the service distribution is exponential type. Secondly, geometric ergodicity is established for a certain Markov chain in which arises in the decentralized control of a multi-access, packet-switched broadcast channel.

Books being placed on a shelf, a book bi is selected with probability pi, it is removed and replaced at the left-hand end of the shelf prior to next removal. The different arrangements of books on the shelf are considered as states in a Markov chain, known as a Tsetlin library. In this paper, we establish the necessary and sufficient condition of transience for some extensions of this chain.

In this paper, we investigate the behaviour of a Gaussian random field after an ‘upcrossing' of a particular level. In Section 1, we briefly discuss model processes and their background, and give a definition of an upcrossing of a level for random fields. A model field is constructed for the random field after an upcrossing of a level by using horizontal window conditioning in Section 2. The final section contains asymptotic distributions for the model field and for the location and height of the ‘closest' maximum to the upcrossing as the level becomes arbitrarily high.

Correction terms are obtained for the Brownian motion approximation to one- and two-barrier first-passage probabilities. These approximations are given in terms of their Laplace transforms, which are formally (and non-rigorously) inverted. Applications to the one-sample Kolmogorov-Smirnov statistic are discussed.

Let K and L be convex bodies in where L can roll freely in K. Suppose that Borel sets are painted in the boundaries of K and L. The probability that after a random rolling of L in K the contact is paint-to-paint is determined and expressed by curvature measures of K and L.

Rates of convergence are derived for the convergence in distribution of renormalised sample maxima to the appropriate extreme-value distribution. Related questions which are discussed include the estimation of the principal error term and the optimality of the renormalising constants. Throughout the paper a close parallel is drawn with the theory of slow variation with remainder. This theory is used in proving most of the results. Some applications are discussed, including some models of importance in reliability.

Motivated by a problem in PERT networks we consider the question of construction of random variables with maximum sums when the marginals are fixed.

Messages are to be transmitted through a series of n nodes linked by communication channels. The lengths of successive messages are independent identically distributed random variables, and the time taken to transmit a message through a channel is determined by its length. Each node has a finite buffer, and when the number of messages at a node reaches the buffer size transmission from the preceding node is interrupted. This paper is concerned with the maximum rate at which messages can pass through the system, called the throughput. We investigate the asymptotic behaviour of throughput as the series length increases, and determine the rate at which buffer sizes should grow to ensure that throughput does not decline to 0.

We consider single-class Markovian queueing networks with state-dependent service rates (the immigration processes of Whittle (1968)). The distance of customer flows from Poisson processes is estimated in both the open and closed cases. The bounds on distances lead to simple criteria for good Poisson approximations. Using the bounds, we give an asymptotic, closed network version of the ‘loop criterion' of Melamed (1979) for an open network. Approximation of two or more flows by independent Poisson processes is also studied.

Consider a path in a multiclass Gordon–Newell network such that a customer present in a node of this path cannot be overtaken by any other customer behind him in a node of this path or by probabilistic influences created by such customers. The passage time through such a path is a mixture of Erlangian distributions, where the mixing distribution is given by the steady state of the network.