We show that a lower bound for the probability that a spatial general epidemic never becomes extinct is given by the percolation probability of an associated bond percolation process.

An upper estimate of the probability density of d-parameter Wiener process with one barrier, already known for d = 2, is extended for general d.

A queueing model of considerable interest in computer engineering is the processor-sharing model in which the server shares its fixed capacity equally among all units present in the system. Here, we derive the mean and the variance of the equilibrium sojourn time, and deduce that the variance of the sojourn time is larger for the processor-sharing model than for the corresponding FCFS model.

Norma McArthur, one of the four founding trustees of the Applied Probability Trust, died at her home in Canberra, Australia, on 17 January 1984. She had fought a long battle with cancer, exhibiting great resilience, clear-sightedness and courage to the last. When we took leave of each other in Canberra in August 1983, she was making plans for our next meeting in July 1984.

In this paper the probability distribution of the discrete busy period of the 〈M, 1, GI/∞, 1〉 queue and some of its stability properties are given.

This note deals with extinction criteria for bisexual Galton–Watson processes with arbitrary mating functions in terms of the averaged reproduction mean per mating unit. It gives a satisfactory answer to a question put forward by Hull (1982).

Let Zn denote the number of cells at time n in a critical discrete-time Galton–Watson branching process with finite offspring variance. Let Martingale arguments are used to show that for some 0<a≦b<∞

The busy period of the GI/D/∞ queue is determined as the time when at least one customer is served. Let v be the number of customers served during this period. The busy period of order n is defined as a busy period for which v ≧ n. In this paper we derive the exact distributions, integral equations, characteristic functions and all moments of those periods. Finally, some properties of the idle periods of order n are established.

Majorisation is used to compare continuous distributions in terms of randomness. General results on randomness in the continuous case are given and these are used to investigate the connection between randomness and parameter values in some well-known families of distributions including the normal and gamma.

Certain first-order autoregressive processes are shown not to be strong mixing. A direct proof is given. This proof gives considerably more insight into the nature of the result than do proofs by contradiction. The result and proof help to clarify the relation between the autoregressive and strong mixing conditions.