Let ε be a (single or composite) pattern defined over a sequence of Bernoulli trials. This article presents a unified approach for the study of the joint distribution of the number Sn of successes (and Fn of failures) and the number Xn of occurrences of ε in a fixed number of trials as well as the joint distribution of the waiting time Tr till the rth occurrence of the pattern and the number STr of successes (and FTr of failures) observed at that time. General formulae are developed for the joint probability mass functions and generating functions of (Xn,Sn), (Tr,STr) (and (Xn,Sn,Fn),(Tr,STr,FTr)) when Xn belongs to the family of Markov chain imbeddable variables of binomial type. Specializing to certain success runs, scans and pattern problems several well-known results are delivered as special cases of the general theory along with some new results that have not appeared in the statistical literature before.

The scientific community is saddened to learn of the death of Professor Georges Matheron on 7 August 2000. As a major figure in applied mathematics over four decades, he leaves an outstanding scientific heritage, covering a wide range of domains where probabilistic tools and models are implemented. He also trained more than one generation of researchers, engineers and teachers. In the fields of theoretical and applied random media and image analysis his work has inspired many researchers worldwide: mathematicians, statisticians, physicists, experimentalists, earth scientists, mining engineers and also users of image analysis and synthesis in all its applications (materials, biology, artificial vision, CAD, remote sensing, geophysics, biometry, image coding etc.).

In this paper we extend the notion of quasi-reversibility and apply it to the study of queueing networks with instantaneous movements and signals. The signals treated here are considerably more general than those in the existing literature. The approach not only provides a unified view for queueing networks with tractable stationary distributions, it also enables us to find several new classes of product form queueing networks, including networks with positive and negative signals that instantly add or remove customers from a sequence of nodes, networks with batch arrivals, batch services and assembly-transfer features, and models with concurrent batch additions and batch deletions along a fixed or a random route of the network.

When the premium rate is a positive absolute constant throughout the time period of observation and the safety loading of the insurance business is positive, a classical result of collective risk theory claims that probabilities of ultimate ruin ψ(u) and of ruin within finite time ψ(t,u) decrease as eϰu with a constant ϰ>0, as the initial risk reserve u increases. This paper establishes uniform approximations to ψ(t,u) with slower rates of decrease when the premium rate depends on u in such a way that the safety loading decreases to zero as u→∞.