This paper describes work on the stochastic modelling of loss networks. Such systems have long been of interest to telephone engineers and are becoming increasingly important as models of computer and information systems. Throughout the century problems from this field have provided an impetus to the development of probability theory, pure and applied. This paper provides an introduction to the area and a review of recent work.

We obtain a diffusion approximation result for processes satisfying equations with past-dependent coefficients. We apply this result to a model of option pricing, in which the underlying asset price volatility depends on past evolution, and obtain a generalized (asymptotic) Black and Scholes formula.

We study the structure of $T^{(k)}_n$, the subtree rooted at $k$ in a random recursive tree of order $n$, under the assumption that $k$ is fixed and $n \rightarrow \infty$. Employing generalized Polya urn models, exact and limiting distributions are derived for the size, the number of leaves and the number of internal nodes of $T^{(k)}_n$. The exact distributions are given by intricate formulas involving Eulerian numbers, but a recursive argument based on the urn model suffices for establishing the first two moments of the above-mentioned random variables. Known results show that the limiting distribution of the size of $T^{(k)}_n$, normalized by dividing by $n$ is $\operatorname{Beta}(1, k - 1)$. A martingale central limit argument is used to show that the difference between the number of leaves and the number of internal nodes of $T^{(k)}_n$, suitably normalized, converges to a mixture of normals with a $\operatorname{Beta}(1, k - 1)$ as the mixing density. The last result allows an easy determination of limiting distributions of suitably normalized versions of the number of leaves and the number of internal nodes of $T^{(k)}_n$.

This article is concerned with the $(r, q)$ inventory model, where demand accumulates continuously, but the demand rate at each instant is determined by an underlying stochastic process. The primary result is the demonstration of a certain insensitivity property, which characterizes the limiting behavior of the model. This property drastically simplifies the computation of performance measures for the system.

Recently there has been a considerable amount of work on the transient behaviour of loss networks in two different limiting regimes. The first of these, which we do not consider here, is when link capacities and offered traffics become large but the number of links remains finite. The second is the diverse routing limit when the number of links increases with the offered load to each link held constant. Thus far, however, all results of this latter type have been for simplified models with exchangeable links. In adopting such a simplified model one loses the inherent graph structure of the original loss network, which appears to be a serious drawback. In this paper we consider a loss network with graph structure and show that, subject to natural constraints on the initial configuration, the model behaves asymptotically exactly like one with exchangeable links. Our result is proved by combining the techniques of Gibbens, Hunt and Kelly with those of Hajek for a problem in random graph theory.

The pseudolikelihood for general spatial point processes including marked point processes is derived, and some of its properties are investigated. In particular, we prove consistency of the maximum pseudolikelihood estimates for Markov processes of finite range.

We introduce a new class of generators of two types: add-with-carry and subtract-with-borrow. Related to lagged-Fibonacci generators, the new class has interesting underlying theory, astonishingly long periods and provable uniformity for full sequences. Among several that we mention, we recommend particularly promising ones that will generate a sequence of $2^{1751}$ bits, or a sequence of $2^{1376}$ 32-bit integers, or a sequence of $2^{931}$ reals with 24-bit fractions--all using simple computer arithmetic (subtraction) and a few memory locations.