A survey is presented of stochastic and deterministic developments in the study of the effects of nearest-neighbour ‘migration’ between spatially separated ‘colonies’. Such processes are of general applicability, and are relevant to any vector process X(t) = (X1(t), · ··, XN(t)) in which the arrival, departure and transfer rates for the states {X(t) = n} may be written in the form α i(ni), βi(ni) and γ ij(ni, nj), respectively, where n = (n1, · ··, nN). Whilst the main body of results are described in terms of birth-death, genetic and epidemic situations, the final section examines within colony interaction in the context of spatial predator-prey processes.

Batches of immigrants arrive in a region at event times of a renewal process and individuals grow according to a Bellman-Harris branching process. Tribal emigration allows the possibility that all descendants of a group of immigrants collectively leave the region at some instant.

A number of results are derived giving conditions for the existence of a limiting distribution for the population size. These conditions can be given either in terms of the immigration distribution or in terms of the distribution of emigration times. Some limit theorems are obtained when the latter conditions are not fulfilled.

Spatial point processes are considered whose points are subjected to certain classes of affine transformations indexed by some variable, T. Under some hypotheses, for large T integrals with respect to such a point process behave approximately as if the process were Poisson. Under stronger hypotheses, the transformed process converges as a process to a Poisson process. The result gives the asymptotic distribution of certain density estimates.

The problem of identifying solutions of general convolution equations relative to a group has been studied in two classical papers by Choquet and Deny (1960) and Deny (1961). Recently, Lau and Rao (1982) have considered the analogous problem relative to a certain semigroup of the real line, which extends the results of Marsaglia and Tubilla (1975) and a lemma of Shanbhag (1977). The extended versions of Deny&s theorem contained in the papers by Lau and Rao, and Shanbhag (which we refer to as LRS theorems) yield as special cases improved versions of several characterizations of exponential, Weibull, stable, Pareto, geometric, Poisson and negative binomial distributions obtained by various authors during the last few years. In this paper we review some of the recent contributions to characterization of probability distributions (whose authors do not seem to be aware of LRS theorems or special cases existing earlier) and show how improved versions of these results follow as immediate corollaries to LRS theorems. We also give a short proof of Lau–Rao theorem based on Deny&s theorem and thus establish a direct link between the results of Deny (1961) and those of Lau and Rao (1982). A variant of Lau–Rao theorem is proved and applied to some characterization problems.

Some simple models are described which may be used for the modelling or generation of sequences of dependent discrete random variates with negative binomial and geometric univariate marginal distributions. The models are developed as analogues of well-known continuous variate models for gamma and negative exponential variates. The analogy arises naturally from a consideration of self-decomposability for discrete random variables. An alternative derivation is also given wherein both the continuous and the discrete variate processes arise simultaneously as measures on a process of overlapping intervals. The former is the process of interval lengths and the latter is a process of counts on these intervals.

The idea is developed of imbedding a given Markov process in a more general Markov process in a definite sense, the sense of weak coupling. It is shown that this gives a natural theory of insensitivity in that, the stronger the balance conditions satisfied by the imbedded process, the weaker the conditions that are required of the imbedding process. The imbeddings associated with a range of balance conditions are discussed.

The theory of allocation indices for defining the optimal policy in multi-armed bandit problems developed by Gittins is presented in the continuous-time case where the projects (or ‘arms’) are strong Markov processes. Complications peculiar to the continuous-time case are discussed. This motivates investigation of whether approximation of the continuous-time problems by discrete-time versions provides a valid technique with convergent allocation indices and optimal expected rewards. Conditions are presented under which the convergence holds.

Simulated annealing is a randomized algorithm which has been proposed for finding globally optimum least-cost configurations in large NP-complete problems with cost functions which may have many local minima. A theoretical analysis of simulated annealing based on its precise model, a time-inhomogeneous Markov chain, is presented. An annealing schedule is given for which the Markov chain is strongly ergodic and the algorithm converges to a global optimum. The finite-time behavior of simulated annealing is also analyzed and a bound obtained on the departure of the probability distribution of the state at finite time from the optimum. This bound gives an estimate of the rate of convergence and insights into the conditions on the annealing schedule which gives optimum performance.

We examine a replacement model for a semi-Markov shock model with additive damage. Sufficient conditions are given for the optimality of control limit policies. The paper generalizes and unifies previous research in the area.

In addition, we investigate in detail the practical modelling and computational aspects of the replacement problem using a semi-Markov modelling structure.

This paper contains two contributions to the insensitivity theory for generalized semi-Markov schemes, namely

1. (1) a detailed account of a close relationship between insensitive schemes and partially balanced discrete-state Markov chains, and

2. (2) an extension of the notion of an insensitive scheme in order to incorporate insensitivity phenomena not previously covered.

The paper deals with the asymptotic behaviour of queues for which the generic sequence is not necessarily stationary but is asymptotically stationary in some sense. The latter property is defined by an appropriate type of convergence of probability distributions of the sequences to the distribution of a stationary sequence We consider six types of convergence of to

The main result is as follows: if the sequence of the distributions converges in one of six ways then the sequence of distributions of the sequences converges in the same way, independently of initial conditions. Furthermore the limiting distribution is the same as the limiting distribution obtained by the weak convergence of the distributions Here wk and w∗k denote the waiting time of the kth unit in the queue generated by (v, u) and (v0, u0) respectively.

The paper is a continuation of [7]. One of the main results is as follows: if the sequence (w, v, u) is asymptotically stationary in some sense then (l, w, v, u) is asymptotically stationary in the same sense. The other main result deals with an asymptotic behaviour of the vector of the queue size and the waiting time in the heavy-traffic situation. This result resembles a formula of the Little type.

A class of fitness matrices whose parameters may be varied to give differing stability structure is shown by Chebyshev&s covariance inequality to possess a variance lower bound for the change in mean fitness.

We present a simple derivation for an approximation to the transient blocking probability of an M/M/N loss system.