P. K. Pollett

Sojourn times in closed queueing networks

This paper obtains the stationary joint distribution of a customers sojourn times along an overtake-free path in a closed multiclass Jackson network. The distribution has a simple representation in terms of the product form distribution for the state of the network at an arrival instant.

On a Model for Interference Between Searching Insect Parasites

The purpose of this paper is to study a stochastic model which assesses the effect of mutual interference on the searching efficiency in populations of insect parasites. By looking carefully at the assumptions which govern the model, I shall explain why the searching efficiency is of the same order as the total number, N, in the population, a conclusion which is consistent with the predictions of population biologists; previous studies have reached the conclusion that the efficiency is of order \/~N . The major results of the paper establish normal approximations for the distribution of the numbers of active parasites. These are valid at all stages of the process, in particular the non-equilibrium phase, where explicit analytic formulae for the state-probabilities are unavailable.

Quasi-stationary distributions for discrete-state models

This paper contains a survey of results related to quasi-stationary distributions, which arise in the setting of stochastic dynamical systems that eventually evanesce, and which may be useful in describing the long-term behaviour of such systems before evanescence. We are concerned mainly with continuous-time Markov chains over a finite or countably infinite state space, since these processes most often arise in applications, but will make reference to results for other processes where appropriate. Next to giving an historical account of the subject, we review the most important results on the existence and identification of quasi-stationary distributions for general Markov chains, and give special attention to birth-death processes and related models. The question of under what circumstances a quasi-stationary distribution, given its existence, is indeed a good descriptor of the long-term behaviour of a system before evanescence, is addressed as well. We conclude with a discussion of computational aspects, with more details given in a web appendix accompanying this paper.

On the Equivalence of Mu-Invariant Measures for the Minimal Process and its Q-Matrix

In this paper we obtain necessary and sufficient conditions for a measure or vector that is [mu]-invariant for a q-matrix, Q, to be [mu]-invariant for the family of transition matrices, {P(t)}, of the minimal process it generates. Sufficient conditions are provided in the case when Q is regular and these are shown not to be necessary. When [mu]-invariant measures and vectors can be identified, they may be used, in certain cases, to determine quasistationary distributions for the process.

Limit theorems for discrete-time metapopulation models

We describe a class of one-dimensional chain binomial models of use in studying metapopulations (population networks). Limit theorems are established for time-inhomogeneous Markov chains that share the salient features of these models. We prove a law of large numbers, which can be used to identify an approximating deterministic trajectory, and a central limit theorem, which establishes that the scaled fluctuations about this trajectory have an approximating autoregressive structure.

CONNECTING REVERSIBLE MARKOV PROCESSES

We provide a framework for interconnecting a collection of reversible Markov processes in such a way that the resulting process has a product-form invariant measure with respect to which the process is reversible. A number of examples are discussed including Kingmans reversible migration process,

Limiting conditional distributions for birth-death processes

In a recent paper one of us identified all of the quasi-stationary distributions for a non-explosive, evanescent birth-death process for which absorption is certain, and established conditions for the existence of the corresponding limiting conditional distributions. Our purpose is to extend these results in a number of directions. We shall consider separately two cases depending on whether or not the process is evanescent. In the former case we shall relax the condition that absorption is certain. Furthermore, we shall allow for the possibility that the minimal process might be explosive, so that the transition rates alone will not necessarily determine the birth-death process uniquely. Although we shall be concerned mainly with the minimal process, our most general results hold for any birth-death process whose transition probabilities satisfy both the backward and the forward Kolmogorov differential equations.

A DESCRIPTION OF THE LONG-TERM BEHAVIOUR OF ABSORBING CONTINUOUS-TIME MARKOV CHAINS USING A CENTRE MANIFOLD

We use the notion of an invariant manifold to describe the long-term behaviour of absorbing continuous-time Markov processes with a denumerable infinity of states. We show that there exists an invariant manifold for the forward differential

Quasistationary distributions for autocatalytic reactions

We provide simple conditions for the existence of quasistationary distributions that can be used to describe the long-term behaviour ofopen autocatalytic reaction systems. We illustrate with reference to a particular example that the quasistationary distribution is close to the usual stationary diffusion approximation.

Preserving Partial Balance in Continuous-Time Markov-Chains

Recently a number of authors have considered general procedures for coupling stochastic systems. If the individual components of a system, when considered in isolation, are found to possess the simplifying feature of either reversibility, quasireversibility or partial balance they can be coupled in such a way that the equilibrium analysis of the system is considerably simpler than one might expect in advance. In particular the system usually exhibits a product-form equilibrium distribution and this is often insensitive to the precise specification of the individual components. It is true, however, that certain kinds of components lose their simplifying feature if the specification of the coupling procedure changes. From a practical point of view it is important, therefore, to determine if, and then under what conditions, the revelant feature is preserved. In this paper we obtain conditions under which partial balance in a component is preserved and these often amount to the requirement that there exists a quantity which is unaffected by the internal workings of the component in question. We give particular attention to the components of a stratified clustering process as these most often suffer from loss of partial balance.