Phil Pollett's Research Pages
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Are you interested in doing an honours project in Stochastic Processes?
I am prepared to supervise one or more honours projects in any area of mathematics or its applications that is cognate to my own research interests. I am also willing to offer students in other disciplines access to my expertise in probability and random processes through joint supervision with a supervisor in another discipline (this is perhaps the only effective means for students enrolled in other departments to gain access to my expertise).
Some specific projects are listed below. Please appreciate that the projects listed below are quite specific, and are described in a fair amount of detail. Of course, if you are interested in the general area of one or other of these projects, then I will almost certainly be able to suggest other projects which you can work on in that area. Projects are listed in roughly increasing order "mathematical technicality": the "real world" at one end, and the "abstract world" at the other (see if you can tell which is which!)
Please contact me for further details.
Markovian models have been proposed as models for an array of biological systems, but their application has been limited, partly due to a lack of clear statistical procedures for model fitting. We will look at methods that address these statistical limitations. We first study a general likelihood based approach where the process is observed at successive, but not necessarily equally spaced, time points (for example, sets of abundancy data collected at various times). We then look at an approach which is simpler in terms of computational implementation, and which is suitable for parameter estimation in density-dependent population models, where the rates of transition are a function of the population density.
We develop and implement simulation methods for estimating (i) extinction probabilities and (ii) the expected time to extinction for a range of population models, being two key measures of population viability. We will exploit recent advances in simulation technology, which, in the present context, identify a related model that is easier to simulate, but which provide more efficient estimators.
We will examine models for populations that occupy several geographically separated regions of habitat (patches). Although the individual patches may become empty through "local" extinction, they may be recolonized through migration from other patches. There is considerable empirical evidence which suggests that a balance between migration and extinction is reached that enables population networks of this kind to persist for long periods. We will develop methods which account for the persistence of these populations and which provide an effective means of studying their long-term behaviour before extinction occurs. Our models will be adjusted to account for environmental effects on patch suitability.
There are many stochastic systems, arising in areas as diverse as wildlife management, chemical kinetics and reliability theory, which eventually "die out", yet appear to be stationary over any reasonable time scale. The notion of a quasi-stationary distribution has proved to be a potent tool in modelling this behaviour. Our aim is to establish workable analytical conditions for the existence of quasi-stationary distributions for Markovian models in terms of their transition rates, as well as develop and implement efficient computational procedures for evaluating them.
This project is concerned with the study of communications networks. In circuit-switched networks with random alternative routing, the system may fluctuate between a "low blocking state", where calls are accepted readily, and a "high blocking state", where the likelihood of a call being accepted can be quite low. The aim is to develop a method that enables one to assess the stability of the two states, in particular, one that allows us to obtain qualitative estimates of the time for which these states persist. It is of considerable practical importance to be able to assess the persistence of the high blocking state, for it can have serious implications for the performance of the network: in the high blocking state a situation develops where large numbers of calls use alternative routes, which demand greater link occupancy than do first-choice routes, and thus new calls are likely to be blocked frequently.
This project is concerned with the performance evaluation of communications networks. For the simplest networks there are explicit analytical formulae for the important measures of performance, but for networks that involve some level of dynamic control, exact analytical methods have achieved only limited success. Under several regimes, the Erlang Fixed Point (EFP) method provides a good approximation for several performance measures, but when these regimes are not operative the method can perform badly. In many cases this is because the key assumption of independent blocking does not hold. The aim is to develop methods for estimating the blocking probabilities that explicitly account for the dependencies between neighbouring links.
The aim of this project is to study a simple stochastic epidemic model, incorporating births into the class individuals susceptible to a given disease. We hope to estimate the mean duration of the epidemic. We will also consider the question of whether or not epidemic ultimately dies out. The limiting behaviour of the epidemic, conditional on non-extinction, will be studied using approximation methods, via two different approaches.
In this project we consider the problem of identifying regions of congestion in closed queueing networks with state-dependent service rates. A particular queue is called a bottleneck if the number of customers in that queue grows without bound as the total number of customers in the network becomes large. The aim is to derive methods for identifying potential bottlenecks, with a view to controlling congestion.
The aim is to develop a theory of quasi-stationary distributions for general level-dependent quasi-birth-and-death processes (QBDs). The idea is to extend earlier work that deals with the level-independent case. The present case allows for greater flexibility in modelling a range of biological phenomena, but evidently requires a more delicate mathematical analysis.
This project concerns evaluating the asymptotic probability that a graph, whose edges are put down at random, is acyclic in the limit as the number of vertices becomes large. Results such as these have found applications in the study of optimal algorithms for minimal perfect hashing.
The general aim is to study the equilibrium behaviour of Markovian models using diffusion approximations, and, specifically, to identify diffusion approximations for a variety processes based on the work of Andrew Barbour and Tom Kurtz. We aim to determined distributional approximations for the concentration of reactants in simple chemical reactions, which are accurate at any stage of the reaction, thus extending the results of Frank Dunstan and John Reynolds, who considered only the "equilibrium phase". We will also consider a problem from parasitology. Our aim will be to study a stochastic model, proposed by Phil Diamond, 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, we hope to 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 reached the conclusion that the efficiency is of order the square root of N.
This project is motivated by a conjecture of Kryscio and Lefèvre (1989) concerning stochastic ordering of the quasi-stationary distribution of the SIS logistic epidemic model. For a continuous-time Markov chain on the positive integers with the origin as an absorbing state, Ferrari, Kesten, Martinez and Picco (1995) studied the existence of quasi-stationary and limiting conditional distributions by characterizing quasi-stationary distributions as fixed points of a transformation F on the space of probability distributions on {1,2,...}. In the case of a birth-death process, one can write down the components of F(v) explicitly for any distribution v. The aim is to use this explicit representation to show that F preserves likelihood-ratio ordering between distributions, and thus settle the Kryscio and Lefèvre's conjecture.
The aim is to derive methods for evaluating the expected value and then the distribution of a path integral for a general Markov chain on a countable state space.
If X(t) is an absorbing birth-death process, then it can be shown that X^{T}(t), the process conditioned on non-absorption by time T, converges weakly to a time-homogeneous Markovian limit Y(t) as T becomes large. We aim to derived necessary and sufficient conditions for the process Y(t) to be non-explosive.
Our aim is to derive necessary and sufficient conditions for the existence of bounded or summable solutions to systems of linear equations associated with Markov chains, based on an extension of a famous result of Harry Reuter (1956), which provides a convenient means of checking various uniqueness criteria for birth-death processes. We aim to extend these uniqueness criteria, thus allowing chains with much more general transition structures to be accommodated. These might include upwardly skip-free chains, such as the birth, death and catastrophe process, and downwardly skip-free chains, such as the Markov branching process and its many generalizations.
The aim is to develop a theory of invariant measures for pure-jump Markov processes by exploiting the construction of certain dual processes. The first step is to identify an appropriate (and suitably general) topology for the state space, needed to effect these constructions. It is expected that the analysis of pure-jump Markov processes will be more delicate than for Markov chains on a countable state space. Several applications will be considered, including the equilibrium analysis of simple stress-release models for seismicity.
This project is of a highly technical nature and is concerned with the analusis of continuous-time Markov chains. In attempting to extend my analytical results on invariant measures to Q-processes (that is, processes with transition rates Q) other than the minimal process, I realized that m-invariant measures for a Q-process are m-subinvariant for Q, but may not be strictly m-invariant, and, moreover, that a m-subinvariant measure for Q may not be m-invariant for any Q-process. We start with a subinvariant measure for a stable, conservative, single exit q-matrix Q and the aim is to provide necessary and sufficient conditions for the existence, and then the uniqueness, of a Q-process for which m is invariant. The important special case concerning the existence of a unique, honest Q-process for which m is invariant, is an important special case. This may prove to be a significant advance in the theory of Markov chains, for it is hoped that the solution to this problem will shed some light on the so-called Modern Construction Problem, where Q is assumed to be an arbitrary q-matrix.
Details of my research interests, a list of my published work, and general information about my research can be found here.
Please e-mail me (pkp@maths.uq.edu.au) to make an appointment. Or, if you want to drop in, the best time to catch me is Thursday mornings. My office is Room 652 of the Priestley Building. Further contact details and a list of my office hours can be found here.
"It is remarkable that a science which began with the consideration of games of chance should have become the most important object of human knowledge."
Pierre Simon, Marquis de Laplace, Théorie Analytique des Probabilités, 1812.
If you have any comments on these pages,
feel free to e-mail
me: pkp@maths.uq.edu.au