Past Seminars

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Approximating an epidemic process by a branching process
A. D. Barbour
University of Zuerich
Start time: 1:15 pm
Date: Friday 29 August 2008
Location: Russell Love Theatre, Richard Berry Bldg

The growth of an epidemic process is often effectively approximated in its early stages by that of an associated branching process, an observation first formalized by Whittle (1955). The same ideas frequently form the basis for describing the local structure of random graphs, including those exhibiting `small world' or power law behaviour. In this talk, we discuss ways of formalizing the approximation, by using stochastic coupling methods.

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A (Gentle) Introduction to Regular Variation and Applications
Daniel Tokarev
MASCOS, University of Melbourne
Start time: 3:15 pm
Date: Wednesday 20 August 2008
Location: Chemical & Biomolecular Engineering Theatre, Room G20, Chemical

Regular Variation theory, introduced by J. Karamata, is concerned with relating asymptotic behaviour of (real) functions and operators, such as Laplace transform, convolution and n-fold functional iterate. It has become an important tool in Probability, Analysis and other fields, and has greatly simplified and clarified much theory, such as in theorems of Abelian and Tauberian type and results on maxima and sums of independent and identically distributed random variables. This talk is intended to introduce the concept of regular variation and some fundamental results and applications in Probability.

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Diffusion Approximation for a Heavily Loaded Multi-User Wireless Communication System with Cooperation
Professor Ruth Williams
University of California, San Diego
Start time: 3:15 pm
Date: Friday 1 August 2008
Location: Theatre 1, ground floor of the ICT Building, 111 Barry St, The University of Melbourne

We consider a model for a cellular wireless communication system in which data is transmitted to multiple users over a common channel. For information theoretic reasons, the rate of transmission over this channel can be enhanced by cooperation. Assuming a fixed channel and that the average arrival rate of data for each user is known, we consider a simple scheduling policy which exploits cooperation and which has been shown to be throughput-optimal under Markovian assumptions. As a measure of performance under this policy, we establish a heavy traffic diffusion approximation for the workload process. This diffusion process is a semimartingale reflecting Brownian motion (SRBM) living in the positive orthant of N-dimensional space (where N is the number of users). Nominally, this SRBM has one direction of reflection associated with each of the 2^{N}-1 boundary faces. However, we show that in fact only those directions associated with the (N-1)-dimensional boundary faces matter in the heavy traffic limit. Based on joint work with Sumit Bhardwaj.

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The pivot algorithm for self-avoiding walks
Nathan Clisby
MASCOS, The University of Melbourne
Start time: 3:15 am
Date: Friday 9 May 2008
Location: Theatre 1, ground floor of the ICT Building, 111 Barry St, The University of Melbourne

The self-avoiding walk (SAW) is an important model in statistical mechanics, as it is a standard model in the study of critical phenomena (phase transitions) and in addition it accurately characterises the excluded volume effect of real polymers (long chain molecules).

The pivot algorithm is a technique with a long history, and is an extremely powerful tool in the study of SAWs. For a number of important quantities (e.g. critical exponents) it is by far the most efficient known method of calculation. It works via a Markov chain where successive SAWs are generated by attempting to 'pivot' part of the walk by rotating or reflecting the walk around a randomly selected pivot point. I will explain how to implement the pivot algorithm and why it is so effective, and then describe my current research: by incorporating additional geometric information while running the Markov chain it is possible to dramatically improve the speed of the algorithm.

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First Passage Densities and Boundary Crossing Probabilities for Diffusion Processes
Andrew Downes
MASCOS, The University of Melbourne
Start time: 3:15 pm
Date: Friday 18 April 2008
Location: Theatre 1, ICT Building, 111 Barry St, Carlton

We consider the boundary crossing problem for time-homogeneous diffusions and general curvilinear boundaries. Bounds are derived for the approximation error of the one-sided (upper) boundary crossing probability when replacing the original boundary by a different one.

In doing so we establish the existence of the first-passage time density and provide an upper bound for this function. In the case of processes with diffusion interval equal to R this is extended to a lower bound, as well as bounds for the first crossing time of a lower boundary. These results are illustrated by numerical examples.

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