# various, see below

*by various, see below*

*Institution:*

*Date: Thu 7th February 2013*

*Time: 10:00 AM*

*Location: Old Geology 2*

*Abstract*: 10.00 – 11.00am

Dr Giang NGUYEN

Title: On weak convergence to Markov-modulated Brownian motion

Abstract: : Markov-modulated Brownian motions, also known as regime-switching Brownian motions, have gained much interests in the past two decades, due to their mathematically interesting nature and their applications in finance. In this talk, we show that each Markov-modulated Brownian motion arises as the limit of a sequence of Markov-modulated linear fluid processes. Furthermore, we prove that the stationary distribution of a reflected Markov-modulated Brownian motion is the limit of the well-analyzed stationary distributions of linear fluid processes. Key matrices in the limiting stationary distribution are the solutions of a new quadratic equation. We describe how to solve this equation efficiently. (Joint work with G. Latouche.)

11.00 – 12noon

Dr Nathan ROSS

Title: Vertex degrees in preferential attachment random graphs

Abstract: Random graph models have recently been used to understand the structure and dynamics of real world networks, for example the "web of human sexual contacts" and the power grid. One family of such models is called preferential attachment random graphs which evolve in time by sequentially adding vertices and edges in a random way so that connections to vertices having high degree are favored. These models were popularized by Barabasi and Albert in 1999 to explain the so-called power law behavior observed in degree distributions of some real world networks such as the graph derived from the world wide web by considering webpages as vertices and hyperlinks between them as edges. In this talk we discuss recent results for preferential attachment graphs which provide rates of convergence for both the distribution of the degree of a fixed vertex (properly scaled) to its distributional limit and the distribution of the degree of a randomly chosen vertex to an appropriate power law. Our results are derived by identifying the distributional limits as unique fixed points of appropriate distributional transformations and then showing that these transformations have a small effect on the respective distributions of interest. This method of proof is useful for obtaining distributional limit and approximation results in other applications and it automatically yields various descriptions and properties of the limiting distributions under consideration. Joint work with Erol Pekoz and Adrian Roellin.

2.00 – 3.00pm

Dr Sophie HAUTPHENNE

Title: "Some problems on the extinction probability of multitype branching processes"

Abstract: Branching processes with finitely many types have been studied extensively; the extinction probability is known to be the minimal nonnegative solution of a finite system of equations, and a simple extinction criterion is given by the spectral radius of the mean progeny matrix.

The analysis of multitype branching processes becomes more complicated when the number of types is allowed to be infinite, or when the branching process undergoes catastrophes or evolves in a random environment. This talk addresses our contribution to the computation of the extinction probability vector, and to the characterization of extinction criteria, for such specific processes.

3.00 – 4.00pm

Dr Leonardo ROJAS-NANDAYAPA

Title: Measuring the improbable: Rare-event simulation for dependent heavy-tailed random variables.

Abstract: Certain real-world events are improbable, but when they occur, can have a profound impact: natural disasters, major accidents, financial crises and system failures are obvious examples. These rare events are often determined by elements which exhibit extreme random behaviour and are often interrelated in complex ways, so it is natural that models involving dependent heavy-tailed random variables are among the most appropriate to analyse them. However, calculating the associated probabilities can turn out to be a very difficult task.

In this talk I will show a collection of efficient Monte Carlo estimators for rare events associated with sums and maxima of dependent heavy-tailed random variables.

First I will show an estimator for the tail probability of the sum of correlated lognormals which is proven to be asymptotically optimal. The key idea here is that by considering a representation in polar coordinates we can condition on the spherical component and integrate the radial distribution. A natural generalization is to consider sums of log-elliptical distributions, and for such case we provide a very mild sufficient condition on the radial distribution for attaining asymptotic optimality.

Next, I will show a second estimator that can be applied under an arbitrary dependence structure provided that one can simulate conditionally on a single random variable taking a very large value. In this case asymptotic optimality is proven for a very large class of heavy-tailed distributions which includes the lognormal and regularly varying distributions.

Finally, I will show an estimator for the tail probability of the maximum of general dependent random variables which is constructed using very simple arguments (the inclusion-exclusion principle and the binomial theorem). I will show that this estimator can produce significant reductions in the asymptotic variance.

This talk is based on some joint works with Søren Asmussen, José Blanchet and Sandeep Juneja.

4.00 – 5.00pm

Dr Stella KAPODISTRIA

Title: Analytic approaches for multi-dimensional queueing systems

Abstract: Spurred by communication systems, computer networks, transportation systems, and manufacturing models, modern Queueing Theory is the discipline devoted to the quantitative and qualitative performance analysis of technical systems. The theory evolved from the study of switching and telephone systems, to the analysis, optimization and decision making in business, commerce, industry, healthcare, public service, engineering, traffic management and many more. Since the 1950's mathematicians and engineers combining techniques from combinatorics, probability theory and complex function analysis have set the foundations of Queueing Theory and have successfully tackled case specific problems. As technology advances, the systems under consideration become more and more complicated bringing about the need of developing new approaches and revising the existing techniques.

The aim of this presentation is to familiarize the audience with certain tools of Queueing Theory (including its latest achievements) and the results of their application. In particular, I will focus on the performance analysis of two-dimensional queueing systems that share resemblance with random walks restricted in the quarter plane. I will illustrate the advantages and limitations of the classical and the alternative methods, whilst solving a functional equation for the generating function Q(x, y) of the joint stationary queue length distribution. I will talk about: (i) Classification of two-dimensional queueing systems and developing exact solution methods, (ii) Direct and indirect techniques and connections with enumerative combinatorics, and (iii) Asymptotic expansions. Moreover, I will demonstrate some key ideas which show promise of successful employment and structural insight for higher-dimensional spaces. I will conclude my talk with possible extensions and my future research plan.