Stochastics - Theoretical and Applied Research



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Project: Bayesian adaptation using Gaussian priors; B. Szabo, PhD

In recent years there has been a huge increase in the use of adaptive Bayesian methods in high-dimensional or nonparametrical statistical problems. The two most popular procedures are the empirical Bayes and the hierarchical full Bayes methods. Our goal is to contribute in the fundamental understanding of the asymptotic behavior of these methods. We focused mainly on the empirical Bayes method, but the derived results can be applied to the hierarchical Bayes method also. We were examining the Gaussian sequence model and showed that by bad choice of the prior distribution, the empirical Bayes method behaves sub-optimally, but with careful choice we can get optimal, adaptive results.

Project: Nematic phase transitions for hard-rod lattice models; S. Taati, PD

Nematic liquid crystals flow like liquids but have anisotropic optical properties as in most crystals.  They are widely used in the liquid crystal displays (LCD).  The molecules of a liquid crystal material resemble long and thin rigid rods.  In the nematic phase, these rods tend to align, while their relative positions remain disordered.  In the lattice model, the molecules are abstracted as n-by-1 rectangles that are placed on the two-dimensional lattice, either horizontally or vertically, without overlapping one another.  In this project, we aim to prove the occurrence of a nematic phase in this model within the framework of statistical mechanics

Project: Random polymers; M. Heydenreich, Assistent to Professor

The research of Markus Heydenreich is centered around critical phenomena of random spatial models. This includes models for porous media and (abstract) polymers. In particular, the self-similar, fractal structures appearing at the critical point are investigated. Special emphasis is on the study of random walk on the incipient infinite cluster, a primary example of random walk on a fractal like structure. This example exhibits anomalous diffusion, which means that the random walk on the incipient infinite cluster (like on many natural fractals) is significantly slower than on Euclidean space.

Project: Multilayered queueing systems; J.P. Dorsman, PhD

The current research project is placed within the domain of shared resource possession in stochastic networks. Recent applications in engineering, business and the public sector led to systems with much more complex, often layered, service architectures, where entities that act as server at one layer can act as customer at a higher layer. The performance analysis and design of these new applications require the development of sophisticated new stochastic network models.  So far, such layered systems have only been analysed approximately and through simulations, and were limited to computer-science problems. However, the mathematical analysis of such systems is extremely challenging and exciting.
In this project, contrary to previous approximation studies, we aim to develop new mathematical tools that will allow for rigorous mathematical insights into the quantitative and qualitative behaviour of layered queueing systems. Up to now, we have studied a class of queueing systems consisting of two layers, where the interaction between the two layers leads to statistical dependence between the queue lengths of the queues within the upper layer. We have derived accurate approximations for the marginal queue length distributions of these queues by modelling the interaction between the layers explicitly. In addition, through the use of the Power-Series algorithm, we have studied the light-traffic behaviour of the joint queue length for these queues. When the heavy-traffic behaviour can be identified as well, a combination of the two will lead to valuable insights in the joint queue length process for the queues in the upper layer.

Project: Modeling brain networks dynamics; B. Rós, PhD

My research is in the area of stochastic modeling and statistical analysis of brain network dynamics. In the project we consider the brain as a system that consists of a set of interconnected specialized regions that give rise to behavior through their mutual interactions. Changes in behavior over time or differences in behavior under different conditions can be due to specific changes or differences in this network. In this project we use mathematical modeling and statistical analysis to identify which regions in the brain cooperate in different situations and under different conditions and how they do this. Our approach is to build a network model consisting of appropriate brain regions as nodes with the edges defined by the relationships between them. This model will be inferred from simultaneously collected spatio-temporal fMRI and EEG data.

Project: Stochastics models for emergency care (NOW/Defense PhD); S. Ding, PhD

My research falls into the area of multi-skill call centers. The objective is to improve the performance of a multi-skill call centers. To this end, a good and accurate forecasting method is of interests. This requires investigation of time-series analysis. Forecast of a call center plays a crucial role in the performance of a call center. I am very interested in evaluating and comparing different forecast methods, and develop a suitable method for our data. When we obtain the call forecast, the next step is to make stuffing for call centers. The number of agents per skill group is the key point in reaching the service level of call centers. In order to derive this number with cost minimization, simulation and approximation will be useful. Besides the theoretical interests of call center problem, it also has a practical usage in the industry.

Project: Network Congestion; N.S. Walton, Ass.Prof

My research is principally concerns congestion occurring in networks. Many congested systems naturally converge to a certain mode of behavior, whether that be equilibrium or non-equilibrium behavior. In a number of circumstances, such behavior can be expressed as the solution to an optimization problem. Although congestion may be unavoidable in a well utilized network, by understanding the nature of such optimization descriptions, we can hope to better understand how to redesign systems to optimize what we want them to optimize.
The tools used in this analysis include to probability, queueing theory, optimization and game theory, and application areas include Internet, telephony and road traffic.

Project: Sandpile models on random graphs; Mrs. W. Ruszel, Ass.Prof.

We study two types of models. The first model is a sandpile model on the complete graph with a uniformly bounded critical height h. The simplest example occurs when this critical height equals 2. Here, the dynamics consist of two types. At each time, if the configuration is stable, then there is a new “particle” dropped into the system at a uniform location, otherwise the unstable site picks h sites at random and distributes the particles to those sites. One of the sites is a sink, where the particle upon entering is lost. We want to study the stationary state of the joint distribution of the number of sites that need to be toppled as well as the number of topplings still to be performed at each time step.  Moreover, we aim to study the tail behavior of the avalanche size, which is defined as the time the system takes to return to a stable configuration after adding a new particle.

The second model we consider is a sandpile model on a random binary tree with branching parameter p. The dynamics is a composition of the sandpile dynamics and the topology of the random graph itself, leading to double randomness. We want to investigate the avalanche sizes in the stationary state depending on p. For p=1 previous results show power-law behavior for the avalanche sizes. We want to investigate weather this happens already for some p smaller than 1.