Description of the three research areas

Performance analysis of information and communication systems (PNA2.1)

Information and communication systems continue to expand rapidly in terms of traffic volume, the number of users, as well as the range of applications. The use of both the Internet and wireless services has experienced an explosive growth. Network operators and service providers anticipate further expansion, fueled by the emergence of all-optical networking as well as the convergence of wireless and Internet access, along with a fundamental trend towards service integration. Future information and communication systems are expected to accommodate a variety of new applications with a diverse range of Quality-of-Service (QoS) requirements.
These observations have raised the need for the development and analysis of quantitative stochastic models to predict and control the QoS of information and communication systems, including wired and wireless networks and large-scale distributed systems. Our main focus is on the development and analysis of queueing theoretic models and methods to predict and control the performance experienced by the user. In addition, we focus on network economics, addressing problems related to pricing and cost allocation in communication systems.

Some publications:

1. S.C. Borst, R. Nunez-Queija and A.P. Zwart (2006). Sojourn time asymptotics in processor-sharing queues. Queueing Systems 53, 31-51.
2. I.M. Verloop, S.C. Borst, R. Nunez-Queija (2006). Delay-based scheduling in bandwidth-sharing networks. In: Proc. ACM Sigmetrics / Performance 2006 Conference, Saint-Malo, France, June 28-30, 365-366.
3. R.D. van der Mei and E.M.M. Winands (2007). Polling models with renewal arrivals: a new method to derive heavy-traffic asymptotics. Performance Evaluation 64, 1029-1040.
4. B.M.M. Gijsen, R.D. van der Mei, P. Engelberts, J.L. van den Berg and K.M.C. van Wingerden (2006). Sojourn-time approximations in queueing networks with feedback. Performance Evaluation 63, 743-758.
5. P.M.D. Lieshout, M.R.H. Mandjes, S.C. Borst (2006). GPS scheduling: selection of optimal weights and comparison with strict priorities. In: Proc. ACM Sigmetrics / Performance 2006 Conerence., Saint-Malo, France, June 28--30, 75—86 (received best student paper award).
    

Probability and spatial stochastics (PNA2.2)

During the last ten years the study of random processes in a spatial context has rapidly intensified. On one hand, there is an increasing motivation to understand such processes, for instance in chemistry and physics (e.g., models of magnetization, polymerization), earth and life sciences (epidemics, nerve systems, forest fires) and engineering (wireless communication networks). On the other hand, these processes give rise to very interesting mathematical problems requiring a rich variety of ideas and techniques. For instance, one of the main breakthroughs in this field, the introduction, development and applications of Stochastic Loewner Evolutions, involves a beautiful mixture of conformal mapping theory, stochastic analysis and interesting combinatorial-geometric arguments.
Much work by PNA2 members on random spatial processes concentrates on various models with a percolation-like flavor; in particular forest-fire models, models of epidemics, certain models of the spread of fluid through a random medium (invasion percolation), and systems of coalescing, randomly moving particles. The group is also active in fundamental research on spectral analysis of random fields, such as fractional Brownian motions, that are (e.g.) used to model traffic phenomena in communication networks with long-range dependence. An important method in the description of such processes is that of Krein's spectral theory of vibrating strings.

Some publications:

1. J. van den Berg and B.N.B. de Lima. Linear lower bounds for $\delta_c(p)$ for a class of 2D self-destructive percolation models. Random Structures and Algorithms 34, 520-526, Wiley (2009). 
2. J. van den Berg and H. Kesten (2002). Randomly coalescing random walks in dimension d ≥ 3. In and Out of Equilibrium (V. Sidoravicius, ed.), Birkhäuser, 1-45.
3. J. van den Berg and R. Brouwer (2006). Self-organized forest-fires near the critical time. Communications in Mathematical Physics 67, 265-277.
4. K. Dzhaparidze, H. van Zanten and P. Zareba (2006). Representations of isotropic Gaussian random fields with homogeneous increments. J. Appl. Math. Stoch. Anal., Art. ID 72731.
5. K. Dzhaparidze and H. van Zanten (2005). Krein's spectral theory and the Paley-Wiener expansion for fractional Brownian motion. Ann. Probab. 33, no. 2, 620-644.

Stochastic geometry (PNA2.3)

Stochastic geometry is concerned with random geometric structures, ranging from simple points or line segments to arbitrary closed sets. Although it has roots in geometric probability and integral geometry, the modern theory of random sets was developed in the seventies, independently by David Kendall in Cambridge and Georges Matheron in Fontainebleau with important contributions from the German school around Professors Mecke and Stoyan. Stochastic geometry techniques can be applied in a wide range of fields for instance image analysis, telecommunication networks, forestry, and environmental research.

Some publications:

1. M.N. van Lieshout (2000). Markov Point Processes and their Applications (London/Singapore: Imperial College Press/World Scientific Publishing).
2. T. Schreiber and M.N.M. van Lieshout. Disagreement loop and path creation/annihilation algoritms for planar Markov fields with applications to image segmentation. Scandinavian Journal of Statistics, to appear.
3. M.N.M. van Lieshout. Applications of stochastic geometry in image analysis. Stochastic geometry: Highlights, interactions and new perspectives, W.S. Kendall and I.S. Molchanov (Eds.), 427-450. Oxford: Clarendon Press, 2009.
4. M.N.M. van Lieshout. Depth map calculation for a variable number of moving objects
   using Markov sequential object processes. IEEE Transactions on Pattern Analysis and Machine Intelligence 30, 1308-1312, 2008.
5. M.N. van Lieshout (2006). Maximum likelihood estimation for random sequential adsorption. Advances in Applied Probability (SGSA) 38, 889-898.
6. R. Kluszczynski, M.N. van Lieshout and T. Schreiber (2006). Image segmentation by polygonal Markov fields. To appear in Annals of the Institute of Statistical Mathematics. Published online, August 19, 2006.