On this page I give you an overview of the type of research I do. If you are looking for my scientific publications, I recommend you visit the page that is devoted to that.

# Overview of my research interests

## Research keywords

Large-Scale Complex Systems; Network Science; Interacting Particle Systems; Rydberg atoms; Clustering / Community Detection

## Typical models I work with

Markov chains; Random graphs; Exploration processes; Random matrices; Stochastic Systems

## Mathematical techniques

Asymptotic analysis / Asymptotic expansions; Fluid limits / Diffusion limits; Optimization / Stochastic optimization; Concentration inequalities; Mixing times; Spectral analyses; Matrix decompositions; Stochastic differential equations; Markov decision processes; Martingales; Variational calculus

## My multidisciplinary background

I tend to work on multidisciplinary projects. My work so far has required an in-depth understanding of the mathematical fields probability and statistics and control theory and optimization, and the physics fields atomic, molecular physics and statistical physics (condensed matter). I hold Master’s degrees in mathematics and physics, and a PhD in mathematics.

## Developing research direction

The research direction I am developing primarily focuses on relating microscopic behaviors to macroscopic behaviors in large, complex stochastic systems. It focuses secondarily on creating efficient and optimal controls that overcome issues associated with the systems’ complexities. The mathematical analyses rely on the application of cutting-edge techniques for dealing with martingales, fluid / diffusion limits, stochastic differential equations and asymptotic behavior.

# My master’s and PhD thesis

My theses deal with large-scale complex systems ranging from communication networks to physical interaction processes, which are all influenced by randomness. In spite of their uncertain behavior and potentially erratic performance, modern society has grown increasingly dependent on such systems. Their inherently complex designs, the driving stochastic processes, and the limited availability of resources all pose great societal and scientific challenges.

The goal of my research was therefore to develop analysis techniques and optimization procedures that are broadly applicable to large-scale complex systems. The focus was on probabilistic models of interacting particle systems, stochastic networks, and service systems. These are all large scale and display fascinating complex behavior. The work was multidisciplinary, and can roughly be divided into three topics.

## Control and optimization of large-scale stochastic networks

I have developed optimization algorithms that are applicable to the whole class of product-form Markov processes. These algorithms can be implemented in an online fashion, that is, in such a way that the individual components of the network make autonomous decisions that ultimately lead to globally optimal network behavior. The algorithm can for instance be used to balance a network of queues, i.e., achieving equal average queue lengths, and does so by solving an inversion problem in an online fashion. Essentially, every node adapts its service rate individually based on the online observation of its own average queue-length. The individual nodes do not need global network information (like the network structure), and even though all nodes influence each other, the algorithm guarantees that the whole network achieves their mutual goal of a balanced operation.

## Ultracold Rydberg gases and quantum engineering

I have studied Rydberg gases: an interacting-particle system that consists of atoms which exhibit strong mutual blockade effects. These gases are studied for their applications in quantum computing and condensed matter physics. I have found that in certain regimes, the complex interactions of these particles can be described using the stochastic processes that also model the behavior of transmitters in wireless networks. This allowed me to identify interesting connections between research fields in physics and mathematics, and to transfer techniques and insights from applied probability to the realm of Rydberg gases. For example, I showed how optimization algorithms for large-scale stochastic networks (described above) can be used to actively engineer the atomic system, and how specially constructed random graphs can give theoretical descriptions of statistical properties of the Rydberg gas.

## Performance analysis and revenue maximization of critically loaded service systems

I have studied large-scale Markovian many-server systems that operate in the Quality-and-Efficiency Driven (QED) regime. These systems dwarf the usual trade-off between high system utilization and short waiting times. In order to achieve these dual goals, the system is scaled so as to approach full utilization, while the number of servers grows simultaneously large, rendering crucial Economies-of-Scale. These QED scaling laws provide a powerful framework for system dimensioning, and I have extended this framework by incorporating scalable admission control schemes and general revenue functions. Additionally, I have identified exactly which nontrivial threshold control policies are optimal in the QED regime for a broad class of revenue functions. This yielded new insights into the relation between the optimal control and revenue structure.

### Bullet-wise summary

To summarize, my theses have:

- developed models for wireless networks that combine high-dimensional Markov processes with techniques from interacting particle systems;
- identified relations between stochastic wireless network models and ultracold Rydberg gases, promoting knowledge exchange between different research fields;
- obtained theoretical descriptions of properties of Rydberg gases by developing stochastic processes that mimic the gas and live on random graphs;
- created optimization algorithms applicable to the whole class of product-form Markov processes that allow for distributed implementation;
- introduced scalable control schemes to queues operating in QED regimes, improving our understanding of certain fundamental queueing systems;
- designed a scalable revenue framework for the QED regime, and established fundamental properties of optimal control schemes that maximize revenue;
- calculated the effects of incorporating higher-order terms into the approximating optimization problems encountered in the QED branch of queueing literature.