Today, I presented several results of my master’s project. After the presentation, I had to defend my thesis titled **Interacting particles in wireless networks and Rydberg gasses**. This finished my graduation project, both for the mathematics and physics department.

Also, you can find a poster on the subject here, which we presented at the Nederlands Mathematisch Congress 2012, held at the Eindhoven University of Technology on the 12th of April, 2012.

## Abstract

Wireless networks and ultracold Rydberg gasses can both be mathematically modeled as interacting particle systems. Their particles (transmitters and atoms) can change between two configurations and exhibit a blockade effect, particles can prevent other particles from changing configuration. This leads to complicated spatial and temporal behavior in both cases. But even

though both systems have similarities, their models are different. Both models are discussed in detail and compared in this thesis, giving insight into both systems. The key difference is shown to be the process with which a particle is assumed to change its configuration. Atoms in Rydberg gasses make transitions because of lasers, well described using coherent dynamics. Transmitters in wireless networks (de)activate according to stochastic processes, better described using incoherent dynamics.

Besides a comparison, this thesis contains several new results. Model extensions are proposed that allow for application beyond the scope of wireless networks and Rydberg gasses. New distributed, random-access algorithms are proposed for distributing the capacity in (wireless) networking systems. The performance of these and existing distributed algorithms [JW10] is evaluated using simulation. Some algorithms turn out to converge extremely slowly, which would cause problems in practice. Ways to improve them are presented. Birth-and-death processes and differential equations are used to describe and approximate the number of active transmitters. One such differential equation is related to Einstein’s rate equations, emphasizing that wireless networks can be interpreted as two-level particle systems. Finally, this thesis gives several suggestions for possible future research. In particular, a new method to create patterns of regularly excited Rydberg atoms in ultracold gasses is suggested.

## Bibliography

- [JW10]
- L. Jiang and J. Walrand.
*A Distributed CSMA Algorithm for Throughput and Utility Maximization in Wireless Networks.*IEEE/ACM Transactions on Networking, 18:960-972, June 2010.

Head on over to My Resume for an overview of my experience and qualifications.