Stochastic Optimization of Large-Scale Complex Systems
I defended my thesis Stochastic Optimization of Large-Scale Complex Systems on January 28th, 2016 at the Eindhoven University of Technology. The defense was a great experience, and I have thoroughly enjoyed my time as a PhD student. The colleagues at the Stochastics Operations Research group and EURandom were great, and I'm also especially grateful to my supervisors Johan van Leeuwaarden and Sem Borst for all the unique opportunities.
Starting February 1st, I will start as a post-doctoral researcher at the KTH Royal Institute of Technology in Stockholm, Sweden.
This thesis develops analysis techniques and optimization procedures that are broadly applicable to large-scale complex systems. The focus is on probabilistic models of interacting particle systems, stochastic networks, and service systems, which are all large systems and display fascinatingly complex behavior. In practice, these systems obey simple local rules leading to Markov processes that are amenable to analyses that shed light on the interplay between the local rules and the global system behavior. Chapter 1 provides an overview of this thesis's content, and illustrates our approaches of analysis and optimization.
Chapter 2 deals with the topic Control and optimization of large-scale stochastic networks. There, we develop optimization algorithms that are applicable to the whole class of product-form Markov processes. These algorithms can be implemented in such a way that individual components of stochastic networks make autonomous decisions that ultimately lead to globally optimal network behavior, and can for instance be used to balance a network of queues, i.e. to achieve equal average queue lengths. The algorithm does so by solving an inversion problem in an online fashion: every queue individually adapts its service rate based on online observations of its own average queue length. The individual queues do not need global network information (like the network structure), and even though all queues influence each other, the algorithm guarantees that the whole network achieves their common goal of a balanced operation.
Throughout Chapters 3, 4, and 5, we discuss the topic Ultracold Rydberg gases and quantum engineering. Rydberg gases consist of atoms that exhibit strong mutual blockade effects, and this gives rise to an interacting particle system with intriguing complex interactions. These particle systems are investigated in laboratory environments because of their application in quantum computing and condensed matter physics. Our research finds 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 allows us to identify interesting connections between the fields of physics and mathematics, and to transfer techniques and insights from applied probability to the realm of Rydberg gases. For example, the optimization algorithms for largescale stochastic networks (described above) can be used to actively engineer the atomic system, and by constructing special random graphs we can give theoretical descriptions of statistical properties of the Rydberg gas.
Chapters 6, 7, and 8 cover the final topic Performance analysis and revenue maximization of critically loaded service systems. There, we consider largescale Markovian many-server systems that operate in the so-called Quality-andEfficiency-Driven (QED) regime and dwarf the usual trade-off between high system utilization and short waiting times. In order to achieve these dual goals, these systems are scaled so as to approach full utilization, while the number of servers grows simultaneously large, rendering crucial Economies-of-Scale. Our research extends the applicability of the QED regime by incorporating scalable admission control schemes and general revenue functions. This also allows us to identify for a broad class of revenue functions exactly which nontrivial threshold control policies are optimal in the QED regime, yielding insight into the relation between the optimal control and revenue structure. By studying the system's precise asymptotic behavior when nearing the QED regime, we are able to also analyze the effectiveness of the QED regime as a framework for system dimensioning.