Scaling limits for modulated infinite-server queues and related stochastic processes

In this dissertation, we develop mathematical models and tools that may help us to gain insight into the behavior of queueing systems with many servers and stochastic processes related to such systems. The distinguishing feature of our models is the presence of an independently evolving random environment to which the queue reacts. More precisely, this means that the parameters of the queue depend on another stochastic process, which is called the background process. This phenomenon is also called modulation and it is used to model scenarios in which there is more variability than captured by traditional queueing models. In the first part of this dissertation, we study the number of customers in a modulated infinite-server queue at a fixed point in time. We derive scaling limits in three different regimes, namely the central-limit regime, the moderate-deviations regime, and the large-deviations regime. In the second part of this dissertation, we investigate the sample-path behavior of certain modulated stochastic processes. We assume that the background process is a Markov chain with a finite state space. In particular, we focus on networks of infinite-server queues, on Erlang models, and on Ornstein-Uhlenbeck processes under Markov modulation. We prove that these processes converge weakly on sample-path level and show that the corresponding limit processes are solutions to certain stochastic differential equations