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# Analysis of Infinite-Server Queueing Models In A Random Environment

In this thesis we study infinite server queues driven by Cox processes in a random environment. More specifically, we consider a Cox/Gt/∞ infinite server queueing model with the arrival rate modeled as a highly fluctuating stochastic process, which arguably takes into account some small time scale variations often observed in practice. We prove a homogenization property for this system, which yields an approximation by an Mt/Gt/∞ queue with some effective parameters. That is, in the fast oscillatory context, we show how the Cox/Gt/∞ queueing system in a random environment can be approximated by a more classical Markovian system. We also establish diffusion approximations to the (re-scaled) number-in-system process by proving functional central limit theorems (FCLTs) using a stochastic homogenization framework. This framework permits the establishment of quenched and annealed limits in a unified manner. At the quantitative level, the so-called subcritical and supercritical regimes indicate the relative dominance between the two underlying stochasticities driving the system: the randomness in the arrival intensity and that in the service times. The results illustrate intricate interactions between the underlying driving forces of the system.