  We study infinite server queues driven by Cox processes in a fast oscillatory
random environment. While exact performance analysis is difficult, we 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, we identity two
parameter regimes, termed subcritical and supercritical indicating the relative
dominance between the two underlying stochasticities driving our system: the
randomness in the arrival intensity and that in the serivce times. We show that
while quenched FCLTs can only be established in the subcritical regime,
annealed FCLTs can be proved in both cases. Furthermore, the limiting
diffusions in the annealed FCLTs display qualitatively different diffusivity
properties in the two regimes, even though the stochastic primitives are
identical. In particular, when the service time distribution is heavy-tailed,
the diffusion is sub- and super-diffusive in the sub- and super-critical cases.
The results illustrate intricate interactions between the underlying driving
forces of our system.
