  This paper addresses the analysis of the queue-length process of
single-server queues under overdispersion, i.e., queues fed by an arrival
process for which the variance of the number of arrivals in a given time window
exceeds the corresponding mean. Several variants are considered, using concepts
as mixing and Markov modulation, resulting in different models with either
endogenously triggered or exogenously triggered random environments. Only in
special cases explicit expressions can be obtained, e.g. when the random
arrival and/or service rate can attain just finitely many values. While for
more general model variants exact analysis is challenging, one ${\it can}$
derive limit theorems in the heavy-traffic regime. In some of our derivations
we rely on evaluating the relevant Laplace transform in the heavy-traffic
scaling using Taylor expansions, whereas other results are obtained by applying
the continuous mapping theorem.
