  Arrival processes to service systems often display fluctuations that are
larger than anticipated under the Poisson assumption, a phenomenon that is
referred to as overdispersion. Motivated by this, we analyze a class of
discrete stochastic models for which we derive heavy-traffic approximations
that are scalable in the system size. Subsequently, we show how this leads to
novel capacity sizing rules that acknowledge the presence of overdispersion.
This, in turn, leads to robust approximations for performance characteristics
of systems that are of moderate size and/or may not operate in heavy traffic.
