Heavy-traffic limit theorems for queueing networks justify that a network in heavy traffic can be approximated by a reflected Brownian motion. We present other types of Markovian approximations for networks with fast (or slow) inputs and services.

The underlying idea is related to the central-limit phenomenon that limit theorems with normal or BM limits typically have analogues with Poisson or infinitely divisible limits. For instance, if $S_n$ is a binomial random variable with parameters $n$ and $p$, then $S_n$ is approximately normal when $n$ is large and $p$ is fixed. Analogously, $S_n$ is approximately Poisson when $n$ is large and $p$ is very small.

Another example is Donsker's functional central limit theorem for sums of random variables. There are analogous functional central limit theorems for sums of sparse point processes that converge to Poisson or infinitely divisible point processes.

I will review a few of these results, and then describe general non-Markovian jump processes that converge to Markov processes. Applications are approximations for fast (or slow) spatial queueing systems and queueing networks.