  Donsker Theorem is perhaps the most famous invariance principle result for
Markov processes. It states that when properly normalized, a random walk
behaves asymptotically like a Brownian motion. This approach can be extended to
general Markov processes whose driving parameters are taken to a limit, which
can lead to insightful results in contexts like large distributed systems or
queueing networks. The purpose of this paper is to assess the rate of
convergence in these so-called diffusion approximations, in a queueing context.
To this end, we extend the functional Stein method introduced for the Brownian
approximation of Poisson processes, to two simple examples: the single-server
queue and the infinite-server queue. By doing so, we complete the recent
applications of Stein's method to queueing systems, with results concerning the
whole trajectory of the considered process, rather than its stationary
distribution.
