  We develop diffusion approximations for parallel-queueing systems with the
randomized longest-queue-first scheduling algorithm by establishing new
mean-field limit theorems as the number of buffers $n\to\infty$. We achieve
this by allowing the number of sampled buffers $d=d(n)$ to depend on the number
of buffers $n$, which yields an asymptotic `decoupling' of the queue length
processes.
  We show through simulation experiments that the resulting approximation is
accurate even for moderate values of $n$ and $d(n)$. To our knowledge, we are
the first to derive diffusion approximations for a queueing system in the
large-buffer mean-field regime. Another noteworthy feature of our scaling idea
is that the randomized longest-queue-first algorithm emulates the
longest-queue-first algorithm, yet is computationally more attractive. The
analysis of the system performance as a function of $d(n)$ is facilitated by
the multi-scale nature in our limit theorems: the various processes we study
have different space scalings. This allows us to show the trade-off between
performance and complexity of the randomized longest-queue-first scheduling
algorithm.
