  We consider an acyclic network of single-server queues with heterogeneous
processing rates. It is assumed that each queue is fed by the superposition of
a large number of i.i.d. Gaussian processes with stationary increments and
positive drifts, which can be correlated across different queues. The flow of
work departing from each server is split deterministically and routed to its
neighbors according to a fixed routing matrix, with a fraction of it leaving
the network altogether.
  We study the exponential decay rate of the probability that the steady-state
queue length at any given node in the network is above any fixed threshold,
also referred to as the "overflow probability". In particular, we first
leverage Schilder's sample-path large deviations theorem to obtain a general
lower bound for the limit of this exponential decay rate, as the number of
Gaussian processes goes to infinity. Then, we show that this lower bound is
tight under additional technical conditions. Finally, we show that if the input
processes to the different queues are non-negatively correlated, non
short-range dependent fractional Brownian motions, and if the processing rates
are large enough, then the asymptotic exponential decay rates of the queues
coincide with the ones of isolated queues with appropriate Gaussian inputs.
