  We consider a two-node fluid network with batch arrivals of random size
having a heavy-tailed distribution. We are interested in the tail asymptotics
for the stationary distribution of a two-dimensional queue-length process. The
tail asymptotics have been well studied for two-dimensional reflecting
processes where jumps have either a bounded or an unbounded light-tailed
distribution. However, presence of heavy tails totally changes the asymptotics.
Here we focus on the case of strong stability where both nodes release fluid
with sufficiently high speeds to minimise their mutual influence. We show that,
like in the one-dimensional case, big jumps provide the main cause for queues
to become large, but now they may have multi-dimensional features. For deriving
these results, we develop an analytic approach that differs from the
traditional tail asymptotic studies, and obtain various weak tail equivalences.
Then, in the case of one-dimensional subexponential jump-size distributions, we
find the exact asymptotics based on the sample-path arguments.
