  We consider a Markov modulated fluid network with a finite number of
stations. We are interested in the tail asymptotics behavior of the stationary
distribution of its buffer content process. Using two different approaches, we
derive upper and lower bounds for the stationary tail decay rate in various
directions. Both approaches are based on a well-known time-evolution formula of
a Markov process, so-called Dynkin's formula, where a key ingredient is a
suitable choice of test functions. Those results show how multidimensional tail
asymptotics can be studied for the more than two-dimensional case, which is
known as a hard problem.
