  Markov modulation is versatile in generalization for making a simple
stochastic model which is often analytically tractable to be more flexible in
application. In this spirit, we modulate a two dimensional reflecting skip-free
random walk in such a way that its state transitions in the boundary faces and
interior of a nonnegative integer quadrant are controlled by Markov chains.
This Markov modulated model is referred to as a 2d-QBD process according to
Ozaw (2013). We are interested in the tail asymptotics of its stationary
distribution, which has been well studied when there is no Markov modulation.
  Ozawa studied this tail asymptotics problem, but his answer is not
analytically tractable. We think this is because Markov modulation is so free
to change a model even if the state space for Markov modulation is finite.
Thus, some structure, say, extra conditions, would be needed to make the Markov
modulation analytically tractable while minimizing its limitation in
application.
  The aim of this paper is to investigate such structure for the tail
asymptotic problem. For this, we study the existence of a right subinvariant
positive vector, called a superharmonic vector, of a nonnegative matrix with
QBD block structure, where each block matrix is finite dimensional. We
characterize this existence under a certain extra assumption. We apply this
characterization to the 2d-QBD process, and derive the tail decay rates of its
marginal stationary distribution in an arbitrary direction. This solves the
tail decay rate problem for a two node generalized Jackson network, which has
been open for many years.
