  Quasi-birth-and-death (QBD) processes with infinite ``phase spaces'' can
exhibit unusual and interesting behavior. One of the simplest examples of such
a process is the two-node tandem Jackson network, with the ``phase'' giving the
state of the first queue and the ``level'' giving the state of the second
queue. In this paper, we undertake an extensive analysis of the properties of
this QBD. In particular, we investigate the spectral properties of Neuts's
R-matrix and show that the decay rate of the stationary distribution of the
``level'' process is not always equal to the convergence norm of R. In fact, we
show that we can obtain any decay rate from a certain range by controlling only
the transition structure at level zero, which is independent of R. We also
consider the sequence of tandem queues that is constructed by restricting the
waiting room of the first queue to some finite capacity, and then allowing this
capacity to increase to infinity. We show that the decay rates for the finite
truncations converge to a value, which is not necessarily the decay rate in the
infinite waiting room case. Finally, we show that the probability that the
process hits level n before level 0 given that it starts in level 1 decays at a
rate which is not necessarily the same as the decay rate for the stationary
distribution.

Probability