We wish to describe how a chosen node in a network of queues over-
loads. The overloaded node may also drive other nodes into overload, but the
remaining "super" stable nodes are only driven into a new steady state with
stochastically larger queues. We model this network of queues as a Markov
additive chain with a boundary. The customers at the "super" stable nodes are
described by a Markov chain, while the other nodes are described by an additive
chain. We use the existence of a harmonic function h for a Markov
additive chain provided by Ney and Nummelin and the asymptotic theory for
Markov additive processes to prove asymptotic results on the mean time for a
specified additive component to hit a high level l. We give the limiting
distribution of the "super" stable nodes at this hitting time. We also give the
steady-state distribution of the "super" stable nodes when the specified
component equals l. The emphasis here is on sharp asymptotics, not rough
asymptotics as in large deviation theory. Moreover, the limiting distributions
are for the unscaled process, not for the fluid limit as in large deviation
theory.