Asymptotics of first passage times for random walk in an orthant

We wish to describe how a chosen node in a network of queues overloads. The overloaded node may also drive other nodes into overload, but the remainin g "super" stable nodes are only driven into a new steady state with stochas tically larger queues. We model this network of queues as a Markov additive chain with a boundary. The customers at the "super" stable nodes are descr ibed 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 add itive processes to prove asymptotic results on the mean time for a specifie d additive component to hit a high level l. We give the limiting distributi on of the "super" stable nodes at this hitting time. We also give the stead y-state distribution of the "super" stable nodes when the specified compone nt equals l, The emphasis here is on sharp asymptotics, not rough asymptoti cs as in large deviation theory. Moreover, the limiting distributions are f or the unsealed process, not for the fluid limit as in large deviation theo ry.