Asymptotics of first passage times for random walk in an orthant

Abstract

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 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.