State of a network when one node overloads

We delve into a couple of topics in the theory of Markov chains and stochastic networks. The properties of a stable Markov chain X = (X1, Xˆ) will be investigated when X1 tends to infinity. We derive the distribution of Xˆ when X 1 passes a threshold for the first time as the threshold tends to infinity. Moreover, the exact asymptotics of the mean time until X 1 reaches the threshold is given. In addition, we present a new approach to determine the exact asymptotics of the X's steady state. The results are applied to an open modified Jackson network with two partially coupled processors. Finally, a ratio limit property is established for a Markovian kernel which has unbounded jumps