On Two-Dimensional Markov Chains in the Positive Quadrant with Partial Spatial Homogeneity
1995, v.1, №2, 267-280
We consider the problem of classifying Markov chains on the quarter plane $Z_+^2$ which possess a property of partial spatial homogeneity. Such chains arise frequently in the study of queueing and loss networks and have been previously studied, notably by Malyshev and Menshikov and by Fayolle. However existing results are either given under restrictive conditions, or lack complete proofs. We take a new approach to the construction of Lyapunov functions for such chains to give proofs which require weaker conditions, are complete in all cases and additionally are considerably simpler. We also describe the application of these results to the study of the dynamic and equilibrium behaviour of large loss networks.
Keywords: Markov chains,random walks,ergodicity,transience,Lyapunov functions,loss networks,queueing networks