Abstract
A simple and quite general approach is proposed to derive criteria for transience and ergodicity of a certain class of irreducibleN-dimensional Markov chains in ℤ N+ assuming a boundedness condition on the second moment of the jumps. The method consists in constructing convenient smooth supermartingales outside some compact set. The Lyapounov functions introduced belong to the set of quadratic forms in ℤ N+ and do not always have a definite sign. Existence and construction of these forms is shown to be basically equivalent to finding vectors satisfying a system of linear inequalities.
Part I is restricted toN=2, in which case a complete characterization is obtained for the type of random walks analyzed by Malyshev and Mensikov, thus relaxing their condition of boundedness of the jumps. The motivation for this work is partly from a large class of queueing systems that give rise to random walks in ℤ N+
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Fayolle, G. On random walks arising in queueing systems: ergodicity and transience via quadratic forms as lyapounov functions — Part I. Queueing Syst 5, 167–183 (1989). https://doi.org/10.1007/BF01149191
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DOI: https://doi.org/10.1007/BF01149191