A linear program approach to ergodicity of M/G/1 type Markov chains with a tree structure

It has been shown recently that the Perron-Frobenius eigenvalue of a nonnegative matrix provides information for a complete classification of M/G/1 type Markov chains with a tree structure. The use of that ergodicity condition depends largely on the computation of a set of nonnegative matrices, which can be quite challenging. In this paper, without using a set of nonnegative matrices, we develop two linear programs whose solutions provide sufficient conditions for ergodicity of the Markov chains of interest. We also introduce a simple approximation to the ergodicity problem. Numerical examples demonstrate that the linear program approach, as well as the approximation approach, can be quite useful.