ABSTRACT
In this survey paper, we report some recent generalizations of matrix-analytic solutions for bivariate Markov chains. In particular, the generalization is for both the GI/M/ 1 and the M/G/1 paradigms. There are two variables in the Markov chain, one of which takes values on the nodes of a d-ary tree. The other is an auxiliary variable, which takes one of m possible values . For the GI/M/1 (M/G/1 ) paradigm, the Markov chain exhibits a skip free to the right (left) property. For both problems with this structure, the steady state probability depends on d matrices , which are solutions of a system of non-linear matrix equations.
