State of the art in the analysis of multidimensional Markov chains

As multidimensional Markov chains are prevalent in the analysis of multiserver systems, several approaches have been proposed for their analysis. A most popular approach is to simply truncate the state space of the Markov chain and analyze the resulting Markov chain, for example, via matrix analytic methods. By contrast, DR reduces a multidimensional Markov chain into a 1D Markov chain, but the state space is not simply truncated. Rather, the infinite portion of the state space is aggregated into a smaller space, capturing detailed behavior (such as the first three moments and correlations of the sojourn time distributions in the infinite portion) of the original multidimensional Markov chain. There are also approaches that can be applied to multidimensional Markov chains without state space truncation; however, these approaches are limited in the class of Markov chains to which they can be applied either due to essential restrictions or due to computational complexity. We will see that these approaches have limitations in the analysis of the RFB and GFB processes.