Quasi lumpability, lower-bounding coupling matrices, and nearly completely decomposable Markov chains

In this paper, it is shown that nearly completely decomposable (NCD) Markov chains are quasi-lumpable. The state space partition is the natural one, and the technique may be used to compute lower and upper bounds on the stationary probability of each NCD block. In doing so, a lower-bounding nonnegative coupling matrix is employed. The nature of the stationary probability bounds is closely related to the structure of this lower-bounding matrix. Irreducible lower-bounding matrices give tighter bounds compared with bounds obtained using reducible lower-bounding matrices. It is also noticed that the quasi-lumped chain of an NCD Markov chain is an ill-conditioned matrix and the bounds obtained generally will not be tight. However, under some circumstances, it is possible to compute the stationary probabilities of some NCD blocks exactly