QBD processes and matrix orthogonal polynomilas: somw new explicit examples

In the case of birth-and-death processes there are a few exactly solvable situations where the n-step transition matrix can be written down using the Karlin-McGregor formula. A few of these come from group representation theory. I plan to show how this can be extended to some instances of QBD processes with an arbitrary finite number of phases. The group involved is the set of all unitary matrices of size N. For a fixed N one gets examples where the number of phases is a free parameter, and there are a few extra parameters to play with. By tunning these parameters one can exhibit examples where states are recurrent or transient. The rather surprising fact that for these examples one can compute everything explicitly raises the issue of finding a possible network application for this piece of mathematics that involves matrix valued orthogonal polynomials. I will give an ab-initio discussion of the examples starting with the case of one phase.