QBD processes and matrix orthogonal polynomilas: somw new explicit examples

Abstract

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.