  We study a family of quasi-birth-and-death (QBD) processes associated with
the so-called first family of Jacobi-Koornwinder bivariate polynomials. These
polynomials are orthogonal on a bounded region typically known as the swallow
tail. We will explicitly compute the coefficients of the three-term recurrence
relations generated by these QBD polynomials and study the conditions under we
can produce families of discrete-time QBD processes. Finally, we show an urn
model associated with one special case of these QBD processes.
