On multiserver retrial queues: history, Okubo-type hypergeometric systems and matrix continued-fractions

We study two families of QBD processes with linear rates: (A) the multiserver retrial queue and its easier relative; and (B) the multiserver M/M/infinity Markov modulated queue. The linear rates imply that the stationary probabilities satisfy a recurrence with linear coefficients; as known from previous work, they yield a ``minimal/non-dominant" solution of this recurrence, which may be computed numerically by matrix continued-fraction methods. Furthermore, the generating function of the stationary probabilities satisfies a linear differential system with polynomial coefficients, which calls for the venerable but still developing theory of holonomic (or D-finite) linear differential systems. We provide a differential system for our generating function that unifies problems (A) and (B), and we also include some additional features and observe that in at least one particular case we get a special ``Okubo-type hypergeometric system", a family that recently spurred considerable interest. The differential system should allow further study of the Taylor coefficients of the expansion of the generating function at three points of interest: 1) the irregular singularity at 0; 2) the dominant regular singularity, which yields asymptotic series via classic methods like the Frobenius vector expansion; and 3) the point 1, whose Taylor series coefficients are the factorial moments.