We derive the transient distribution and periodic family of asymptotic distributions and the transient and periodic moments for the quasi-birth-and-death processes with time-varying periodic rates. The distributions and moments are given in terms of integral equations involving the related random-walk process. The method is a straight-forward application of generating functions.
Margolius, Barbara Haas, "Transient and Periodic Solution to The Time-Inhomogeneous Quasi-Birth Death Process" (2007). Mathematics Faculty Publications. 244.
The final publication is available at Springer via http://dx.doi.org/10.1007/s11134-007-9027-8