Generating function approach for discrete queueing analysis with decomposable arrival and service Markov chains

The author uses a generating function approach with spectral decomposition for discrete queuing analysis with arrival and service Markov chains (MCs). The complexity of this approach lies in the construction of eigenvalues and eigenvectors for both arrival and service generating function matrices. A MC is called decomposable if it can be decomposed into a set of smaller MCs, where each of them has a number of states less than five. For decomposable arrival and service MCs, it is shown how to construct steady-state queuing solutions on the basis of simple Kronecker product properties. The evaluation of each individual root is well decomposed in a simple convergent form. As an example, the author has constructed solutions for those MCs decomposed in units of heterogeneous two-state MCs. In numerical studies the significant effect of large time-varying scales of arrival/service processes on queue length distribution has been explored