An eigenvalue approach to discrete-time MAP/G/1 queues

In this paper, we demonstrate how one can effectively analyze the operational behavior of the discrete-time MAP/G/1 queue with various control policies. For this purpose, we take the single-vacation system as an example which becomes the basic model for single-machine production system with maintenance. We employ a hybrid approach with supplementary variables, eigenvalue and eigenvectors, and matrix analytic method. We derive the vector generating functions of the queue lengths and show that the queue length vector generating function factors into two components, one of which is the queue length at an arbitrary time during the idle period.