Eigenvalue expression for a batch Markovian process

Consider a batch Markovian arrival process as the counting process of an underlying Markov process representing the state of environment. Such a process is useful for representing correlated inputs for example. They are used both as a modeling tool and as a theoretical device to represent and approximate superposition of input processes and complex large systems. Our objective is to consider the first and second moments of the counting process depending on time and state. Assuming that the probability generating functions of batch size are analytic, and that eigenvalues of the infinitesimal generator are simple, we derive an analytic diagonalization for the matrix generating function of the counting process. Our main result gives the time-dependent form of the first and second factorial moments of the counting process, which is represented by eigenvalues and eigenvectors of the matrix generating function of the batch size.