The superposition of Markovian arrival processes: moments and the minimal Laplace transform

The superposition of two independent Markovian arrival processes (MAPs) is also a Markovian arrival process of which the Markovian representation is given as the Kronecker sum of the transition rate matrices of the component processes. The moments of stationary intervals of the superposition can be obtained by differentiating the Laplace transform (LT) given in terms of the transition rate matrices. In this paper, we propose a streamlined procedure to determine the minimal LT of the merged process in terms of the minimal LT coefficients of the component processes. Combined with the closed-form transformation between moments and LT coefficients, our result enables us to determine the moments of the superposed process based on the moments of the component processes. The main contribution is that the whole procedure can be implemented without explicit Markovian representations. In order to transform the minimal LT coefficients of the component processes into the minimal LT representation of the merged process, we also introduce another minimal representation. A numerical example is provided to illustrate the procedure.