Strong convergence of a class of non-homogeneous Markov arrival processes to a Poisson process

In this paper, we are concerned with a time-inhomogeneous version of the Markovian arrival process. Under the assumption that the environment process is asymptotically time-homogeneous, we discuss a Poisson approximation of the counting process of arrivals when the arrivals are rare. We provide a rate of convergence for the distance in variation. Poisson-type approximation for the process resulting of a special marking procedure of the arrivals is outlined.