Lumping Markov Chains with Silent Steps

A silent step in a dynamic system is a step that is considered unobservable and that can be eliminated. We define a Markov chain with silent steps as a class of Markov chains parameterized with a special real number tau. When tau goes to infinity silent steps become immediate, i.e. timeless, and therefore unobservable. To facilitate the elimination of these steps while preserving performance measures, we introduce a notion of lumping for the new setting. To justify the lumping we first extend the standard notion of ordinary lumping to the setting of discontinuous Markov chains, processes that can do infinitely many transitions in finite time. Then, we give a direct connection between the two lumpings for the case when tau is infinite. The results of this paper can serve as a correctness criterion and a method for the elimination of silent (tau) steps in Markovian process algebras