J. Markovski, A. Sokolova, N. Trcka and E.P. de Vink
Compositionality for Markov Reward Chains with Fast Transitions

A parallel composition is defined for Markov reward chains with fast
transitions and for discontinuous Markov reward chains. In this setting,
compositionality with respect to the relevant aggregation preorders is
established.  For Markov reward chains with fast transitions the preorders are
tau-lumping and tau-reduction. Discontinuous Markov reward chains are 'limits'
of Markov reward chains with fast transitions, and have related notions of
lumping and reduction. In total, four compositionality results are shown. In
addition, the two parallel operators are related by a continuity property.