A parallel composition is defined for Markov reward chains with stochastic discontinuity, and with fast and silent transitions. In this setting, compositionality with respect to the relevant aggregation preorders is established. For Markov reward chains with fast transitions the preorders are t-lumping and t-reduction. Discontinuous Markov reward chains are ‘limits’ of Markov reward chains with fast transitions, and have related notions of lumping and reduction. Markov reward chains with silent transitions are equivalence classes of Markov reward chains with fast transitions and come equipped with the lifted preorders t~-lumping and t~-reduction. In total, six compositionality results are presented. Additionally, the parallel operators involved are related by a continuity result.
Markovski, J., Sokolova, A., Trcka, N., & Vink, de, E. P. (2009). Compositionality for Markov reward chains with fast and silent transitions. Performance Evaluation, 66(8), 435-452. https://doi.org/10.1016/j.peva.2009.01.001