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Compositionality for Markov Reward Chains 
with Fast and Silent Transitions
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by Jasen Markovski, Ana Sokolova, Nikola Trcka, and Erik de Vink
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Abstract:
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A parallel composition is defined for Markov reward chains with
stochastic discontinuity, 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
$\tau$-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. Markov reward
chains with silent transitions are equivalence classes of Markov
reward chains with fast transitions and are equipped with the lifted
preorders $\tau_\sim$-lumping and $\tau_\sim$-reduction.  In total,
six compositionality results are presented. Additionally, the
parallel operators are related by a continuity result.
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