On the tradeoff between compositionality and exactness in weak bisimilarity for integrated-time Markovian process calculi

Integrated-time Markovian process calculi rely on actions whose durations are quantified by exponentially distributed random variables. The Markovian bisimulation equivalences defined so far for these calculi treat exponentially timed internal actions like all the other actions, because each such action has a nonzero duration and hence can be observed if it is executed between a pair of exponentially timed noninternal actions. However, no difference may be noted, at stationary state, between a sequence of exponentially timed internal actions and a single exponentially timed internal action, if their expected durations and execution probabilities coincide, a fact exploited in Hillston’s weak isomorphism. We show that Milner’s approach can be adapted on the basis of this fact, so to derive a weak bisimulation equivalence for integrated-time Markovian process calculi, up to a tradeoff between compositionality and exactness inherent to the Markovian setting. The resulting weak Markovian bisimulation equivalence induces a pseudo-aggregation that is exact at stationary state for all the considered processes, but turns out to be a congruence only over sequential processes. To achieve compositionality over concurrent processes, we need to enhance the abstraction capability of the equivalence in the presence of interleaved computations. However, the corresponding pseudo-aggregation turns out to be exact at stationary state only for a subset of concurrent processes. In addition to this tradeoff, we present, for the first equivalence, a sound and complete axiomatization over sequential processes, which is instrumental to characterize pseudo-aggregations, and a polynomial-time equivalence-checking algorithm, which can be exploited for the compositional minimization of concurrent processes.