Implementation relations and probabilistic schedulers in the distributed test architecture

•New (probabilistic) type of scheduler.•Three new implementation relations for distributed testing of PIOTSs.•Analysis with respect to relations with deterministic schedulers.•The strongest relations are equivalent.•The weakest relations differ.

We present a complete framework to formally test systems with distributed ports where some choices are probabilistically quantified while other choices are non-deterministic. We define different implementation relations, that is, relations that state what it means for a system to be a valid implementation of a specification. We also study how these relate. In order to define these implementation relations we use probabilistic schedulers, a more powerful version, including probabilistic choices, of a notion of scheduler introduced in our previous work. Probabilistic schedulers, when applied to either a specification or an implementation, resolve all the possible non-determinism, so that we can compare purely probabilistic systems.