  FIFO automata are finite state machines communicating through FIFO queues.
They can be used for instance to model distributed protocols. Due to the
unboundedness of the FIFO queues, several verification problems are undecidable
for these systems. In order to model-check such systems, one may look for
decidable subclasses of FIFO systems. Binary half-duplex systems are systems of
two FIFO automata exchanging over a half-duplex channel. They were studied by
Cece and Finkel who established the decidability in polynomial time of several
properties. These authors also identified some problems in generalising
half-duplex systems to multi-party communications. We introduce greedy systems,
as a candidate to generalise binary half-duplex systems. We show that greedy
systems retain the same good properties as binary half-duplex systems, and
that, in the setting of mailbox communications, greedy systems are quite
closely related to a multiparty generalisation of half-duplex systems.
