  We study networks of processes that all execute the same finite protocol and
communicate synchronously in two different ways: a process can broadcast one
message to all other processes or send it to at most one other process. In both
cases, if no process can receive the message, it will still be sent. We
establish a precise complexity class for two coverability problems with a
parameterised number of processes: the state coverability problem and the
configuration coverability problem. It is already known that these problems are
Ackermann-hard (but decidable) in the general case. We show that when the
protocol is Wait-Only, i.e., it has no state from which a process can send and
receive messages, the complexity drops to P and PSPACE, respectively.
